[{"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "In an experiment aimed at studying the effect of advertising on eating behavior in children, a group of 500 children, 7 to 11 years old, were randomly assigned to two different groups. After randomization, each child was asked to watch a cartoon in a private room, containing a large bowl of Goldfish crackers. The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant. In most studies, in most experiments, the threshold that they think about is the probability of something statistically significant, if the probability of that happening by chance is less than 5%, so this is less than 1%. So I would definitely say that the experiment is significant."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean of the 100 injected rats response times is 1.05 seconds, with a sample standard deviation of 0.5 seconds. Do you think that the drug has an effect on response time? So to do this, we're going to set up two hypotheses. We're going to say, the first hypothesis is, we're going to call it the null hypothesis, and that is that the drug has no effect on response time. And your null hypothesis is always going to be, you can view it as a status quo. You assume that whatever you're researching has no effect. So drug has no effect."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to say, the first hypothesis is, we're going to call it the null hypothesis, and that is that the drug has no effect on response time. And your null hypothesis is always going to be, you can view it as a status quo. You assume that whatever you're researching has no effect. So drug has no effect. Or another way to think about it is that the mean of the rats taking the drug should be the mean with the drug is still going to be 1.2 seconds, even with the drug. So that's essentially saying it has no effect, because we know that if you don't give the drug, the mean response time is 1.2 seconds. Now, what you want is an alternative hypothesis."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So drug has no effect. Or another way to think about it is that the mean of the rats taking the drug should be the mean with the drug is still going to be 1.2 seconds, even with the drug. So that's essentially saying it has no effect, because we know that if you don't give the drug, the mean response time is 1.2 seconds. Now, what you want is an alternative hypothesis. The hypothesis is, no, I think the drug actually does do something. So the alternative hypothesis right over here is that the drug has an effect. Or another way to think about it is that the mean does not equal 1.2 seconds when the drug is given."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what you want is an alternative hypothesis. The hypothesis is, no, I think the drug actually does do something. So the alternative hypothesis right over here is that the drug has an effect. Or another way to think about it is that the mean does not equal 1.2 seconds when the drug is given. So how do we think about this? How do we know whether we should accept the alternative hypothesis or whether we should just default to the null hypothesis because the data isn't convincing? And the way we're going to do it in this video, and this is really the way it's done in pretty much all of science, is you say, OK, let's assume that the null hypothesis is true."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way to think about it is that the mean does not equal 1.2 seconds when the drug is given. So how do we think about this? How do we know whether we should accept the alternative hypothesis or whether we should just default to the null hypothesis because the data isn't convincing? And the way we're going to do it in this video, and this is really the way it's done in pretty much all of science, is you say, OK, let's assume that the null hypothesis is true. If the null hypothesis was true, what is the probability that we would have gotten these results with the sample? And if that probability is really, really small, then the null hypothesis probably isn't true. We could probably reject the null hypothesis and we'll say, well, we kind of believe in the alternative hypothesis."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the way we're going to do it in this video, and this is really the way it's done in pretty much all of science, is you say, OK, let's assume that the null hypothesis is true. If the null hypothesis was true, what is the probability that we would have gotten these results with the sample? And if that probability is really, really small, then the null hypothesis probably isn't true. We could probably reject the null hypothesis and we'll say, well, we kind of believe in the alternative hypothesis. So let's think about that. Let's assume that the null hypothesis is true. And so if we assume the null hypothesis is true, let's try to figure out the probability that we would have actually gotten this result, that we would have actually gotten a sample mean of 1.05 seconds with a standard deviation of 0.5 seconds."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could probably reject the null hypothesis and we'll say, well, we kind of believe in the alternative hypothesis. So let's think about that. Let's assume that the null hypothesis is true. And so if we assume the null hypothesis is true, let's try to figure out the probability that we would have actually gotten this result, that we would have actually gotten a sample mean of 1.05 seconds with a standard deviation of 0.5 seconds. So I want to see, if we assume the null hypothesis is true, I want to figure out the probability. And actually, what we're going to do is not just figure out the probability of this, the probability of getting something like this or even more extreme than this. So how likely of an event is that?"}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if we assume the null hypothesis is true, let's try to figure out the probability that we would have actually gotten this result, that we would have actually gotten a sample mean of 1.05 seconds with a standard deviation of 0.5 seconds. So I want to see, if we assume the null hypothesis is true, I want to figure out the probability. And actually, what we're going to do is not just figure out the probability of this, the probability of getting something like this or even more extreme than this. So how likely of an event is that? And to think about that, let's just think about the sampling distribution if we assume the null hypothesis. So the sampling distribution is like this. It'll be a normal distribution."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how likely of an event is that? And to think about that, let's just think about the sampling distribution if we assume the null hypothesis. So the sampling distribution is like this. It'll be a normal distribution. We have a good number of samples. We have 100 samples here. So this is the sampling distribution."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It'll be a normal distribution. We have a good number of samples. We have 100 samples here. So this is the sampling distribution. It will have a mean. And now, if we assume the null hypothesis, that the drug has no effect, the mean of our sampling distribution will be the same thing as the mean of the population distribution, which would be equal to 1.2 seconds. Now, what is the standard deviation of our sampling distribution?"}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the sampling distribution. It will have a mean. And now, if we assume the null hypothesis, that the drug has no effect, the mean of our sampling distribution will be the same thing as the mean of the population distribution, which would be equal to 1.2 seconds. Now, what is the standard deviation of our sampling distribution? The standard deviation of our sampling distribution should be equal to the standard deviation of the population distribution divided by the square root of our sample size, so divided by the square root of 100. We do not know what the standard deviation of the entire population is. So what we're going to do is estimate it with our sample standard deviation."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what is the standard deviation of our sampling distribution? The standard deviation of our sampling distribution should be equal to the standard deviation of the population distribution divided by the square root of our sample size, so divided by the square root of 100. We do not know what the standard deviation of the entire population is. So what we're going to do is estimate it with our sample standard deviation. And that's a reasonable thing to do, especially because we have a nice sample size, a sample size greater than 100. So this is going to be a pretty good approximator for this over here. So we could say that this is going to be approximately equal to our sample standard deviation divided by the square root of 100, which is going to be equal to our sample standard deviation is 0.5 seconds."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what we're going to do is estimate it with our sample standard deviation. And that's a reasonable thing to do, especially because we have a nice sample size, a sample size greater than 100. So this is going to be a pretty good approximator for this over here. So we could say that this is going to be approximately equal to our sample standard deviation divided by the square root of 100, which is going to be equal to our sample standard deviation is 0.5 seconds. And we want to divide that by square root of 100 is 10. So 0.5 divided by 10 is 0.05. So the standard deviation of our sampling distribution is going to be, and we'll put a little hat over it to show that we approximated the population standard deviation with the sample standard deviation."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we could say that this is going to be approximately equal to our sample standard deviation divided by the square root of 100, which is going to be equal to our sample standard deviation is 0.5 seconds. And we want to divide that by square root of 100 is 10. So 0.5 divided by 10 is 0.05. So the standard deviation of our sampling distribution is going to be, and we'll put a little hat over it to show that we approximated the population standard deviation with the sample standard deviation. So it is going to be equal to 0.5 divided by 10, so 0.05. And so what is the probability of getting 1.05 seconds? Or another way to think about it is, how many standard deviations away from this mean is 1.05 seconds?"}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the standard deviation of our sampling distribution is going to be, and we'll put a little hat over it to show that we approximated the population standard deviation with the sample standard deviation. So it is going to be equal to 0.5 divided by 10, so 0.05. And so what is the probability of getting 1.05 seconds? Or another way to think about it is, how many standard deviations away from this mean is 1.05 seconds? And what is the probability of getting a result at least that many standard deviations away from the mean? So let's figure out how many standard deviations away from the mean that is. And essentially, we're just figuring out a z-score for this result right over there."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way to think about it is, how many standard deviations away from this mean is 1.05 seconds? And what is the probability of getting a result at least that many standard deviations away from the mean? So let's figure out how many standard deviations away from the mean that is. And essentially, we're just figuring out a z-score for this result right over there. So let me pick a nice color. I haven't used orange yet. So our z-score, you could even view it as a z-statistic."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And essentially, we're just figuring out a z-score for this result right over there. So let me pick a nice color. I haven't used orange yet. So our z-score, you could even view it as a z-statistic. It's being derived from these other sample statistics. So our z-statistic, how far are we away from the mean? Well, the mean is 1.2."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our z-score, you could even view it as a z-statistic. It's being derived from these other sample statistics. So our z-statistic, how far are we away from the mean? Well, the mean is 1.2. And we are at 1.05, so I'll put that less, just so that it'll be a positive distance. So that's how far away we are. And if we want it in terms of standard deviations, we want to divide it by our best estimate of the sampling distributions standard deviation, which is this 0.05."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the mean is 1.2. And we are at 1.05, so I'll put that less, just so that it'll be a positive distance. So that's how far away we are. And if we want it in terms of standard deviations, we want to divide it by our best estimate of the sampling distributions standard deviation, which is this 0.05. So this is 0.05. And what is this going to be equal to? z, this result right here, 1.05 seconds."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we want it in terms of standard deviations, we want to divide it by our best estimate of the sampling distributions standard deviation, which is this 0.05. So this is 0.05. And what is this going to be equal to? z, this result right here, 1.05 seconds. 1.2 minus 1.05 is 0.15. So this is 0.15 in the numerator divided by 0.05 in the denominator. And so this is going to be 3."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "z, this result right here, 1.05 seconds. 1.2 minus 1.05 is 0.15. So this is 0.15 in the numerator divided by 0.05 in the denominator. And so this is going to be 3. So this result right here is 3 standard deviations away from the mean. So let me draw this. This is the mean."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so this is going to be 3. So this result right here is 3 standard deviations away from the mean. So let me draw this. This is the mean. If I did one standard deviation, two standard deviations, three standard deviations. That's in the positive direction. Actually, let me draw it a little bit different than that."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the mean. If I did one standard deviation, two standard deviations, three standard deviations. That's in the positive direction. Actually, let me draw it a little bit different than that. This wasn't a nicely drawn bell curve. But I'll do one standard deviation. Two standard deviations."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me draw it a little bit different than that. This wasn't a nicely drawn bell curve. But I'll do one standard deviation. Two standard deviations. And then three standard deviations in the positive direction. And then we have one standard deviation, two standard deviations, and three standard deviations in the negative direction. So this result right here, 1.05 seconds that we got for our 100 rat sample is right over here."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two standard deviations. And then three standard deviations in the positive direction. And then we have one standard deviation, two standard deviations, and three standard deviations in the negative direction. So this result right here, 1.05 seconds that we got for our 100 rat sample is right over here. Three standard deviations below the mean. Now what is the probability of getting a result this extreme by chance? And when I talk about this extreme, it could be either a result less than this or a result that extreme in the positive direction, more than three standard deviations."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this result right here, 1.05 seconds that we got for our 100 rat sample is right over here. Three standard deviations below the mean. Now what is the probability of getting a result this extreme by chance? And when I talk about this extreme, it could be either a result less than this or a result that extreme in the positive direction, more than three standard deviations. So this is essentially, if we think about the probability of getting a result more extreme than this result right over here, we're thinking about this area under the bell curve, both in the negative direction or in the positive direction. What is the probability of that? Well, we know from the empirical rule that 99.7% of the probability is within three standard deviations."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And when I talk about this extreme, it could be either a result less than this or a result that extreme in the positive direction, more than three standard deviations. So this is essentially, if we think about the probability of getting a result more extreme than this result right over here, we're thinking about this area under the bell curve, both in the negative direction or in the positive direction. What is the probability of that? Well, we know from the empirical rule that 99.7% of the probability is within three standard deviations. So this thing right here, you could look it up on a z-table as well, but three standard deviations is a nice clean number that doesn't hurt to remember. So we know that this area right here, I'm doing in this reddish orange, that area right over there is 99.7%. So what is left for these two magenta or pink areas?"}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we know from the empirical rule that 99.7% of the probability is within three standard deviations. So this thing right here, you could look it up on a z-table as well, but three standard deviations is a nice clean number that doesn't hurt to remember. So we know that this area right here, I'm doing in this reddish orange, that area right over there is 99.7%. So what is left for these two magenta or pink areas? Well, if these are 99.7%, then both of these combined are going to be 0.3%. So both of these combined are 0.3%. Or if we wrote it as a decimal, it would be 0.003 of the total area under the curve."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is left for these two magenta or pink areas? Well, if these are 99.7%, then both of these combined are going to be 0.3%. So both of these combined are 0.3%. Or if we wrote it as a decimal, it would be 0.003 of the total area under the curve. So to answer our question, if we assume that the drug has no effect, the probability of getting a sample this extreme or actually more extreme than this is only 0.3%, less than 1 in 300. So if the null hypothesis was true, there's only a 1 in 300 chance that we would have gotten a result this extreme or more. So at least from my point of view, this result seems to favor the alternative hypothesis."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if we wrote it as a decimal, it would be 0.003 of the total area under the curve. So to answer our question, if we assume that the drug has no effect, the probability of getting a sample this extreme or actually more extreme than this is only 0.3%, less than 1 in 300. So if the null hypothesis was true, there's only a 1 in 300 chance that we would have gotten a result this extreme or more. So at least from my point of view, this result seems to favor the alternative hypothesis. I'm going to reject the null hypothesis. I don't know 100% sure, but if the null hypothesis was true, there's only a 1 in 300 chance of getting this. So I'm going to go with the alternative hypothesis."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So at least from my point of view, this result seems to favor the alternative hypothesis. I'm going to reject the null hypothesis. I don't know 100% sure, but if the null hypothesis was true, there's only a 1 in 300 chance of getting this. So I'm going to go with the alternative hypothesis. And just to give you a little bit of some of the name or the labels you might see in some statistics or in some research papers, this value, the probability of getting a result more extreme than this, given the null hypothesis, is called a p-value. So the p-value here, and this really just stands for probability value, the p-value right over here is 0.003. So there's a very, very small probability that we could have gotten this result if the null hypothesis was true."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm going to go with the alternative hypothesis. And just to give you a little bit of some of the name or the labels you might see in some statistics or in some research papers, this value, the probability of getting a result more extreme than this, given the null hypothesis, is called a p-value. So the p-value here, and this really just stands for probability value, the p-value right over here is 0.003. So there's a very, very small probability that we could have gotten this result if the null hypothesis was true. So we will reject it. And in general, most people have some type of a threshold here. If you have a p-value less than 5%, which means less than 1 in 20 shot, they'll say, you know what, I'm going to reject the null hypothesis."}, {"video_title": "Hypothesis testing and p-values   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a very, very small probability that we could have gotten this result if the null hypothesis was true. So we will reject it. And in general, most people have some type of a threshold here. If you have a p-value less than 5%, which means less than 1 in 20 shot, they'll say, you know what, I'm going to reject the null hypothesis. There's less than a 1 in 20 chance of getting that result. Here we got much less than 1 in 20. So this is a very strong indicator that the null hypothesis is incorrect and the drug definitely has some effect."}, {"video_title": "Generalizing k scores in n attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we said in that circumstance, if it's a 70% chance of making it, well that means that you have a one minus 70%, or 30% chance of missing. And we said if you took six attempts, the probability of you getting exactly, making two of the baskets, exactly two scores, and I call them scores instead of making it just because I wanted making and missing, have different letters in that video, we said, well, there's six choose two different ways of making two, exactly two out of the six free throws, and then the probability of any one of those ways is going to be making it twice, which is 0.7 squared, and missing it four times, so 0.3 to the fourth power. So this was just one particular situation, but we could generalize based on the logic that we had in that video. In fact, let's do that. So if I were to generalize it, if I were to say the probability, the probability, it's the exact same logic, of exactly, now let's say K, let me do this in a color, an interesting color, so let me do it in this orangish brown color, K shots, or exactly K scores, I'll call making a free throw a score, we'll just assume you got a point for it. So exactly two K scores, in N attempts, let's just say in N attempts, in N, and let me go back to that green color, N attempts, is going to be equal to, well, how many ways can you pick K things out of N, or N choose K, N choose K, and then, actually let's just generalize it even more. Let's just say that you have, your free throw probability is P. So let's say P is, so for this situation right over here, since we generalized it fully, let's say that P is the probability of making a free throw."}, {"video_title": "Generalizing k scores in n attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "In fact, let's do that. So if I were to generalize it, if I were to say the probability, the probability, it's the exact same logic, of exactly, now let's say K, let me do this in a color, an interesting color, so let me do it in this orangish brown color, K shots, or exactly K scores, I'll call making a free throw a score, we'll just assume you got a point for it. So exactly two K scores, in N attempts, let's just say in N attempts, in N, and let me go back to that green color, N attempts, is going to be equal to, well, how many ways can you pick K things out of N, or N choose K, N choose K, and then, actually let's just generalize it even more. Let's just say that you have, your free throw probability is P. So let's say P is, so for this situation right over here, since we generalized it fully, let's say that P is the probability of making a free throw. Actually, since I already have a P here, let me just say F is equal to the probability of making a free throw, or you could say your probability of scoring, if you call a score making a free throw. So if F is your probability of making a free throw, so if you want N scores, then this is going to be, this is going to be, well, it's going to be F to the N power, and then you're going to have, and then you're going to miss the remainder, sorry, F to the K power, because you're making exactly K scores. So F to the K power, and then the remainder, so the N minus K attempts, you're going to miss it."}, {"video_title": "Generalizing k scores in n attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's just say that you have, your free throw probability is P. So let's say P is, so for this situation right over here, since we generalized it fully, let's say that P is the probability of making a free throw. Actually, since I already have a P here, let me just say F is equal to the probability of making a free throw, or you could say your probability of scoring, if you call a score making a free throw. So if F is your probability of making a free throw, so if you want N scores, then this is going to be, this is going to be, well, it's going to be F to the N power, and then you're going to have, and then you're going to miss the remainder, sorry, F to the K power, because you're making exactly K scores. So F to the K power, and then the remainder, so the N minus K attempts, you're going to miss it. So it's going to be that probability of missing, and the probability of missing is going to be one minus F, so it's going to be times one minus F to the N minus K power, to the N minus K power. And just, if you like, or I encourage you, pause the video, and just make sure you understand the parallels between this example where I had a set where, I guess our F was 70%, our F was 70%, one minus F, or our F was.7, and one minus F would be.3, and we were seeing, how do we get two scores in six attempts? And here we're saying K scores in N attempts, and this is just a general way to think about it."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say that I am a drug company and I have come up with a medicine that I think will help folks with diabetes. And in particular, I think it will help reduce their hemoglobin A1C levels. And for those of you who aren't familiar with what hemoglobin A1C is, I encourage you, we have a video on that on Khan Academy, but the general idea is if you have high blood sugar over roughly a three month period of time, high blood sugar, and I could say high average blood sugar, you're going to have a high A1C, a high hemoglobin A1C level. And if you have a low average blood sugar over roughly a three month time, you're going to have a lower hemoglobin A1C. So if taking the pill seems to lower folks' A1C levels more than is likely to happen due to randomly or due to other variables, well then that means that your new pill might be effective at controlling folks' diabetes. So in this situation, when we're constructing an experiment to test this, we would say that whether or not you are taking the pill, this is the explanatory variable, explanatory variable, and the thing that it is affecting, the thing that you're hoping has some response, in this case, the A1C levels are your indicator of whether it has helped controlling the blood sugar, we call that the response variable. That right over there is the response variable."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And if you have a low average blood sugar over roughly a three month time, you're going to have a lower hemoglobin A1C. So if taking the pill seems to lower folks' A1C levels more than is likely to happen due to randomly or due to other variables, well then that means that your new pill might be effective at controlling folks' diabetes. So in this situation, when we're constructing an experiment to test this, we would say that whether or not you are taking the pill, this is the explanatory variable, explanatory variable, and the thing that it is affecting, the thing that you're hoping has some response, in this case, the A1C levels are your indicator of whether it has helped controlling the blood sugar, we call that the response variable. That right over there is the response variable. So how are we actually going to conduct this experiment? Well, let's say that we have a group of folks, let's say that we have been given a group of 100 folks who need to control their diabetes. So 100 people here who need to control their diabetes."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "That right over there is the response variable. So how are we actually going to conduct this experiment? Well, let's say that we have a group of folks, let's say that we have been given a group of 100 folks who need to control their diabetes. So 100 people here who need to control their diabetes. And we say, all right, well, let's take half of this group and put them into, I guess you could say, a treatment group and another half and put them into a control group and see if the treatment group, the one that actually gets my pill, is going to improve their A1C levels in a way that seems like it would not be just random chance. So let's do that. So we're going to have a control group."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So 100 people here who need to control their diabetes. And we say, all right, well, let's take half of this group and put them into, I guess you could say, a treatment group and another half and put them into a control group and see if the treatment group, the one that actually gets my pill, is going to improve their A1C levels in a way that seems like it would not be just random chance. So let's do that. So we're going to have a control group. So this is my control group, control. And this is the treatment group. This is the treatment, treatment group."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we're going to have a control group. So this is my control group, control. And this is the treatment group. This is the treatment, treatment group. And you might say, okay, we'll just give these folks, the treatment group, the pill, and then we won't give the pill that I created to the control group. But that might introduce a psychological aspect that maybe the benefit of the pill is just people feeling, hey, I'm taking something that'll control my diabetes. Maybe that psychologically affects their blood sugar in some way, and this is actually possible."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "This is the treatment, treatment group. And you might say, okay, we'll just give these folks, the treatment group, the pill, and then we won't give the pill that I created to the control group. But that might introduce a psychological aspect that maybe the benefit of the pill is just people feeling, hey, I'm taking something that'll control my diabetes. Maybe that psychologically affects their blood sugar in some way, and this is actually possible. Maybe it makes them act healthier in certain ways. Maybe that makes them act unhealthier in certain ways because they're like, oh, I have a pill to control my diabetes, my blood sugar. I can go eat more sweets now and it'll control it."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe that psychologically affects their blood sugar in some way, and this is actually possible. Maybe it makes them act healthier in certain ways. Maybe that makes them act unhealthier in certain ways because they're like, oh, I have a pill to control my diabetes, my blood sugar. I can go eat more sweets now and it'll control it. And so to avoid that, in order for just the very fact that someone says, hey, I think I'm taking a medicine, I might behave in a different way or it might even psychologically affect my body in a certain way, what we want to do is give both groups a pill. And we want to do it in a way that neither group knows which pill they're getting. So what we would do here is we would give this group a placebo, a placebo, and this group would actually get the medicine, the medicine."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "I can go eat more sweets now and it'll control it. And so to avoid that, in order for just the very fact that someone says, hey, I think I'm taking a medicine, I might behave in a different way or it might even psychologically affect my body in a certain way, what we want to do is give both groups a pill. And we want to do it in a way that neither group knows which pill they're getting. So what we would do here is we would give this group a placebo, a placebo, and this group would actually get the medicine, the medicine. But those pills should look the same and people should not know which group they are in. And that is a, when we do that, that is a blind experiment, experiment. Now, you might have heard about double blind experiments."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So what we would do here is we would give this group a placebo, a placebo, and this group would actually get the medicine, the medicine. But those pills should look the same and people should not know which group they are in. And that is a, when we do that, that is a blind experiment, experiment. Now, you might have heard about double blind experiments. Well, that would be the case where not only do people not know which group they're in, but even their physician or the person who's administering the experiment, they don't know which one they're giving. They don't know if they're giving the placebo or the actual medicine to the group. So let's say we want to do that."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now, you might have heard about double blind experiments. Well, that would be the case where not only do people not know which group they're in, but even their physician or the person who's administering the experiment, they don't know which one they're giving. They don't know if they're giving the placebo or the actual medicine to the group. So let's say we want to do that. So we could do double, double blind experiment. So even the person giving the pill doesn't know which pill they're giving. And you might say, well, why is that important?"}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say we want to do that. So we could do double, double blind experiment. So even the person giving the pill doesn't know which pill they're giving. And you might say, well, why is that important? Well, if the physician knows it might, or the person administering or interfacing with the patient, they might give a tell somehow. They might not put as much emphasis on the importance of taking the pill if it's a placebo. They might by accident give away some type of information."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you might say, well, why is that important? Well, if the physician knows it might, or the person administering or interfacing with the patient, they might give a tell somehow. They might not put as much emphasis on the importance of taking the pill if it's a placebo. They might by accident give away some type of information. So to avoid that type of thing happening, you would have a, you could do a double blind. And there's even, some people talk about a triple blind experiment where even the people analyzing the data don't know which group was the control group and which group was the treatment group. And once again, that's another way to avoid bias."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "They might by accident give away some type of information. So to avoid that type of thing happening, you would have a, you could do a double blind. And there's even, some people talk about a triple blind experiment where even the people analyzing the data don't know which group was the control group and which group was the treatment group. And once again, that's another way to avoid bias. So now that we've kind of figured out, we have a control group, we have a treatment group, we're using A1c as our response variable. So we would want to measure folks A1c levels, their hemoglobin A1c levels before they get either the placebo or the medicine. And then maybe after three months, we would measure their A1c after."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, that's another way to avoid bias. So now that we've kind of figured out, we have a control group, we have a treatment group, we're using A1c as our response variable. So we would want to measure folks A1c levels, their hemoglobin A1c levels before they get either the placebo or the medicine. And then maybe after three months, we would measure their A1c after. But the next question is, how do you divvy these 100 people up into these two groups? And you might say, well, I would want to do it randomly. And you would be right, because if you didn't do it randomly, if you put all the men here and all the women here, well, that might, first of all, sex might explain it, or the behavior of men versus women might explain the differences or the non-differences you see in A1c level if you get a lot of people of one age or one part of the country or one type of dietary habits."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And then maybe after three months, we would measure their A1c after. But the next question is, how do you divvy these 100 people up into these two groups? And you might say, well, I would want to do it randomly. And you would be right, because if you didn't do it randomly, if you put all the men here and all the women here, well, that might, first of all, sex might explain it, or the behavior of men versus women might explain the differences or the non-differences you see in A1c level if you get a lot of people of one age or one part of the country or one type of dietary habits. You don't want that. So in order to avoid having an imbalance of some of those lurking variables, you would want to randomly sample. And we've done multiple videos already on ways to randomly sample."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you would be right, because if you didn't do it randomly, if you put all the men here and all the women here, well, that might, first of all, sex might explain it, or the behavior of men versus women might explain the differences or the non-differences you see in A1c level if you get a lot of people of one age or one part of the country or one type of dietary habits. You don't want that. So in order to avoid having an imbalance of some of those lurking variables, you would want to randomly sample. And we've done multiple videos already on ways to randomly sample. So you're going to randomly sample and put people into either groups. You know, a very simple way of doing that, you could give everyone here a number from one to 100, use a random number generator to do that, and then, you know, if, well, or you could use a random number generator to pick 50 names to put in the control group or 50 names to put in the treatment group, and then everyone else gets put in the other group. Now, to avoid a situation, you know, just randomly, by doing a random sample, you might have a situation where there's some probability that you disproportionately have more men in one group or more women in another group."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And we've done multiple videos already on ways to randomly sample. So you're going to randomly sample and put people into either groups. You know, a very simple way of doing that, you could give everyone here a number from one to 100, use a random number generator to do that, and then, you know, if, well, or you could use a random number generator to pick 50 names to put in the control group or 50 names to put in the treatment group, and then everyone else gets put in the other group. Now, to avoid a situation, you know, just randomly, by doing a random sample, you might have a situation where there's some probability that you disproportionately have more men in one group or more women in another group. And to avoid that, you could do, really, a version of stratified sampling that we've talked about in other videos, which is you could do what's called a block design for your random assignment, where you actually split everyone into men and women. And it might be 50-50, or it might even be, you know, just randomly here. You got, you know, 60 women, 60 women and 40 men."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now, to avoid a situation, you know, just randomly, by doing a random sample, you might have a situation where there's some probability that you disproportionately have more men in one group or more women in another group. And to avoid that, you could do, really, a version of stratified sampling that we've talked about in other videos, which is you could do what's called a block design for your random assignment, where you actually split everyone into men and women. And it might be 50-50, or it might even be, you know, just randomly here. You got, you know, 60 women, 60 women and 40 men. And what you do here is you say, okay, let's randomly take 30 of these women and put them in the control group and 30 of the women and put them in the treatment group. And let's put, randomly, 20 of the men in the control group and 20 of the men in the treatment group. And that way, someone's sex is less likely to introduce bias into what actually happens here."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You got, you know, 60 women, 60 women and 40 men. And what you do here is you say, okay, let's randomly take 30 of these women and put them in the control group and 30 of the women and put them in the treatment group. And let's put, randomly, 20 of the men in the control group and 20 of the men in the treatment group. And that way, someone's sex is less likely to introduce bias into what actually happens here. So once again, doing this is called a block design, really a version of stratified sampling, block design. And there might be other lurking variables that you want to make sure it doesn't just show up here randomly. And so you might want to, there's other ways of randomly assigning."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And that way, someone's sex is less likely to introduce bias into what actually happens here. So once again, doing this is called a block design, really a version of stratified sampling, block design. And there might be other lurking variables that you want to make sure it doesn't just show up here randomly. And so you might want to, there's other ways of randomly assigning. Now, once you do this, you see what was the change in A1c. If you see that, hey, you know, the change in A1c, well, one, if you see there's no difference in A1c levels between these two groups, and you're like, hey, there's a good probability that my pill does nothing, even, and once again, it's all about probabilities. There's some chance that you were just unlucky and it might be a very small chance and that's why you want to do this with a good number of people."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so you might want to, there's other ways of randomly assigning. Now, once you do this, you see what was the change in A1c. If you see that, hey, you know, the change in A1c, well, one, if you see there's no difference in A1c levels between these two groups, and you're like, hey, there's a good probability that my pill does nothing, even, and once again, it's all about probabilities. There's some chance that you were just unlucky and it might be a very small chance and that's why you want to do this with a good number of people. And as we forward our statistics understandings, we will better understand at what threshold levels do we think the probability is high or low enough for us to really feel good about our findings. But let's say that you do see, let's say that you do see an improvement. You need to think about is that improvement, could that have happened due to random chance or is it very unlikely that that happened due purely to random chance?"}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "There's some chance that you were just unlucky and it might be a very small chance and that's why you want to do this with a good number of people. And as we forward our statistics understandings, we will better understand at what threshold levels do we think the probability is high or low enough for us to really feel good about our findings. But let's say that you do see, let's say that you do see an improvement. You need to think about is that improvement, could that have happened due to random chance or is it very unlikely that that happened due purely to random chance? And if it was very unlikely that it happened due purely to random chance, then you would feel pretty good, and other people, when you publish the results, would feel pretty good about your medicine. Now, even then, you know, science is not done. No one will say that they're 100% sure that your medicine is good."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You need to think about is that improvement, could that have happened due to random chance or is it very unlikely that that happened due purely to random chance? And if it was very unlikely that it happened due purely to random chance, then you would feel pretty good, and other people, when you publish the results, would feel pretty good about your medicine. Now, even then, you know, science is not done. No one will say that they're 100% sure that your medicine is good. There still might have been some lurking variables that we did not, that our experiment did not properly adjust for, that just when we even did this block design, we might have disproportionately gotten randomly older people in one of the groups or the other or people from one part of the country in one group or another. So there's always things to think about. And the most important thing to think about, even if you did this as good as you could, you still, some random chance might have given you a false positive or a, you know, you got good results even though it was random, or a false negative."}, {"video_title": "Introduction to experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "No one will say that they're 100% sure that your medicine is good. There still might have been some lurking variables that we did not, that our experiment did not properly adjust for, that just when we even did this block design, we might have disproportionately gotten randomly older people in one of the groups or the other or people from one part of the country in one group or another. So there's always things to think about. And the most important thing to think about, even if you did this as good as you could, you still, some random chance might have given you a false positive or a, you know, you got good results even though it was random, or a false negative. You got bad results even though it was, even though it was actually random. And so a very important idea in experiments, and this is in science in general, is that this experiment, you should document it well, and it should be, it should be, it should, is that it, the process of replication. Other people should be able to replicate this experiment and hopefully get consistent results."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "According to the pictograph below, how many survey respondents have type O positive blood? How many have O negative blood? So a pictograph is really just a way of representing data with pictures that are somehow related to the data. So in this case, they give us little pictures of, I'm assuming, blood drops right over here. And then they tell us that each blood drop, each blood drop in this pictograph represents eight people. So you can kind of view that as a scale of these graphs. Each of these say eight people."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So in this case, they give us little pictures of, I'm assuming, blood drops right over here. And then they tell us that each blood drop, each blood drop in this pictograph represents eight people. So you can kind of view that as a scale of these graphs. Each of these say eight people. So for example, if you say how many people have A positive, it would be one, two, three, four, five, six, seven blood drops. But each of those blood drops represent eight people. So it would be 56 people have type A positive."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Each of these say eight people. So for example, if you say how many people have A positive, it would be one, two, three, four, five, six, seven blood drops. But each of those blood drops represent eight people. So it would be 56 people have type A positive. But let's answer the actual question that they're asking us. How many survey respondents have type O positive? O positive."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So it would be 56 people have type A positive. But let's answer the actual question that they're asking us. How many survey respondents have type O positive? O positive. So this is O, and then we care about O positive. So we have one blood drop, two, three. Let me do this in a different color."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "O positive. So this is O, and then we care about O positive. So we have one blood drop, two, three. Let me do this in a different color. We have one, two, three, four, five, six, seven, eight. So we have eight drops. I'll put those in quotes, because it's pictures of drops."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let me do this in a different color. We have one, two, three, four, five, six, seven, eight. So we have eight drops. I'll put those in quotes, because it's pictures of drops. And then the scale is eight people, eight people, let me write it this way, times eight people per drop. Eight people per drop. And so eight times eight, and actually even the drops, you can view them as canceling out if you view them as units, so drops, drops."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "I'll put those in quotes, because it's pictures of drops. And then the scale is eight people, eight people, let me write it this way, times eight people per drop. Eight people per drop. And so eight times eight, and actually even the drops, you can view them as canceling out if you view them as units, so drops, drops. Eight times eight is equal to 64 people. So they could have written literally the number 64 right over here. 64 people have type O positive blood."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And so eight times eight, and actually even the drops, you can view them as canceling out if you view them as units, so drops, drops. Eight times eight is equal to 64 people. So they could have written literally the number 64 right over here. 64 people have type O positive blood. Now let's think about the O negative case. O negative blood. Well this is O, and then within the blood group O, this is O negative, and we have one drops, two drops."}, {"video_title": "Reading pictographs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "64 people have type O positive blood. Now let's think about the O negative case. O negative blood. Well this is O, and then within the blood group O, this is O negative, and we have one drops, two drops. So we have two drops, two drops, times eight people per drop. Eight people per drop. And so two times eight, each of these represent eight."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Welcome back. Now let's do a problem that involves almost everything we've learned so far about probability and combinations and conditional probability. So let's say I have a bag again. And in that bag I have 5 fair coins. And I have 10 unfair coins. And a fair coin, of course, there's a 50-50 chance of getting heads or tails. And the unfair coin, let's say that there is a 80% chance of getting a heads for any one of those coins."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in that bag I have 5 fair coins. And I have 10 unfair coins. And a fair coin, of course, there's a 50-50 chance of getting heads or tails. And the unfair coin, let's say that there is a 80% chance of getting a heads for any one of those coins. And that there is a 20% chance of getting tails, right? Because it's going to either be heads or tails. So my question is, what happens is I put my hand in the bag and my eyes are closed."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the unfair coin, let's say that there is a 80% chance of getting a heads for any one of those coins. And that there is a 20% chance of getting tails, right? Because it's going to either be heads or tails. So my question is, what happens is I put my hand in the bag and my eyes are closed. And I picked out a coin. And then I flip it six times. And it turns out that I got 5 out of, well, let's say, yeah."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So my question is, what happens is I put my hand in the bag and my eyes are closed. And I picked out a coin. And then I flip it six times. And it turns out that I got 5 out of, well, let's say, yeah. Well, let's say I got 4 out of 6 heads. That's the result I got. What I want to know is, what is the probability that I picked out a fair coin given that I got 4 out of 6 heads?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it turns out that I got 5 out of, well, let's say, yeah. Well, let's say I got 4 out of 6 heads. That's the result I got. What I want to know is, what is the probability that I picked out a fair coin given that I got 4 out of 6 heads? So before moving on, let's do a little bit of a review of Bayes' theorem. And I think that'll give us a good framework for the rest of this problem. So Bayes' theorem, and let me do it in this corner up here."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I want to know is, what is the probability that I picked out a fair coin given that I got 4 out of 6 heads? So before moving on, let's do a little bit of a review of Bayes' theorem. And I think that'll give us a good framework for the rest of this problem. So Bayes' theorem, and let me do it in this corner up here. Bayes' theorem tells us the probability of both A and B happening, that upside down U is just intersection in set theory, but it's essentially saying it's a set of events in which both A and B occur. That's equal to the probability of A occurring given B times the probability of B, which is also equal to the probability of B occurring given A times the probability of A. I think this should make some intuition for you. If it doesn't, it might be a good idea to watch the conditional probability videos."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So Bayes' theorem, and let me do it in this corner up here. Bayes' theorem tells us the probability of both A and B happening, that upside down U is just intersection in set theory, but it's essentially saying it's a set of events in which both A and B occur. That's equal to the probability of A occurring given B times the probability of B, which is also equal to the probability of B occurring given A times the probability of A. I think this should make some intuition for you. If it doesn't, it might be a good idea to watch the conditional probability videos. But what we can do is we can rearrange this equation right here. If we just divide both sides by the probability of B, we get the probability, and I'll do this in a vibrant color, the probability of A given B is equal to the probability of B given A times the probability of A divided by the probability of B. I just took this equation, divided both sides by the probability of B, and I got this. So what is A and B in the problem we're trying to figure out?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it doesn't, it might be a good idea to watch the conditional probability videos. But what we can do is we can rearrange this equation right here. If we just divide both sides by the probability of B, we get the probability, and I'll do this in a vibrant color, the probability of A given B is equal to the probability of B given A times the probability of A divided by the probability of B. I just took this equation, divided both sides by the probability of B, and I got this. So what is A and B in the problem we're trying to figure out? We want to try to figure out the probability that I picked out a fair coin given that I got 4 out of 6 heads. So in this situation, A is that I got a fair coin. A is equal to picked fair coin."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is A and B in the problem we're trying to figure out? We want to try to figure out the probability that I picked out a fair coin given that I got 4 out of 6 heads. So in this situation, A is that I got a fair coin. A is equal to picked fair coin. And then B is equal to 4 out of 6 heads. So in order to figure out the probability that I picked a fair coin given that I got 4 out of 6 heads, I have to know the probability of getting 4 out of 6 heads given that I picked the fair coin times the probability of picking out a fair coin divided by the probability of getting 4 out of 6 heads in general. So this is probably the hardest part to figure out, and we will, along the way, we'll actually probably figure out the top two terms."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "A is equal to picked fair coin. And then B is equal to 4 out of 6 heads. So in order to figure out the probability that I picked a fair coin given that I got 4 out of 6 heads, I have to know the probability of getting 4 out of 6 heads given that I picked the fair coin times the probability of picking out a fair coin divided by the probability of getting 4 out of 6 heads in general. So this is probably the hardest part to figure out, and we will, along the way, we'll actually probably figure out the top two terms. So what's the probability of B, or the probability of getting 4 out of 6 heads? Let's see what happens. Right when I put my hand into the bag and I pick out a coin, there's a 5 in 10 chance, or 5 in 15 chance, right, there are 15 total coins, that I pick a fair coin."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is probably the hardest part to figure out, and we will, along the way, we'll actually probably figure out the top two terms. So what's the probability of B, or the probability of getting 4 out of 6 heads? Let's see what happens. Right when I put my hand into the bag and I pick out a coin, there's a 5 in 10 chance, or 5 in 15 chance, right, there are 15 total coins, that I pick a fair coin. So 5 in 15, that's the same thing as 1 third, that I pick a fair coin. And then there's a 2 thirds chance that I pick a unfair coin. Now if I pick a fair coin, given that I have a fair coin, what is the probability given the fair coin, what is the probability that I get 4 out of 6 heads?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Right when I put my hand into the bag and I pick out a coin, there's a 5 in 10 chance, or 5 in 15 chance, right, there are 15 total coins, that I pick a fair coin. So 5 in 15, that's the same thing as 1 third, that I pick a fair coin. And then there's a 2 thirds chance that I pick a unfair coin. Now if I pick a fair coin, given that I have a fair coin, what is the probability given the fair coin, what is the probability that I get 4 out of 6 heads? Well once again, let's think about the previous several videos. What's the probability of getting any one particular combination of 4 out of 6 heads? So for example, it could be heads, tails, heads, tails, heads, heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now if I pick a fair coin, given that I have a fair coin, what is the probability given the fair coin, what is the probability that I get 4 out of 6 heads? Well once again, let's think about the previous several videos. What's the probability of getting any one particular combination of 4 out of 6 heads? So for example, it could be heads, tails, heads, tails, heads, heads. It could be, I don't know, it could be the first 4 heads, heads, heads, heads, heads, tails, tails. Right, and there are a bunch of these, and we, once again, will use the binomial coefficient, or we'll use our knowledge of combinations to figure out how many different combinations there are. But what's the probability of each of these combinations?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, it could be heads, tails, heads, tails, heads, heads. It could be, I don't know, it could be the first 4 heads, heads, heads, heads, heads, tails, tails. Right, and there are a bunch of these, and we, once again, will use the binomial coefficient, or we'll use our knowledge of combinations to figure out how many different combinations there are. But what's the probability of each of these combinations? Well, what's the probability of heads? That's 0.5 times 0.5 times 0.5 times 0.5. And then the probability of tails, since it's a fair coin, is also 0.5, times 0.5 times 0.5."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what's the probability of each of these combinations? Well, what's the probability of heads? That's 0.5 times 0.5 times 0.5 times 0.5. And then the probability of tails, since it's a fair coin, is also 0.5, times 0.5 times 0.5. So each of these, there's a 1 half chance of getting a heads times a 1 half chance of a tails times a 1 half chance of a heads times a 1 half chance of a tails, et cetera, et cetera. So each of these are essentially 1 half times 1 half, 6 times. So the probability of each of the combinations is 1 half to the 6th power."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the probability of tails, since it's a fair coin, is also 0.5, times 0.5 times 0.5. So each of these, there's a 1 half chance of getting a heads times a 1 half chance of a tails times a 1 half chance of a heads times a 1 half chance of a tails, et cetera, et cetera. So each of these are essentially 1 half times 1 half, 6 times. So the probability of each of the combinations is 1 half to the 6th power. And so how many combinations are there like this, where you get, out of the 6 flips, you're choosing, you're essentially choosing 4 heads. You're choosing, I'm, once again, the god of probability. I am picking 4, exactly 4, of the 6 heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of each of the combinations is 1 half to the 6th power. And so how many combinations are there like this, where you get, out of the 6 flips, you're choosing, you're essentially choosing 4 heads. You're choosing, I'm, once again, the god of probability. I am picking 4, exactly 4, of the 6 heads. Sorry, I'm picking 4 of exactly 6 of the flips to end up heads. I'm choosing which of the flips get selected, so to speak. So it's essentially, there are going to be, out of 6 flips, I'm choosing, as the god of probability, 4 to be heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I am picking 4, exactly 4, of the 6 heads. Sorry, I'm picking 4 of exactly 6 of the flips to end up heads. I'm choosing which of the flips get selected, so to speak. So it's essentially, there are going to be, out of 6 flips, I'm choosing, as the god of probability, 4 to be heads. So that's the number of combinations, the number of unique combinations, where you have 4 out of 6 heads, times the probability of each of the combinations, which is 1 half to the 6th power. Well, let's 6 choose 4. That's 6 factorial over 4 factorial times 6 minus 4 factorial, so that's 2 factorial."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's essentially, there are going to be, out of 6 flips, I'm choosing, as the god of probability, 4 to be heads. So that's the number of combinations, the number of unique combinations, where you have 4 out of 6 heads, times the probability of each of the combinations, which is 1 half to the 6th power. Well, let's 6 choose 4. That's 6 factorial over 4 factorial times 6 minus 4 factorial, so that's 2 factorial. And that's times 1 half to the 6th. And I'll switch colors again, just to stop the monotony. And that equals, let's see, 6 times 5 times 4 times 3 times 2, we don't have to write the 1 times 1, I'll do it anyway."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's 6 factorial over 4 factorial times 6 minus 4 factorial, so that's 2 factorial. And that's times 1 half to the 6th. And I'll switch colors again, just to stop the monotony. And that equals, let's see, 6 times 5 times 4 times 3 times 2, we don't have to write the 1 times 1, I'll do it anyway. Over 4 factorial, 4 times 3 times 2 times 1, and then 2 factorial, 2 times 1. So that cancels with that. The 1 we can ignore."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that equals, let's see, 6 times 5 times 4 times 3 times 2, we don't have to write the 1 times 1, I'll do it anyway. Over 4 factorial, 4 times 3 times 2 times 1, and then 2 factorial, 2 times 1. So that cancels with that. The 1 we can ignore. 2, divide both sides by, the numerator and denominator by 2, and this becomes a 3. So this becomes 15. So this equals 15 times 1 half to the 6th."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "The 1 we can ignore. 2, divide both sides by, the numerator and denominator by 2, and this becomes a 3. So this becomes 15. So this equals 15 times 1 half to the 6th. What's 1 half to the 6th? That's 1 over 64, right? So 1 over 64, so it becomes 15 over 64."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this equals 15 times 1 half to the 6th. What's 1 half to the 6th? That's 1 over 64, right? So 1 over 64, so it becomes 15 over 64. So the probability of getting 4 out of 6 heads, given a fair coin, is 15 out of 64. So this is the probability of 4 out of 6 heads, given a fair coin. And if you look at it, based on our definition of B and A, this is the probability of B given A."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 1 over 64, so it becomes 15 over 64. So the probability of getting 4 out of 6 heads, given a fair coin, is 15 out of 64. So this is the probability of 4 out of 6 heads, given a fair coin. And if you look at it, based on our definition of B and A, this is the probability of B given A. B is 4 out of 6 heads, given a fair coin. Fair enough. So let's figure out the probability of, because there's two ways of getting 4 out of 6 heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you look at it, based on our definition of B and A, this is the probability of B given A. B is 4 out of 6 heads, given a fair coin. Fair enough. So let's figure out the probability of, because there's two ways of getting 4 out of 6 heads. One, that we picked a fair coin, and then times 15 out of 64, and then there's a probability that we picked an unfair coin. So what's the probability of the unfair coin? Of getting 4 out of 6 heads, given the unfair coin?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's figure out the probability of, because there's two ways of getting 4 out of 6 heads. One, that we picked a fair coin, and then times 15 out of 64, and then there's a probability that we picked an unfair coin. So what's the probability of the unfair coin? Of getting 4 out of 6 heads, given the unfair coin? Well, once again, what's the probability of each of the combinations where you get 4 out of 6? So in this situation, let's do the same one. Heads, tails, heads, tails, heads, heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Of getting 4 out of 6 heads, given the unfair coin? Well, once again, what's the probability of each of the combinations where you get 4 out of 6? So in this situation, let's do the same one. Heads, tails, heads, tails, heads, heads. That's 4 out of 6 heads. But in this situation, it's not a 50% chance of getting heads, it's 80%. So it would be 0.8 times 0.2 times 0.8 times 0.2 times 0.8 times 0.8."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Heads, tails, heads, tails, heads, heads. That's 4 out of 6 heads. But in this situation, it's not a 50% chance of getting heads, it's 80%. So it would be 0.8 times 0.2 times 0.8 times 0.2 times 0.8 times 0.8. Essentially, we have this multiplication. We can rearrange it, because it doesn't matter what order you multiply things in. So it's 0.8 to the fourth power times 0.2 squared."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it would be 0.8 times 0.2 times 0.8 times 0.2 times 0.8 times 0.8. Essentially, we have this multiplication. We can rearrange it, because it doesn't matter what order you multiply things in. So it's 0.8 to the fourth power times 0.2 squared. And it doesn't matter. Any of the unique combinations will each have the same probability, because we can just rearrange the order in which we multiply. And then how many of these combinations are there?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 0.8 to the fourth power times 0.2 squared. And it doesn't matter. Any of the unique combinations will each have the same probability, because we can just rearrange the order in which we multiply. And then how many of these combinations are there? If we are, once again, the god of probability, and out of 6 flips, we are choosing 4 that are going to end up heads. How many ways can I pick a group of 4? Well, once again, that's times 6 choose 4."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then how many of these combinations are there? If we are, once again, the god of probability, and out of 6 flips, we are choosing 4 that are going to end up heads. How many ways can I pick a group of 4? Well, once again, that's times 6 choose 4. And we figured out what that is. 6 choose 4 is 15. So this equals 15 times 0.8 to the fourth times 0.2 squared."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, once again, that's times 6 choose 4. And we figured out what that is. 6 choose 4 is 15. So this equals 15 times 0.8 to the fourth times 0.2 squared. So this is the probability of 4 out of 6 heads given an unfair coin. So what's the total probability of getting 4 out of 6 heads? Well, it's going to be the probability of getting the fair coin, which is 1 third, times the probability of getting 4 out of 6 heads given the fair coin."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this equals 15 times 0.8 to the fourth times 0.2 squared. So this is the probability of 4 out of 6 heads given an unfair coin. So what's the total probability of getting 4 out of 6 heads? Well, it's going to be the probability of getting the fair coin, which is 1 third, times the probability of getting 4 out of 6 heads given the fair coin. And that's this, 15 over 64, plus the probability of getting an unfair coin, 2 thirds, times the probability of getting 4 out of 6 heads given the unfair coin. And that's what we figured out here. Times 15 times 0.8 to the fourth times 0.2 squared."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's going to be the probability of getting the fair coin, which is 1 third, times the probability of getting 4 out of 6 heads given the fair coin. And that's this, 15 over 64, plus the probability of getting an unfair coin, 2 thirds, times the probability of getting 4 out of 6 heads given the unfair coin. And that's what we figured out here. Times 15 times 0.8 to the fourth times 0.2 squared. And this is the probability of getting 4 out of 6 heads. And let's figure out what that is. Well, this will cancel out with this."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Times 15 times 0.8 to the fourth times 0.2 squared. And this is the probability of getting 4 out of 6 heads. And let's figure out what that is. Well, this will cancel out with this. This becomes 5 out of 64. That's easy enough. 2 thirds times 15, that's 10."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this will cancel out with this. This becomes 5 out of 64. That's easy enough. 2 thirds times 15, that's 10. And now we just have to figure out what that is. Let's see. I'm going to go over the time limit to see if being a YouTube partner allows me to go over the time limit."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "2 thirds times 15, that's 10. And now we just have to figure out what that is. Let's see. I'm going to go over the time limit to see if being a YouTube partner allows me to go over the time limit. Let's see. 0.8 times 0.8 is equal to. And then times 0.2 squared."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to go over the time limit to see if being a YouTube partner allows me to go over the time limit. Let's see. 0.8 times 0.8 is equal to. And then times 0.2 squared. So times 0.2 is equal to 0.016. So that's that. And then we say times 10, right?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then times 0.2 squared. So times 0.2 is equal to 0.016. So that's that. And then we say times 10, right? Because 2 thirds times 15. So times 10 is equal to 16.384%. So this term right here, let me write that down."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we say times 10, right? Because 2 thirds times 15. So times 10 is equal to 16.384%. So this term right here, let me write that down. And I'll switch colors again. This is 0.16384. And we're going to add that to 5 divided by 64."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this term right here, let me write that down. And I'll switch colors again. This is 0.16384. And we're going to add that to 5 divided by 64. So let's see. 5 divided by 64 is equal to 0.07, whatever, whatever. Plus 0.16384 is equal to 0.241965."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to add that to 5 divided by 64. So let's see. 5 divided by 64 is equal to 0.07, whatever, whatever. Plus 0.16384 is equal to 0.241965. So that's the probability. Not knowing which coin I picked out, that's the probability of getting 4 out of 6 heads. When you combine it, it could be 1 third chance fair, 2 thirds chance unfair."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 0.16384 is equal to 0.241965. So that's the probability. Not knowing which coin I picked out, that's the probability of getting 4 out of 6 heads. When you combine it, it could be 1 third chance fair, 2 thirds chance unfair. So that's 24.19. I'm keeping the precision just because it might come in useful later. Percent chance."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "When you combine it, it could be 1 third chance fair, 2 thirds chance unfair. So that's 24.19. I'm keeping the precision just because it might come in useful later. Percent chance. So that's the probability of B. So let's see if we can clean this up a little bit, just because I don't think we need all of this writing now. I think we're ready to substitute into our Bayes formula, which we, Bayes' theorem that we re-derived."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Percent chance. So that's the probability of B. So let's see if we can clean this up a little bit, just because I don't think we need all of this writing now. I think we're ready to substitute into our Bayes formula, which we, Bayes' theorem that we re-derived. Recording longer videos is dangerous, because if I make a mistake, that's more time wasted. I don't want to delete anything that could be useful. OK."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I think we're ready to substitute into our Bayes formula, which we, Bayes' theorem that we re-derived. Recording longer videos is dangerous, because if I make a mistake, that's more time wasted. I don't want to delete anything that could be useful. OK. So let's see if we can solve the probability that we picked a fair coin given that we got 4 out of 6 heads. So that is going to be equal to, by Bayes' theorem, which should make some sense to you, that is equal to the probability of B given A. So it's the probability that we get 4 out of 6 heads given a fair coin times the probability of a fair coin over the probability of getting 4 out of 6 heads either way."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "OK. So let's see if we can solve the probability that we picked a fair coin given that we got 4 out of 6 heads. So that is going to be equal to, by Bayes' theorem, which should make some sense to you, that is equal to the probability of B given A. So it's the probability that we get 4 out of 6 heads given a fair coin times the probability of a fair coin over the probability of getting 4 out of 6 heads either way. 4 out of 6 heads. So 4 out of 6 heads given a fair coin, we figured that over here, that's 15 over 64. So this equals 15 over 64."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's the probability that we get 4 out of 6 heads given a fair coin times the probability of a fair coin over the probability of getting 4 out of 6 heads either way. 4 out of 6 heads. So 4 out of 6 heads given a fair coin, we figured that over here, that's 15 over 64. So this equals 15 over 64. What's the probability that we picked a fair coin? Well, there's 15 coins, and 5 of them are fair, so it's 5 out of 15, so it's 1 third. And what's the probability that, in general, we picked 4 out of 6 heads?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this equals 15 over 64. What's the probability that we picked a fair coin? Well, there's 15 coins, and 5 of them are fair, so it's 5 out of 15, so it's 1 third. And what's the probability that, in general, we picked 4 out of 6 heads? Well, that's this number, 0.241965. So this equals, let's see, this is equal to 5 over 64 divided by 0.241965. And what is that equal to?"}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what's the probability that, in general, we picked 4 out of 6 heads? Well, that's this number, 0.241965. So this equals, let's see, this is equal to 5 over 64 divided by 0.241965. And what is that equal to? That's 5 divided by 64 is equal to that, divided by 0.241965 is equal to 32.3%. So that's amazing, or relatively amazing. We've now, it's a little bit less than a 1 third shot that we picked the fair coin given that we got 4 out of 6 heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what is that equal to? That's 5 divided by 64 is equal to that, divided by 0.241965 is equal to 32.3%. So that's amazing, or relatively amazing. We've now, it's a little bit less than a 1 third shot that we picked the fair coin given that we got 4 out of 6 heads. And what's interesting is, the 4 out of 6 heads, it kind of decreased the probability that we got a fair coin, right? Because before having any data on what happens when we flip it, we would have had a 1 third probability, which is 33.3, right? But given that we got more heads than tails, kind of the universe of probability is telling us that, well, if you got more heads than tails, that makes it a little bit more likely that you picked the unfair coin, which is a little bit more weighted to heads."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "We've now, it's a little bit less than a 1 third shot that we picked the fair coin given that we got 4 out of 6 heads. And what's interesting is, the 4 out of 6 heads, it kind of decreased the probability that we got a fair coin, right? Because before having any data on what happens when we flip it, we would have had a 1 third probability, which is 33.3, right? But given that we got more heads than tails, kind of the universe of probability is telling us that, well, if you got more heads than tails, that makes it a little bit more likely that you picked the unfair coin, which is a little bit more weighted to heads. But it's saying it's not that much more likely, because this isn't that unusual of a result to get even with a fair coin. And so that's why it became a little bit less likely to get a fair coin. I will, actually, let me give you a bit of an intuition visually, kind of with set theory, on why that makes sense."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "But given that we got more heads than tails, kind of the universe of probability is telling us that, well, if you got more heads than tails, that makes it a little bit more likely that you picked the unfair coin, which is a little bit more weighted to heads. But it's saying it's not that much more likely, because this isn't that unusual of a result to get even with a fair coin. And so that's why it became a little bit less likely to get a fair coin. I will, actually, let me give you a bit of an intuition visually, kind of with set theory, on why that makes sense. So if we go back to Bayes' theorem, let's just say that this is our, this is the universe of all of the events, right? That's all of the universe. There's roughly a 1 third chance that I picked a fair coin."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I will, actually, let me give you a bit of an intuition visually, kind of with set theory, on why that makes sense. So if we go back to Bayes' theorem, let's just say that this is our, this is the universe of all of the events, right? That's all of the universe. There's roughly a 1 third chance that I picked a fair coin. So roughly 1 third of this will be fair. This is fair. This is unfair."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's roughly a 1 third chance that I picked a fair coin. So roughly 1 third of this will be fair. This is fair. This is unfair. And then if I picked a fair coin, we figured out that it was roughly a 15 out of 64 shot that I get 4 out of 6 heads, so maybe that's this little section of the, let me do it in a different color, that's this section. And then we figured out if we have an unfair coin, I forgot what the exact number is, but there was some probability that we get 4 out of 6 heads. It's actually a little bit bigger."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is unfair. And then if I picked a fair coin, we figured out that it was roughly a 15 out of 64 shot that I get 4 out of 6 heads, so maybe that's this little section of the, let me do it in a different color, that's this section. And then we figured out if we have an unfair coin, I forgot what the exact number is, but there was some probability that we get 4 out of 6 heads. It's actually a little bit bigger. It's like that. So this is getting 4 out of 6 heads given you got an unfair coin. This is getting 4 out of 6 heads given that you got a fair coin."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually a little bit bigger. It's like that. So this is getting 4 out of 6 heads given you got an unfair coin. This is getting 4 out of 6 heads given that you got a fair coin. And then this whole area is a probability that you get 4 out of 6 heads. So all Bayes' theorem told us is, look, we got 4 out of 6 heads, so we're in this universe where we got 4 out of 6 heads. And if we got 4 out of 6 heads, 1 third of this universe, roughly, or 32.3% of this subset of 4 out of 6 heads, intersects with the fair coin universe."}, {"video_title": "Conditional probability and combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is getting 4 out of 6 heads given that you got a fair coin. And then this whole area is a probability that you get 4 out of 6 heads. So all Bayes' theorem told us is, look, we got 4 out of 6 heads, so we're in this universe where we got 4 out of 6 heads. And if we got 4 out of 6 heads, 1 third of this universe, roughly, or 32.3% of this subset of 4 out of 6 heads, intersects with the fair coin universe. So this 32.3% is essentially this fraction of the total probability of getting 4 out of 6 heads. Anyway, hopefully that gave you a little bit of intuition, and I hope that YouTube lets me publish this video because I'm on my 17th minute. I'll see you in the next video."}, {"video_title": "Writing hypotheses for a significance test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "They want to test if this is convincing evidence that the mean amount for bottles in this batch is different than the target value of 500 milliliters. Let mu be the mean amount of liquid in each bottle in the batch. Write an appropriate set of hypotheses for their significance test, for the significance test that the quality-control expert is running. So pause this video and see if you can do that. All right, now let's do this together. So first, you're going to have two hypotheses. You're gonna have your null hypothesis and your alternative hypothesis."}, {"video_title": "Writing hypotheses for a significance test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "So pause this video and see if you can do that. All right, now let's do this together. So first, you're going to have two hypotheses. You're gonna have your null hypothesis and your alternative hypothesis. Your null hypothesis is going to be a hypothesis about the population parameter that you care about, and it's going to assume kind of the status quo, no news here. And so the parameter that we care about is the mean amount of liquid in the bottles in the batch. So that's mu right over there."}, {"video_title": "Writing hypotheses for a significance test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "You're gonna have your null hypothesis and your alternative hypothesis. Your null hypothesis is going to be a hypothesis about the population parameter that you care about, and it's going to assume kind of the status quo, no news here. And so the parameter that we care about is the mean amount of liquid in the bottles in the batch. So that's mu right over there. And what would be the assumption that that would be, the no news here? Well, it would be 500 milliliters. That's the target value."}, {"video_title": "Writing hypotheses for a significance test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "So that's mu right over there. And what would be the assumption that that would be, the no news here? Well, it would be 500 milliliters. That's the target value. So it's reasonable to say, well, you know, the null is it's doing what it's supposed to, that where the actual mean for the batch is actually what the target needs to be, is actually 500 milliliters. Some of you might have said, hey, wait, didn't they say the amounts in the sample had a mean of 503 milliliters? Why isn't this 503?"}, {"video_title": "Writing hypotheses for a significance test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "That's the target value. So it's reasonable to say, well, you know, the null is it's doing what it's supposed to, that where the actual mean for the batch is actually what the target needs to be, is actually 500 milliliters. Some of you might have said, hey, wait, didn't they say the amounts in the sample had a mean of 503 milliliters? Why isn't this 503? Remember, your hypothesis is going to be about the population parameter, your assumption about the population parameter. This 503 milliliters right over here, this is a sample statistic. This is a sample mean that's trying to estimate this thing right over here."}, {"video_title": "Writing hypotheses for a significance test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Why isn't this 503? Remember, your hypothesis is going to be about the population parameter, your assumption about the population parameter. This 503 milliliters right over here, this is a sample statistic. This is a sample mean that's trying to estimate this thing right over here. When we do our significance test, we're going to incorporate this 533 milliliters. We're going to think about, well, what's the probability of getting a sample statistic, a sample mean, this far or further away from the assumed mean if we assume that the null hypothesis is true, and if that probability is below a threshold, our significance level, then we reject the null hypothesis, and it would suggest the alternative. But if we're just trying to generate or write a set of hypotheses, this would be our null hypothesis."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this says a manager oversees 11 female employees and nine male employees. They need to pick three of these employees to go on a business trip, so the manager places all 20 names in a hat and chooses at random. Let x equal the number of female employees chosen. So they're going to do three trials, and on each of those trials, you could say success is if they pick a female employee, and then the random variable x is the number of females out of those three. Is x a binomial variable? Why or why not? So pause this video and see if you can work through this on your own."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So they're going to do three trials, and on each of those trials, you could say success is if they pick a female employee, and then the random variable x is the number of females out of those three. Is x a binomial variable? Why or why not? So pause this video and see if you can work through this on your own. All right, now let's go through each choice. Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable. I disagree with this."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So pause this video and see if you can work through this on your own. All right, now let's go through each choice. Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable. I disagree with this. Each trial is being classified as a success or failure. It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that would be a success, so each trial is being classified as a success or failure. So this one over here isn't true."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "I disagree with this. Each trial is being classified as a success or failure. It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that would be a success, so each trial is being classified as a success or failure. So this one over here isn't true. There is no fixed number of trials, so x is not a binomial variable. There is a fixed number of trials. They're doing three trials."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this one over here isn't true. There is no fixed number of trials, so x is not a binomial variable. There is a fixed number of trials. They're doing three trials. They're picking three hats, three names out of a hat. The trials are not independent, so x is not a binomial variable. So this is interesting."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "They're doing three trials. They're picking three hats, three names out of a hat. The trials are not independent, so x is not a binomial variable. So this is interesting. So for example, trial one, what's the probability of success? Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the probability of success given success in, I'll say, trial one, T1? Well, if you succeeded in trial one, that means that there's now only 10 female names in the hat out of 19, and if you don't have success in trial one, then you will have, then it'll be 11 out of 19, so your probability does change based on previous outcomes, and so the trials are not independent, and so x is not a binomial variable."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this is interesting. So for example, trial one, what's the probability of success? Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the probability of success given success in, I'll say, trial one, T1? Well, if you succeeded in trial one, that means that there's now only 10 female names in the hat out of 19, and if you don't have success in trial one, then you will have, then it'll be 11 out of 19, so your probability does change based on previous outcomes, and so the trials are not independent, and so x is not a binomial variable. So this one is true. The trials are not independent, so that violates that condition for being a binomial variable. In order to be a binomial variable, all your trials have to be independent of each other, and so we'd rule this last one out because this last one says that x has a binomial distribution, or it is or does meet all the conditions for being a binomial variable."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if you succeeded in trial one, that means that there's now only 10 female names in the hat out of 19, and if you don't have success in trial one, then you will have, then it'll be 11 out of 19, so your probability does change based on previous outcomes, and so the trials are not independent, and so x is not a binomial variable. So this one is true. The trials are not independent, so that violates that condition for being a binomial variable. In order to be a binomial variable, all your trials have to be independent of each other, and so we'd rule this last one out because this last one says that x has a binomial distribution, or it is or does meet all the conditions for being a binomial variable. Let's do another example. So here we have different scenarios, and I have the conditions for binomial variable written right over here, and so once again, pause the video and look at each of these scenarios for random variables, and look at these conditions, and think about whether these random variables are binomial or not. So let's look at the first one."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "In order to be a binomial variable, all your trials have to be independent of each other, and so we'd rule this last one out because this last one says that x has a binomial distribution, or it is or does meet all the conditions for being a binomial variable. Let's do another example. So here we have different scenarios, and I have the conditions for binomial variable written right over here, and so once again, pause the video and look at each of these scenarios for random variables, and look at these conditions, and think about whether these random variables are binomial or not. So let's look at the first one. In a game involving a standard deck of 52 playing cards, an individual randomly draws seven cards without replacement. Let y be equal to the number of aces drawn. Well, in our introductory video to binomial variables, we talked about if we're doing without replacement, your probability of getting an ace on a given trial, where trial is you're taking a card out of the deck, it's going to be dependent on whether you got aces in previous trials, because if you got an ace in a previous trial, well, that ace, then you're gonna have fewer aces in the decks."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let's look at the first one. In a game involving a standard deck of 52 playing cards, an individual randomly draws seven cards without replacement. Let y be equal to the number of aces drawn. Well, in our introductory video to binomial variables, we talked about if we're doing without replacement, your probability of getting an ace on a given trial, where trial is you're taking a card out of the deck, it's going to be dependent on whether you got aces in previous trials, because if you got an ace in a previous trial, well, that ace, then you're gonna have fewer aces in the decks. So the trials in this case are not independent. Not independent, not independent trials. Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, in our introductory video to binomial variables, we talked about if we're doing without replacement, your probability of getting an ace on a given trial, where trial is you're taking a card out of the deck, it's going to be dependent on whether you got aces in previous trials, because if you got an ace in a previous trial, well, that ace, then you're gonna have fewer aces in the decks. So the trials in this case are not independent. Not independent, not independent trials. Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials. The probability of getting an ace on each trial would be the same, but not when you have without replacement. So this is not binomial right over here, because you don't have independent trials. The second scenario."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials. The probability of getting an ace on each trial would be the same, but not when you have without replacement. So this is not binomial right over here, because you don't have independent trials. The second scenario. 60% of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The second scenario. 60% of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other. So whether one plant lives isn't dependent on whether another plant lives. Let T equal the number of tomato plants that live. All right, so let's look at the conditions."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Assume that the plants live independently of each other. So whether one plant lives isn't dependent on whether another plant lives. Let T equal the number of tomato plants that live. All right, so let's look at the conditions. The outcome of each trial can be classified as either success or failure. So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it dies. So we have either success or failure."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "All right, so let's look at the conditions. The outcome of each trial can be classified as either success or failure. So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it dies. So we have either success or failure. Each trial is independent of the others. They tell us the plants live independently of each other. So whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So we have either success or failure. Each trial is independent of the others. They tell us the plants live independently of each other. So whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies. So each trial is independent of the others. There is a fixed number of trials. Yes, we have 16 right over there."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies. So each trial is independent of the others. There is a fixed number of trials. Yes, we have 16 right over there. The probability P of success on each trial remains constant. Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Yes, we have 16 right over there. The probability P of success on each trial remains constant. Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here. So this one is binomial. Now let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double, two of the same face values."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it meets all of the conditions right over here. So this one is binomial. Now let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double, two of the same face values. Let x equal the number of rolls in one turn. So you're gonna keep rolling until they roll a double. Well, the thing that jumps out at me is that you don't have a fixed number of trials, not fixed number of trials."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double, two of the same face values. Let x equal the number of rolls in one turn. So you're gonna keep rolling until they roll a double. Well, the thing that jumps out at me is that you don't have a fixed number of trials, not fixed number of trials. You could say each trial, each roll is a trial. Success is getting a double, which has a fixed probability. Whether or not you get a double on each trial is gonna be independent of the previous roll."}, {"video_title": "Recognizing binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the thing that jumps out at me is that you don't have a fixed number of trials, not fixed number of trials. You could say each trial, each roll is a trial. Success is getting a double, which has a fixed probability. Whether or not you get a double on each trial is gonna be independent of the previous roll. So it meets all the other constraints, but it does not meet that there's a fixed number of trials. You're gonna keep someone, there's some chance you might have to roll 20 times, or 200 times, or who knows however many times, until they roll a double. And so this violates there's a fixed number of trials."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe it's not your whole school, maybe it's just your grade. So there's 80 students in your population, and you want to get an estimate of the average height in your population. And you think it's too hard for you to go and measure the height of all 80 students, so you decide to find a simple, or take a simple random sample. You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their average height, and say, well, that's probably a pretty good estimate for the population parameter, for the average height of the entire population. So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their average height, and say, well, that's probably a pretty good estimate for the population parameter, for the average height of the entire population. So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and then pick them out. So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and then pick them out. So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right, so that's our bowl. And so all the pieces of paper go in there."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right, so that's our bowl. And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random number generator. And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random number generator. And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random. You might see, you might see something like random. Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You might see something like random. You might see, you might see something like random. Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it could even pick out a random number between one and 80, including one and 80, is you would maybe line up all the students' names alphabetically. And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it could even pick out a random number between one and 80, including one and 80, is you would maybe line up all the students' names alphabetically. And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our number of digits. And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our number of digits. And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers. And so let's say that's our randomly generated list of numbers, and it keeps going well beyond this. And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers. And so let's say that's our randomly generated list of numbers, and it keeps going well beyond this. And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59. Is 59 between one and 80?"}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59. Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits. The next two digits are 83."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits. The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits. So we get a five and a nine."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again. So we keep moving on."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range. We haven't picked that yet."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we keep moving on. Then we get a 37. Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero. Once again, not in our range."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We haven't picked that yet. We do that. Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range. 23, it's in our range, and we haven't picked it yet."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, not in our range. I think you see where this is going. 91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going. And this isn't an exhaustive list of all of the different ways that you can get random numbers, but it starts to give you some techniques in your toolkit."}, {"video_title": "Techniques for generating a simple random sample   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going. And this isn't an exhaustive list of all of the different ways that you can get random numbers, but it starts to give you some techniques in your toolkit. And you might say, oh, well, why don't I just randomly come up with some numbers in my head? And I would really suggest that you don't do that because humans are famously bad at being truly random. And you might wanna do something like even use something that you think is a random process, but you realize later that it wasn't as random as you thought."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "So what he does here is a simulation. It has a population that has a uniform distribution. So he says, I used a flat probabilistic distribution from 0 to 100 for my population. Then we start sampling from that population. We're going to use samples of size 50. And what we do is for each of those samples, we calculate the sample variance based on dividing by n by dividing by n minus 1 and n minus 2. And as we keep having more and more and more samples, we take the mean of the variances calculated in different ways."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Then we start sampling from that population. We're going to use samples of size 50. And what we do is for each of those samples, we calculate the sample variance based on dividing by n by dividing by n minus 1 and n minus 2. And as we keep having more and more and more samples, we take the mean of the variances calculated in different ways. And we figure out what those means converge to. So that's a sample. Here's another sample."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And as we keep having more and more and more samples, we take the mean of the variances calculated in different ways. And we figure out what those means converge to. So that's a sample. Here's another sample. Here's another sample. If I sample here, then now I'm adding a bunch. And I'm sampling continuously."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Here's another sample. Here's another sample. If I sample here, then now I'm adding a bunch. And I'm sampling continuously. And you saw something very interesting happen. When I divide by n, I get my sample variance is still, even when I'm taking the mean of many, many, many, many sample variances that I've already taken, I'm still underestimating the true variance. When I divide by n minus 1, it looks like I'm getting a pretty good estimate."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And I'm sampling continuously. And you saw something very interesting happen. When I divide by n, I get my sample variance is still, even when I'm taking the mean of many, many, many, many sample variances that I've already taken, I'm still underestimating the true variance. When I divide by n minus 1, it looks like I'm getting a pretty good estimate. The mean of all of my sample variances is really converged to the true variance. When I divided by n minus 2 just for kicks, it's pretty clear that I overestimated with my mean of my sample variances. I overestimated the true variance."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "When I divide by n minus 1, it looks like I'm getting a pretty good estimate. The mean of all of my sample variances is really converged to the true variance. When I divided by n minus 2 just for kicks, it's pretty clear that I overestimated with my mean of my sample variances. I overestimated the true variance. So this gives us a pretty good sense that n minus 1 is the right thing to do. Now, this is another way and another interesting way of visualizing it. In the horizontal axis right over here, we're comparing each plot as one of our samples."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "I overestimated the true variance. So this gives us a pretty good sense that n minus 1 is the right thing to do. Now, this is another way and another interesting way of visualizing it. In the horizontal axis right over here, we're comparing each plot as one of our samples. And how far to the right is, how much more is that sample mean than the true mean? And when we go to the left, it's how much less is the sample mean than the true mean? So for example, this sample right over here, it's all the way over to the right."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "In the horizontal axis right over here, we're comparing each plot as one of our samples. And how far to the right is, how much more is that sample mean than the true mean? And when we go to the left, it's how much less is the sample mean than the true mean? So for example, this sample right over here, it's all the way over to the right. The sample mean there was a lot more than the true mean. Sample mean here was a lot less than the true mean. Sample mean here, only a little bit more than the true mean."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "So for example, this sample right over here, it's all the way over to the right. The sample mean there was a lot more than the true mean. Sample mean here was a lot less than the true mean. Sample mean here, only a little bit more than the true mean. In the vertical axis, using this denominator, dividing by n, we calculate two different variances. One variance we use the sample mean. The other variance we use the population mean."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Sample mean here, only a little bit more than the true mean. In the vertical axis, using this denominator, dividing by n, we calculate two different variances. One variance we use the sample mean. The other variance we use the population mean. And this, in the vertical axis, we compare the difference between the mean calculated with the sample mean versus the mean calculated with the population mean. So for example, this point right over here, when we calculate our mean with our sample mean, which is the normal way we do it, it significantly underestimates what the mean would have been if somehow we knew what the population mean was and we could calculate it that way. And you get this really interesting shape."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "The other variance we use the population mean. And this, in the vertical axis, we compare the difference between the mean calculated with the sample mean versus the mean calculated with the population mean. So for example, this point right over here, when we calculate our mean with our sample mean, which is the normal way we do it, it significantly underestimates what the mean would have been if somehow we knew what the population mean was and we could calculate it that way. And you get this really interesting shape. And it's something to think about. And he recommends thinking about why or what kind of a shape this actually is. The other interesting thing is, when you look at it this way, it's pretty clear this entire graph is sitting below the horizontal axis."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And you get this really interesting shape. And it's something to think about. And he recommends thinking about why or what kind of a shape this actually is. The other interesting thing is, when you look at it this way, it's pretty clear this entire graph is sitting below the horizontal axis. So we're always, when we calculate our sample variance using this formula, when we use the sample mean to do it, which we typically do, we're always getting a lower variance than when we use the population mean. Now this over here, when we divide by n minus 1, we're not always underestimating. Sometimes we're overestimating it."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "The other interesting thing is, when you look at it this way, it's pretty clear this entire graph is sitting below the horizontal axis. So we're always, when we calculate our sample variance using this formula, when we use the sample mean to do it, which we typically do, we're always getting a lower variance than when we use the population mean. Now this over here, when we divide by n minus 1, we're not always underestimating. Sometimes we're overestimating it. And when you take the mean of all of these variances, you converge. And here we're overestimating it a little bit more. And just to be clear, what we're talking about in these three graphs, let me take a screenshot of it and explain it in a little bit more depth."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Sometimes we're overestimating it. And when you take the mean of all of these variances, you converge. And here we're overestimating it a little bit more. And just to be clear, what we're talking about in these three graphs, let me take a screenshot of it and explain it in a little bit more depth. So just to be clear, in this red graph right over here, let me do this in a color close to, at least, so this orange, what this distance is for each of these samples, we're calculating the sample variance using, so let me, using the sample mean. And in this case, we are using n as our denominator, in this case right over here. And from that, we're subtracting the sample variance, or I guess you could call this some kind of pseudo-sample variance, if we somehow knew the population mean."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And just to be clear, what we're talking about in these three graphs, let me take a screenshot of it and explain it in a little bit more depth. So just to be clear, in this red graph right over here, let me do this in a color close to, at least, so this orange, what this distance is for each of these samples, we're calculating the sample variance using, so let me, using the sample mean. And in this case, we are using n as our denominator, in this case right over here. And from that, we're subtracting the sample variance, or I guess you could call this some kind of pseudo-sample variance, if we somehow knew the population mean. This isn't something that you see a lot in statistics, but it's a gauge of how much we are underestimating our sample variance, given that we don't have the true population mean at our disposal. And so this is the distance we're calculating. And you see we are always underestimating."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And from that, we're subtracting the sample variance, or I guess you could call this some kind of pseudo-sample variance, if we somehow knew the population mean. This isn't something that you see a lot in statistics, but it's a gauge of how much we are underestimating our sample variance, given that we don't have the true population mean at our disposal. And so this is the distance we're calculating. And you see we are always underestimating. Here, we overestimate a little bit, and we also underestimate. But when you take the mean, and when you average them all out, it converges to the actual value. So here, we're dividing by n minus 1."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It is the number of successes after n trials, after n trials, where the probability of success, the probability of success, success for each trial is p. And this is a safe, this is a reasonable way to describe really any random, any binomial variable. We're assuming that each of these trials are independent. The probability stays constant. We have a finite number of trials right over here. Each trial results in either a very clear success or failure. So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable? What would the expected value, expected value of x be equal to?"}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We have a finite number of trials right over here. Each trial results in either a very clear success or failure. So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable? What would the expected value, expected value of x be equal to? And I will just cut to the chase and tell you the answer, and then later in this video, we'll prove it to ourselves a little bit more mathematically. The expected value of x, it turns out, is just going to be equal to the number of trials times the probability of success for each of those trials. And so if you wanted to make that a little bit more concrete, imagine if a trial is a free throw, taking a shot from the free throw line."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "What would the expected value, expected value of x be equal to? And I will just cut to the chase and tell you the answer, and then later in this video, we'll prove it to ourselves a little bit more mathematically. The expected value of x, it turns out, is just going to be equal to the number of trials times the probability of success for each of those trials. And so if you wanted to make that a little bit more concrete, imagine if a trial is a free throw, taking a shot from the free throw line. Success, success is made shot, so you actually make the shot, the ball went in the basket. Your probability is, let me do this yellow color, your probability, this would be your free throw percentage, so let's say it's 30% or 0.3. And let's say, for the sake of argument, that we're taking 10 free throws, so n is equal to 10."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so if you wanted to make that a little bit more concrete, imagine if a trial is a free throw, taking a shot from the free throw line. Success, success is made shot, so you actually make the shot, the ball went in the basket. Your probability is, let me do this yellow color, your probability, this would be your free throw percentage, so let's say it's 30% or 0.3. And let's say, for the sake of argument, that we're taking 10 free throws, so n is equal to 10. So this is making it all a lot more concrete. So in this particular scenario, your expected value, your expected value, if x is the number of made free throws after taking 10 free throws with a free throw percentage of 30%, well, based on what I just told you, it'd be n times b. It would be the number of trials times the probability of success in any one of those trials times 0.3, which is just going to be, of course, equal to three."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And let's say, for the sake of argument, that we're taking 10 free throws, so n is equal to 10. So this is making it all a lot more concrete. So in this particular scenario, your expected value, your expected value, if x is the number of made free throws after taking 10 free throws with a free throw percentage of 30%, well, based on what I just told you, it'd be n times b. It would be the number of trials times the probability of success in any one of those trials times 0.3, which is just going to be, of course, equal to three. Now, does that make intuitive sense? Well, if you're taking 10 shots with a 30% free throw percentage, it actually does feel natural that I would expect to make three shots. Now, with that out of the way, let's make ourselves feel good about this mathematically, and we're gonna leverage some of our expected value properties."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It would be the number of trials times the probability of success in any one of those trials times 0.3, which is just going to be, of course, equal to three. Now, does that make intuitive sense? Well, if you're taking 10 shots with a 30% free throw percentage, it actually does feel natural that I would expect to make three shots. Now, with that out of the way, let's make ourselves feel good about this mathematically, and we're gonna leverage some of our expected value properties. In particular, we're gonna leverage the fact that if I have the expected value of the sum of two independent random variables, let's say x plus y, it's going to be equal to the expected value of x plus the expected value of y that we talk about in other videos. And so, assuming this right over here, let's construct a new random variable. Let's call our random variable y."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now, with that out of the way, let's make ourselves feel good about this mathematically, and we're gonna leverage some of our expected value properties. In particular, we're gonna leverage the fact that if I have the expected value of the sum of two independent random variables, let's say x plus y, it's going to be equal to the expected value of x plus the expected value of y that we talk about in other videos. And so, assuming this right over here, let's construct a new random variable. Let's call our random variable y. And we know the following things about y. The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for this random variable. And so, you might be seeing where this is going."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let's call our random variable y. And we know the following things about y. The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for this random variable. And so, you might be seeing where this is going. You could view this random variable, it's really representing one trial. It becomes one in a success, zero when you don't have a success. And so, you could view our original random variable x right over here as being equal to y plus y, and we're gonna have 10 of these."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so, you might be seeing where this is going. You could view this random variable, it's really representing one trial. It becomes one in a success, zero when you don't have a success. And so, you could view our original random variable x right over here as being equal to y plus y, and we're gonna have 10 of these. So, we're gonna have 10 y's. In the concrete sense, you could view the random variable y as equaling one if you make a free throw, and equaling zero if you don't make a free throw. It's really just representing one of those trials, and you can view x as the sum of n of those trials."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so, you could view our original random variable x right over here as being equal to y plus y, and we're gonna have 10 of these. So, we're gonna have 10 y's. In the concrete sense, you could view the random variable y as equaling one if you make a free throw, and equaling zero if you don't make a free throw. It's really just representing one of those trials, and you can view x as the sum of n of those trials. Well, actually, let me be very clear here. I immediately went to the concrete, but I really should be saying n y's, because I wanna stay general right over here. So, there are n n y's right over here."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It's really just representing one of those trials, and you can view x as the sum of n of those trials. Well, actually, let me be very clear here. I immediately went to the concrete, but I really should be saying n y's, because I wanna stay general right over here. So, there are n n y's right over here. This was just a particular example, but I am going to try to stay general for the rest of the video, because now we are really trying to prove this result right over here. So, let's just take the expected value of both sides. So, what is it going to be?"}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So, there are n n y's right over here. This was just a particular example, but I am going to try to stay general for the rest of the video, because now we are really trying to prove this result right over here. So, let's just take the expected value of both sides. So, what is it going to be? So, we get the expected value of x is equal to, well, it's the expected value of all of this thing, but by that property right over here, this is going to be the expected value of y plus the expected value of y, plus, and we're gonna do this n times, plus the expected value of y, and we're gonna have n of these. So, we have n. And so, you could rewrite this as being equal to, so this is our n right over here. This is n times the expected value of y."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So, what is it going to be? So, we get the expected value of x is equal to, well, it's the expected value of all of this thing, but by that property right over here, this is going to be the expected value of y plus the expected value of y, plus, and we're gonna do this n times, plus the expected value of y, and we're gonna have n of these. So, we have n. And so, you could rewrite this as being equal to, so this is our n right over here. This is n times the expected value of y. Now, what is the expected value of y? Well, this is pretty straightforward. We can actually just do it directly."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This is n times the expected value of y. Now, what is the expected value of y? Well, this is pretty straightforward. We can actually just do it directly. The expected value of y, let me just write it over here. The expected value of y is just the probability weighted outcomes, and since there's only two discrete outcomes here, it's pretty easy to calculate. We have a probability of p of getting a one, so it's p times one, plus we have a probability of one minus p of getting a zero."}, {"video_title": "Expected value of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We can actually just do it directly. The expected value of y, let me just write it over here. The expected value of y is just the probability weighted outcomes, and since there's only two discrete outcomes here, it's pretty easy to calculate. We have a probability of p of getting a one, so it's p times one, plus we have a probability of one minus p of getting a zero. Well, what does this simplify to? Well, zero times anything, that's zero, and then you have one times p. This is just equal to p. So, expected value of y is just equal to p, and so there you have it. We get the expected value of x is 10 times the expected value, or the expected value of x is n times the expected value of y, and the expected value of y is p, so the expected value of x is equal to np."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You start off with any crazy distribution. It doesn't have to be crazy. It could be a nice normal distribution, but to really make the point that you don't have to have a normal distribution, I like to use crazy ones. So let's say you have some kind of crazy distribution that looks something like that. It could look like anything. So we've seen multiple times you take samples from this crazy distribution. So let's say you were to take samples of, let's say n is equal to 10."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say you have some kind of crazy distribution that looks something like that. It could look like anything. So we've seen multiple times you take samples from this crazy distribution. So let's say you were to take samples of, let's say n is equal to 10. So we take 10 instances of this random variable, average them out, and then plot our average. We plot our average. We get one instance there."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say you were to take samples of, let's say n is equal to 10. So we take 10 instances of this random variable, average them out, and then plot our average. We plot our average. We get one instance there. We keep doing that. We do that again. We take 10 samples from this random variable, average them, plot them again."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We get one instance there. We keep doing that. We do that again. We take 10 samples from this random variable, average them, plot them again. You plot it again, and eventually you do this a gazillion times, in theory infinite number of times, and you're going to approach the sampling distribution of the sample mean. And n equal 10, it's not gonna be a perfect normal distribution, but it's gonna be close. It'd be perfect only if n was infinity."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We take 10 samples from this random variable, average them, plot them again. You plot it again, and eventually you do this a gazillion times, in theory infinite number of times, and you're going to approach the sampling distribution of the sample mean. And n equal 10, it's not gonna be a perfect normal distribution, but it's gonna be close. It'd be perfect only if n was infinity. But let's say we eventually, all of our samples, you know, we get a lot of averages that are there, that stacks up there, that stacks up there, and eventually we'll approach something that looks something like that. And we've seen from the last video that, one, if, let's say we were to do it again, and this time let's say that n is equal to 20. One, the distribution that we get is going to be more normal, and maybe in future videos we'll delve even deeper into things like kurtosis and skew."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It'd be perfect only if n was infinity. But let's say we eventually, all of our samples, you know, we get a lot of averages that are there, that stacks up there, that stacks up there, and eventually we'll approach something that looks something like that. And we've seen from the last video that, one, if, let's say we were to do it again, and this time let's say that n is equal to 20. One, the distribution that we get is going to be more normal, and maybe in future videos we'll delve even deeper into things like kurtosis and skew. But it's gonna be more normal, but even more important, or I guess even more obviously to us, and we saw that in the experiment, it's gonna have a lower standard deviation. So they're all gonna have the same mean. Let's say the mean here is, you know, I don't know, let's say the mean here is five."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "One, the distribution that we get is going to be more normal, and maybe in future videos we'll delve even deeper into things like kurtosis and skew. But it's gonna be more normal, but even more important, or I guess even more obviously to us, and we saw that in the experiment, it's gonna have a lower standard deviation. So they're all gonna have the same mean. Let's say the mean here is, you know, I don't know, let's say the mean here is five. Then the mean here is also gonna be five. The mean of our sampling distribution of the sample mean is gonna be five. And it doesn't matter what our n is."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say the mean here is, you know, I don't know, let's say the mean here is five. Then the mean here is also gonna be five. The mean of our sampling distribution of the sample mean is gonna be five. And it doesn't matter what our n is. If our n is 20, it's still gonna be five, but our standard deviation is gonna be less in either of these scenarios. And we saw that just by experimenting. It might look like this."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it doesn't matter what our n is. If our n is 20, it's still gonna be five, but our standard deviation is gonna be less in either of these scenarios. And we saw that just by experimenting. It might look like this. It's gonna be more normal, but it's gonna have a tighter standard deviation. So maybe it'll look like that. And if we did it with a even larger sample size, let me do that in a different color."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It might look like this. It's gonna be more normal, but it's gonna have a tighter standard deviation. So maybe it'll look like that. And if we did it with a even larger sample size, let me do that in a different color. If we did that with an even larger sample size, n is equal to 100, what we're gonna get is something that fits the normal distribution even better. We take 100 instances of this random variable, average them, plot it. 100 instances of this random variable, average them, plot it."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we did it with a even larger sample size, let me do that in a different color. If we did that with an even larger sample size, n is equal to 100, what we're gonna get is something that fits the normal distribution even better. We take 100 instances of this random variable, average them, plot it. 100 instances of this random variable, average them, plot it. We just keep doing that. If we keep doing that, what we're gonna have is something that's even more normal than either of these. So it's gonna be a much closer fit to a true normal distribution."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "100 instances of this random variable, average them, plot it. We just keep doing that. If we keep doing that, what we're gonna have is something that's even more normal than either of these. So it's gonna be a much closer fit to a true normal distribution. But even more obvious to the human eye, it's gonna be even tighter. So it's going to be a very low standard deviation. It's gonna look something like that."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's gonna be a much closer fit to a true normal distribution. But even more obvious to the human eye, it's gonna be even tighter. So it's going to be a very low standard deviation. It's gonna look something like that. And I'll show you that on the simulation app in the next, or probably later in this video. So two things happen. As you increase your sample size for every time you do the average, two things are happening."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's gonna look something like that. And I'll show you that on the simulation app in the next, or probably later in this video. So two things happen. As you increase your sample size for every time you do the average, two things are happening. You're becoming more normal, and your standard deviation is getting smaller. So the question might arise, is there a formula? So if I know the standard deviation, so this is my standard deviation of just my original probability density function."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "As you increase your sample size for every time you do the average, two things are happening. You're becoming more normal, and your standard deviation is getting smaller. So the question might arise, is there a formula? So if I know the standard deviation, so this is my standard deviation of just my original probability density function. This is the mean of my original probability density function. So if I know the standard deviation, and I know n, n is gonna change depending on how many samples I'm taking every time I do a sample mean. If I know my standard deviation, or maybe if I know my variance, the variance is just the standard deviation squared."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I know the standard deviation, so this is my standard deviation of just my original probability density function. This is the mean of my original probability density function. So if I know the standard deviation, and I know n, n is gonna change depending on how many samples I'm taking every time I do a sample mean. If I know my standard deviation, or maybe if I know my variance, the variance is just the standard deviation squared. If you don't remember that, you might wanna review those videos. But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm gonna take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are? And so this, sorry, the standard deviation of these distributions, and to make, so you don't get confused between that and that, and let me say the variance."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I know my standard deviation, or maybe if I know my variance, the variance is just the standard deviation squared. If you don't remember that, you might wanna review those videos. But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm gonna take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are? And so this, sorry, the standard deviation of these distributions, and to make, so you don't get confused between that and that, and let me say the variance. If you know the variance, you can figure out the standard deviation. One is just the square root of the other. So this is the variance of our original distribution."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so this, sorry, the standard deviation of these distributions, and to make, so you don't get confused between that and that, and let me say the variance. If you know the variance, you can figure out the standard deviation. One is just the square root of the other. So this is the variance of our original distribution. Now, to show that this is the variance of our sampling distribution of our sample mean, we'll write it right here. This is the variance of our mean, of our sample mean. Remember, the sample, our true mean is this, that the Greek letter mu is your true mean."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the variance of our original distribution. Now, to show that this is the variance of our sampling distribution of our sample mean, we'll write it right here. This is the variance of our mean, of our sample mean. Remember, the sample, our true mean is this, that the Greek letter mu is your true mean. This is equal to the mean, while an X with a line over it means sample mean. Sample mean. So here, what we're saying is this is the variance of our sample means."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, the sample, our true mean is this, that the Greek letter mu is your true mean. This is equal to the mean, while an X with a line over it means sample mean. Sample mean. So here, what we're saying is this is the variance of our sample means. Now, this is gonna be a true distribution. This isn't an estimate. This is, there's some, you know, if we magically knew this distribution, there's some true variance here."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here, what we're saying is this is the variance of our sample means. Now, this is gonna be a true distribution. This isn't an estimate. This is, there's some, you know, if we magically knew this distribution, there's some true variance here. And of course, the mean, so this has a mean. This right here, we can just get our notation right. This is the mean of the sampling distribution of the sampling mean."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is, there's some, you know, if we magically knew this distribution, there's some true variance here. And of course, the mean, so this has a mean. This right here, we can just get our notation right. This is the mean of the sampling distribution of the sampling mean. So this is the mean of our means. It just happens to be the same thing. This is the mean of our sample means."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the mean of the sampling distribution of the sampling mean. So this is the mean of our means. It just happens to be the same thing. This is the mean of our sample means. It's gonna be the same thing as that, especially if we do the trial over and over again. But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N? And it turns out there is."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the mean of our sample means. It's gonna be the same thing as that, especially if we do the trial over and over again. But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N? And it turns out there is. And I'm not gonna do a proof here. I really wanna give you the intuition of it. And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an N of two or an N of five."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it turns out there is. And I'm not gonna do a proof here. I really wanna give you the intuition of it. And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an N of two or an N of five. You're just very unlikely to be far away, right, if you took 100 trials as opposed to taking five. So I think you know that, in some way, it should be inversely proportional to N. The larger your N, the smaller standard deviation. And actually, it turns out it's about as simple as possible."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an N of two or an N of five. You're just very unlikely to be far away, right, if you took 100 trials as opposed to taking five. So I think you know that, in some way, it should be inversely proportional to N. The larger your N, the smaller standard deviation. And actually, it turns out it's about as simple as possible. It's one of those magical things about mathematics. And I'll prove it to you one day. I want to give you a working knowledge first."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, it turns out it's about as simple as possible. It's one of those magical things about mathematics. And I'll prove it to you one day. I want to give you a working knowledge first. In statistics, I'm always struggling whether I should be formal and giving you rigorous proofs, but I've kind of come to the conclusion that it's more important to get the working knowledge first in statistics. And then later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you. But I think experimental proofs are kind of all you need for right now, using those simulations to show that they're really true."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I want to give you a working knowledge first. In statistics, I'm always struggling whether I should be formal and giving you rigorous proofs, but I've kind of come to the conclusion that it's more important to get the working knowledge first in statistics. And then later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you. But I think experimental proofs are kind of all you need for right now, using those simulations to show that they're really true. So it turns out that the variance of your sampling distribution of your sample mean is equal to the variance of your original distribution, that guy right there, divided by N. That's all it is. So if this up here has a variance of, let's say this up here has a variance of 20, I'm just making that number up, then, and then let's say your N is 20, then the variance of your sampling distribution of your sample mean for N of 20, well, you're just gonna take that, the variance up here, your variance is 20, divided by your N, 20. So here, your variance is going to be 20 divided by 20, which is equal to one."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I think experimental proofs are kind of all you need for right now, using those simulations to show that they're really true. So it turns out that the variance of your sampling distribution of your sample mean is equal to the variance of your original distribution, that guy right there, divided by N. That's all it is. So if this up here has a variance of, let's say this up here has a variance of 20, I'm just making that number up, then, and then let's say your N is 20, then the variance of your sampling distribution of your sample mean for N of 20, well, you're just gonna take that, the variance up here, your variance is 20, divided by your N, 20. So here, your variance is going to be 20 divided by 20, which is equal to one. This is the variance of your original probability distribution, and this is your N. What's your standard deviation gonna be? What's gonna be the square root of that? Standard deviation is gonna be the square root of one, well, that's also going to be one."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here, your variance is going to be 20 divided by 20, which is equal to one. This is the variance of your original probability distribution, and this is your N. What's your standard deviation gonna be? What's gonna be the square root of that? Standard deviation is gonna be the square root of one, well, that's also going to be one. So we could also write this. We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the sampling distribution of the sample mean, it's often called the standard deviation of the mean, and it's also called, I'm gonna write this down, the standard error of the mean, standard error of the mean. All of these things that I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Standard deviation is gonna be the square root of one, well, that's also going to be one. So we could also write this. We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the sampling distribution of the sample mean, it's often called the standard deviation of the mean, and it's also called, I'm gonna write this down, the standard error of the mean, standard error of the mean. All of these things that I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean. That's why this is confusing, because you use the word mean and sample over and over again, and if it confuses you, let me know, I'll do another video or pause and repeat it, whatever. But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of your original, of your original function, of your original probability density function, which could be very non-normal, divided by the square root of n. I just took the square root of both sides of this equation. I personally, I like to remember this, that the variance is just inversely proportional to n, and then I like to go back to this, because this is very simple in my head."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of these things that I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean. That's why this is confusing, because you use the word mean and sample over and over again, and if it confuses you, let me know, I'll do another video or pause and repeat it, whatever. But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of your original, of your original function, of your original probability density function, which could be very non-normal, divided by the square root of n. I just took the square root of both sides of this equation. I personally, I like to remember this, that the variance is just inversely proportional to n, and then I like to go back to this, because this is very simple in my head. You just take the variance divided by n. Oh, and if I want the standard deviation, I just take the square roots of both sides, and I get this formula. So here, the standard deviation, when n is 20, the standard deviation of the sampling distribution of the sample mean is gonna be one. Here, when n is 100, well, our variance, our variance here, when n is equal to 100, so our variance of the sampling mean of the sample distribution, or our variance of the mean, of the sample mean, we could say, is going to be equal to 20."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I personally, I like to remember this, that the variance is just inversely proportional to n, and then I like to go back to this, because this is very simple in my head. You just take the variance divided by n. Oh, and if I want the standard deviation, I just take the square roots of both sides, and I get this formula. So here, the standard deviation, when n is 20, the standard deviation of the sampling distribution of the sample mean is gonna be one. Here, when n is 100, well, our variance, our variance here, when n is equal to 100, so our variance of the sampling mean of the sample distribution, or our variance of the mean, of the sample mean, we could say, is going to be equal to 20. This guy's variance divided by n. So it equals, n is 100, so it equals 1 5th. Now, this guy's standard deviation, or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is gonna be the square root of that. So one over the square root of five."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here, when n is 100, well, our variance, our variance here, when n is equal to 100, so our variance of the sampling mean of the sample distribution, or our variance of the mean, of the sample mean, we could say, is going to be equal to 20. This guy's variance divided by n. So it equals, n is 100, so it equals 1 5th. Now, this guy's standard deviation, or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is gonna be the square root of that. So one over the square root of five. And so, this guy's a little bit under 1 1 2 standard deviation while this guy had a standard deviation of one. So you see, it's definitely thinner. Now, I know what you're saying."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So one over the square root of five. And so, this guy's a little bit under 1 1 2 standard deviation while this guy had a standard deviation of one. So you see, it's definitely thinner. Now, I know what you're saying. Well, Sal, you just gave a formula. I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, I know what you're saying. Well, Sal, you just gave a formula. I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation. So I'll, just for fun, let me make a, I'll just mess with this distribution a little bit. So that's my new distribution. And let me take an n of, let me take two things that's easy to take the square root of, because if we're looking at standard deviation."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's see if we can prove it to ourselves using the simulation. So I'll, just for fun, let me make a, I'll just mess with this distribution a little bit. So that's my new distribution. And let me take an n of, let me take two things that's easy to take the square root of, because if we're looking at standard deviation. So let's take, we'll take an n of 16, and an n of 25. And let's, well, I'll do a, let's do 10,000 trials. So in this case, every one of the trials, we're gonna take 16 samples from here, average them, plot it here, and then do a frequency plot."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let me take an n of, let me take two things that's easy to take the square root of, because if we're looking at standard deviation. So let's take, we'll take an n of 16, and an n of 25. And let's, well, I'll do a, let's do 10,000 trials. So in this case, every one of the trials, we're gonna take 16 samples from here, average them, plot it here, and then do a frequency plot. Here, we're gonna do 25 at a time, and then average them. I'll do it once animated, just to remember. So I'm taking 16 samples, plot it there."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in this case, every one of the trials, we're gonna take 16 samples from here, average them, plot it here, and then do a frequency plot. Here, we're gonna do 25 at a time, and then average them. I'll do it once animated, just to remember. So I'm taking 16 samples, plot it there. I take 16 samples, as described by this probability density function, or 25 now, plot it down here. Now, if I do that 10,000 times, what do I get? What do I get?"}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm taking 16 samples, plot it there. I take 16 samples, as described by this probability density function, or 25 now, plot it down here. Now, if I do that 10,000 times, what do I get? What do I get? All right, so here, you know, just visually, you can tell just when n was larger, the standard deviation here is smaller. This is more squeezed together. But actually, let's write, let's write this stuff down."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What do I get? All right, so here, you know, just visually, you can tell just when n was larger, the standard deviation here is smaller. This is more squeezed together. But actually, let's write, let's write this stuff down. Let's see if I can remember it here. Here, n is, so in this random distribution I made, my standard deviation was 9.3. I'm gonna remember these."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But actually, let's write, let's write this stuff down. Let's see if I can remember it here. Here, n is, so in this random distribution I made, my standard deviation was 9.3. I'm gonna remember these. Our standard deviation for the original thing was 9.3. And so, standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it, if it conforms to our formula."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm gonna remember these. Our standard deviation for the original thing was 9.3. And so, standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it, if it conforms to our formula. So I'm gonna take this offscreen for a second, and I'm gonna go back and do some mathematics. So I have this on my other screen, so I can remember those numbers. So in the trial we just did, my wacky distribution had a standard deviation of 9.3."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see if it, if it conforms to our formula. So I'm gonna take this offscreen for a second, and I'm gonna go back and do some mathematics. So I have this on my other screen, so I can remember those numbers. So in the trial we just did, my wacky distribution had a standard deviation of 9.3. When n is equal to, let me do this in another color. When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things, we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. We experimentally determined it to be 2.33."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in the trial we just did, my wacky distribution had a standard deviation of 9.3. When n is equal to, let me do this in another color. When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things, we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. We experimentally determined it to be 2.33. And then when n is equal to 25, when n is equal to 25, we got the standard error of the mean being equal to 1.87. Let's see if it conforms to our formulas. So we know that the variance, or we could almost say the variance of the mean, or the standard error, well, you know, the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both sides, and then you get standard error of the mean is equal to standard deviation of your original distribution divided by the square root of n. So let's see if this works out for these two things."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We experimentally determined it to be 2.33. And then when n is equal to 25, when n is equal to 25, we got the standard error of the mean being equal to 1.87. Let's see if it conforms to our formulas. So we know that the variance, or we could almost say the variance of the mean, or the standard error, well, you know, the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both sides, and then you get standard error of the mean is equal to standard deviation of your original distribution divided by the square root of n. So let's see if this works out for these two things. So if I were to take 9.3, so let me do this case. So 9.3 divided by the square root of 16, right? N is 16."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we know that the variance, or we could almost say the variance of the mean, or the standard error, well, you know, the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both sides, and then you get standard error of the mean is equal to standard deviation of your original distribution divided by the square root of n. So let's see if this works out for these two things. So if I were to take 9.3, so let me do this case. So 9.3 divided by the square root of 16, right? N is 16. So divided by the square root of 16, which is four, what do I get? So 9.3 divided by four. Let me get a little calculator out here."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "N is 16. So divided by the square root of 16, which is four, what do I get? So 9.3 divided by four. Let me get a little calculator out here. Let's see, we have, let me clear it out. We wanted to divide 9.3 divided by four. 9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me get a little calculator out here. Let's see, we have, let me clear it out. We wanted to divide 9.3 divided by four. 9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32. 2.32. So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32. 2.32. So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials. Maybe right after this, I'll see what happens if we did 20,000 or 30,000 trials where we take samples of 16 and average them. Now let's look at this. Here, we would take 9.3."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This was after 10,000 trials. Maybe right after this, I'll see what happens if we did 20,000 or 30,000 trials where we take samples of 16 and average them. Now let's look at this. Here, we would take 9.3. So let me draw a little line here. Maybe scroll over, that might be better. So we take our standard deviation of our original distribution."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here, we would take 9.3. So let me draw a little line here. Maybe scroll over, that might be better. So we take our standard deviation of our original distribution. So just that formula that we derived right here would tell us that our standard error should be equal to the standard deviation of our original distribution, 9.3, divided by the square root of n, divided by the square root of 25, right? Four was just the square root of 16. So this is equal to 9.3 divided by five, and let's see if it's 1.87."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we take our standard deviation of our original distribution. So just that formula that we derived right here would tell us that our standard error should be equal to the standard deviation of our original distribution, 9.3, divided by the square root of n, divided by the square root of 25, right? Four was just the square root of 16. So this is equal to 9.3 divided by five, and let's see if it's 1.87. So let me get my calculator back. So if I take 9.3 divided by five, what do I get? 1.86, which is very close to 1.87."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to 9.3 divided by five, and let's see if it's 1.87. So let me get my calculator back. So if I take 9.3 divided by five, what do I get? 1.86, which is very close to 1.87. So we got, we got in this case, 1.86. 1.86. So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "1.86, which is very close to 1.87. So we got, we got in this case, 1.86. 1.86. So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect. Let's do another 10,000. So you got another 10,000 trials. Well, we're still in the ballpark."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect. Let's do another 10,000. So you got another 10,000 trials. Well, we're still in the ballpark. We're not gonna, maybe I can't hope to get the exact number, you know, rounded or whatever. But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampling distribution sampling mean, is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided by your sample size. By the number of samples you take for when, for every basket that you average, I guess is the best way to think about it."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we're still in the ballpark. We're not gonna, maybe I can't hope to get the exact number, you know, rounded or whatever. But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampling distribution sampling mean, is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided by your sample size. By the number of samples you take for when, for every basket that you average, I guess is the best way to think about it. And you know, sometimes this can get confusing because you are taking samples of averages based on samples. So when someone says sample size, you're like, is sample size the number of times I took averages or the number of things I'm taking averages of each time? And you know, it doesn't hurt to clarify that."}, {"video_title": "Standard error of the mean   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "By the number of samples you take for when, for every basket that you average, I guess is the best way to think about it. And you know, sometimes this can get confusing because you are taking samples of averages based on samples. So when someone says sample size, you're like, is sample size the number of times I took averages or the number of things I'm taking averages of each time? And you know, it doesn't hurt to clarify that. Normally when they talk about sample size, they're talking about n. And at least in my head, when I think of the trials as you take a sample size of 16, you average it, that's one trial and you plot it. Then you do it again and you do another trial and you do it over and over again. But anyway, hopefully this makes everything clear and then you now also understand how to get to the standard error of the mean."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it reads, Harry Potter is at Ollivander's Wand Shop. As we all know, the wand must choose the wizard. So Harry cannot make the choice himself. He interprets the wand selection as a random process so he can compare the probabilities of different outcomes. The wood types available are holly, elm, maple, and wenge, wenge, wenge? The core materials on offer are phoenix feather, unicorn hair, dragon scale, raven feather, and thestral tail. All right."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "He interprets the wand selection as a random process so he can compare the probabilities of different outcomes. The wood types available are holly, elm, maple, and wenge, wenge, wenge? The core materials on offer are phoenix feather, unicorn hair, dragon scale, raven feather, and thestral tail. All right. Based on the sample space of possible outcomes listed below, what is more likely? And so we see here we have four different types of woods for the wand, and then each of those could be combined with five different types of core. Phoenix feather, unicorn hair, dragon scale, raven feather, and thestral tail."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right. Based on the sample space of possible outcomes listed below, what is more likely? And so we see here we have four different types of woods for the wand, and then each of those could be combined with five different types of core. Phoenix feather, unicorn hair, dragon scale, raven feather, and thestral tail. And so that gives us four different woods, and each of those can be combined for five different cores, 20 possible outcomes. And they don't say it here, but the way they're talking, I guess we can, I'm gonna go with the assumption that they're equally likely outcomes, although it would have been nice if they said that these are all equally likely, but these are the 20 outcomes. And so which of these are more likely?"}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Phoenix feather, unicorn hair, dragon scale, raven feather, and thestral tail. And so that gives us four different woods, and each of those can be combined for five different cores, 20 possible outcomes. And they don't say it here, but the way they're talking, I guess we can, I'm gonna go with the assumption that they're equally likely outcomes, although it would have been nice if they said that these are all equally likely, but these are the 20 outcomes. And so which of these are more likely? The wand that selects Harry will be made of holly or unicorn hair. So how many of those outcomes involve this? So holly are these five outcomes, and if you said holly or unicorn hair, it's gonna be these five outcomes plus, well this one involves unicorn hair, but we've already included this one."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so which of these are more likely? The wand that selects Harry will be made of holly or unicorn hair. So how many of those outcomes involve this? So holly are these five outcomes, and if you said holly or unicorn hair, it's gonna be these five outcomes plus, well this one involves unicorn hair, but we've already included this one. But the other ones that's not included for the holly that involve unicorn hair are the elm unicorn, the maple unicorn, and the wenge unicorn. So it's these five plus these three right over here. So eight of these 20 outcomes."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So holly are these five outcomes, and if you said holly or unicorn hair, it's gonna be these five outcomes plus, well this one involves unicorn hair, but we've already included this one. But the other ones that's not included for the holly that involve unicorn hair are the elm unicorn, the maple unicorn, and the wenge unicorn. So it's these five plus these three right over here. So eight of these 20 outcomes. And if these are all equally likely outcomes, that means there's an 8 20th probability of a wand that will be made of holly or unicorn hair. So this is 8 20th, so that's the same thing as 4 10th, so 40% chance. Now the wand that selects Harry will be made of holly and unicorn hair."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So eight of these 20 outcomes. And if these are all equally likely outcomes, that means there's an 8 20th probability of a wand that will be made of holly or unicorn hair. So this is 8 20th, so that's the same thing as 4 10th, so 40% chance. Now the wand that selects Harry will be made of holly and unicorn hair. Well holly and unicorn hair, that's only one out of the 20 outcomes. So this of course is going to be a higher probability. It actually includes this outcome, and then seven other outcomes."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now the wand that selects Harry will be made of holly and unicorn hair. Well holly and unicorn hair, that's only one out of the 20 outcomes. So this of course is going to be a higher probability. It actually includes this outcome, and then seven other outcomes. So this is, the first choice includes the outcome for the second choice, plus seven other outcomes. So this is definitely going to be a higher probability. Let's do a couple more of these, or at least one more of these."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "It actually includes this outcome, and then seven other outcomes. So this is, the first choice includes the outcome for the second choice, plus seven other outcomes. So this is definitely going to be a higher probability. Let's do a couple more of these, or at least one more of these. You and a friend are playing fire, water, sponge. I've never played that game. In this game, each of the two players chooses fire, water, or sponge."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do a couple more of these, or at least one more of these. You and a friend are playing fire, water, sponge. I've never played that game. In this game, each of the two players chooses fire, water, or sponge. Both players reveal their choice at the same time, and the winner is determined based on the choices. I guess this is like rock, paper, scissors. Fire beats sponge by burning it."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "In this game, each of the two players chooses fire, water, or sponge. Both players reveal their choice at the same time, and the winner is determined based on the choices. I guess this is like rock, paper, scissors. Fire beats sponge by burning it. Sponge beats water by soaking it up. And water beats fire by putting it out. All right, well, it kind of makes sense."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Fire beats sponge by burning it. Sponge beats water by soaking it up. And water beats fire by putting it out. All right, well, it kind of makes sense. If both players choose the same object, it is a tie. All the possible outcomes of the game are listed below. If we take outcomes one, three, four, five, seven, and eight as a subset of the sample space, which of the following statements, which of the statements below, describe this subset?"}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, well, it kind of makes sense. If both players choose the same object, it is a tie. All the possible outcomes of the game are listed below. If we take outcomes one, three, four, five, seven, and eight as a subset of the sample space, which of the following statements, which of the statements below, describe this subset? So let's look at the outcomes that they have over here. Well, it makes sense that there are nine possible outcomes, because for each of the three choices I can make, there's going to be three choices that my friend can make. So three times three is nine."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we take outcomes one, three, four, five, seven, and eight as a subset of the sample space, which of the following statements, which of the statements below, describe this subset? So let's look at the outcomes that they have over here. Well, it makes sense that there are nine possible outcomes, because for each of the three choices I can make, there's going to be three choices that my friend can make. So three times three is nine. Let's see, they've highlighted these red outcomes, outcome one, three, four, five, seven, and eight. So let's see what's common about them. Outcome one, fire, I get fire, friend gets water."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So three times three is nine. Let's see, they've highlighted these red outcomes, outcome one, three, four, five, seven, and eight. So let's see what's common about them. Outcome one, fire, I get fire, friend gets water. OK, so let's see, my friend would win. Outcome three, I pick fire, my friend does sponge. So actually, I would win that one."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Outcome one, fire, I get fire, friend gets water. OK, so let's see, my friend would win. Outcome three, I pick fire, my friend does sponge. So actually, I would win that one. And then outcome four, water, fire. And then outcome five, water, sponge. Sponge, huh, these are all, I don't see a pattern just yet."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So actually, I would win that one. And then outcome four, water, fire. And then outcome five, water, sponge. Sponge, huh, these are all, I don't see a pattern just yet. Let's look at the choices. The subset consists of all outcomes where your friend does not win. All outcomes where your friend does not win."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sponge, huh, these are all, I don't see a pattern just yet. Let's look at the choices. The subset consists of all outcomes where your friend does not win. All outcomes where your friend does not win. Well, that's not true, because look, outcome one, my friend wins. Water puts out fire. So we're not going to select this first choice."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All outcomes where your friend does not win. Well, that's not true, because look, outcome one, my friend wins. Water puts out fire. So we're not going to select this first choice. So let's see, the subset consists of all the outcomes where your friend wins or there's a tie. So let's see, where the friend wins or there's a tie. Well, outcome three, this is an outcome where I would win, or you or whoever your is, whoever they're talking about."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're not going to select this first choice. So let's see, the subset consists of all the outcomes where your friend wins or there's a tie. So let's see, where the friend wins or there's a tie. Well, outcome three, this is an outcome where I would win, or you or whoever your is, whoever they're talking about. This is one where the friend doesn't win, because fire burns sponge. So I'm not going to select that one either. Choice three, the subset consists of all of the outcome where you win or there is a tie."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, outcome three, this is an outcome where I would win, or you or whoever your is, whoever they're talking about. This is one where the friend doesn't win, because fire burns sponge. So I'm not going to select that one either. Choice three, the subset consists of all of the outcome where you win or there is a tie. Well, we just said outcome one, I don't win that. My friend wins that. Water puts out the fire."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Choice three, the subset consists of all of the outcome where you win or there is a tie. Well, we just said outcome one, I don't win that. My friend wins that. Water puts out the fire. Now let's look at the last choice. The subset consists of all of the outcomes where there is not a tie. All right, so this is interesting."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Water puts out the fire. Now let's look at the last choice. The subset consists of all of the outcomes where there is not a tie. All right, so this is interesting. Because look, outcome two, there is a tie. Outcome six, there is a tie. Outcome nine, there is a tie."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, so this is interesting. Because look, outcome two, there is a tie. Outcome six, there is a tie. Outcome nine, there is a tie. And there's actually only three scenarios where there's a tie. Either it's fire, fire, water, water, or sponge, sponge. And those are the ones that are not selected."}, {"video_title": "Describing subsets of sample spaces exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Outcome nine, there is a tie. And there's actually only three scenarios where there's a tie. Either it's fire, fire, water, water, or sponge, sponge. And those are the ones that are not selected. So all of these, someone is going to win. Outcome one, three, four, five, seven, or eight. So definitely, definitely go with that one."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that the set B, let me do this in a different color. Let's say that the set B is composed of 1, 7, and 18. And let's say that the set C is composed of 18, 7, 1, and 19. Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset. And this is a notation right over here. This is a subset."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset. And this is a notation right over here. This is a subset. B is a subset of A. So let me write that down. B is subset of A."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is a subset. B is a subset of A. So let me write that down. B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further. And we could say that B is a strict, or sometimes said, a proper subset of A. And the way you do that is, essentially, you can almost imagine that this is kind of a less than or equal sign."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further. And we could say that B is a strict, or sometimes said, a proper subset of A. And the way you do that is, essentially, you can almost imagine that this is kind of a less than or equal sign. And then you kind of cross out the equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member of A, but everything that's in A is not a member of B. So let me write this."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the way you do that is, essentially, you can almost imagine that this is kind of a less than or equal sign. And then you kind of cross out the equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member of A, but everything that's in A is not a member of B. So let me write this. This is B. B is a strict or proper subset. So for example, we can write that A is a subset of A."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write this. This is B. B is a strict or proper subset. So for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see. C contains a 1. It contains a 7. It contains an 18."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's see. C contains a 1. It contains a 7. It contains an 18. So every member of B is indeed a member of C. So this right over here is true. Now, can we write that C is a subset of A? Let's see."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "It contains an 18. So every member of B is indeed a member of C. So this right over here is true. Now, can we write that C is a subset of A? Let's see. Every element of C needs to be in A. So A has an 18. It has a 7."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see. Every element of C needs to be in A. So A has an 18. It has a 7. It has a 1. But it does not have a 19. So once again, this right over here is false."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "It has a 7. It has a 1. But it does not have a 19. So once again, this right over here is false. Now, we could write B is a subset of C, or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this, and then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B. And this is just another way of saying that B is a subset of A."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once again, this right over here is false. Now, we could write B is a subset of C, or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this, and then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B. And this is just another way of saying that B is a subset of A. But the way you could think about this is A contains every element that is in B. And it might contain more. It might contain exactly every element."}, {"video_title": "Subset, strict subset, and superset   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is just another way of saying that B is a subset of A. But the way you could think about this is A contains every element that is in B. And it might contain more. It might contain exactly every element. Because you can kind of view this as you kind of have the equal symbol there if you were to view this as greater than or equal. They're not quite exactly the same thing. But we know already that we could also write A is a strict superset of B, which means that A contains everything B has and then some."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the last video, we came up with a 95% confidence interval for the mean weight loss between the low-fat group and the control group. In this video, I actually want to do a hypothesis test to see really the test if this data makes us believe that the low-fat diet actually does anything at all. And to do that, let's set up our null and alternative hypotheses. So our null hypothesis should be that, hey, this low-fat diet does nothing. And if the low-fat diet does nothing, that means that the mean, the population mean on our low-fat diet minus the population mean on our control should be equal to 0. And this is a completely equivalent statement to saying that the mean of the sampling distribution of our low-fat diet minus the mean of the sampling distribution of our control should be equal to 0. And that's because we've seen this multiple times."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our null hypothesis should be that, hey, this low-fat diet does nothing. And if the low-fat diet does nothing, that means that the mean, the population mean on our low-fat diet minus the population mean on our control should be equal to 0. And this is a completely equivalent statement to saying that the mean of the sampling distribution of our low-fat diet minus the mean of the sampling distribution of our control should be equal to 0. And that's because we've seen this multiple times. The mean of your sampling distribution is going to be the same thing as your population mean. So this is the same thing as that. That is the same thing as that."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's because we've seen this multiple times. The mean of your sampling distribution is going to be the same thing as your population mean. So this is the same thing as that. That is the same thing as that. Or another way of saying it is if we think about the distribution, if we think about the mean of the distribution of the difference of the sample means, and we focused on this in the last video, that that should be equal to 0. Because this thing right over here is the same thing as that right over there. So that is our null hypothesis."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is the same thing as that. Or another way of saying it is if we think about the distribution, if we think about the mean of the distribution of the difference of the sample means, and we focused on this in the last video, that that should be equal to 0. Because this thing right over here is the same thing as that right over there. So that is our null hypothesis. And our alternative hypothesis is just going to be, our alternative hypothesis, I'll write it over here, our alternative hypothesis is just that it actually does do something, that our mean, and actually let's say that it actually has an improvement. So that would mean that we have more weight loss. So if we have the mean of group 1, the population mean of group 1 minus the population mean of group 2 should be greater than 0."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that is our null hypothesis. And our alternative hypothesis is just going to be, our alternative hypothesis, I'll write it over here, our alternative hypothesis is just that it actually does do something, that our mean, and actually let's say that it actually has an improvement. So that would mean that we have more weight loss. So if we have the mean of group 1, the population mean of group 1 minus the population mean of group 2 should be greater than 0. So this is going to be a one-tailed distribution. Or another way we could view it is that the mean of the difference of the distributions, x1 minus x2, is going to be greater than 0. These are equivalent statements."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if we have the mean of group 1, the population mean of group 1 minus the population mean of group 2 should be greater than 0. So this is going to be a one-tailed distribution. Or another way we could view it is that the mean of the difference of the distributions, x1 minus x2, is going to be greater than 0. These are equivalent statements. Because we know that this is the same thing as this, which is the same thing as this, which is what I wrote right over here. Now, to do any type of hypothesis test, we have to decide on a level of significance. We have to say, what we're going to do is, we're going to assume that our null hypothesis is correct."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are equivalent statements. Because we know that this is the same thing as this, which is the same thing as this, which is what I wrote right over here. Now, to do any type of hypothesis test, we have to decide on a level of significance. We have to say, what we're going to do is, we're going to assume that our null hypothesis is correct. And then given, with that assumption that the null hypothesis is correct, we're going to see what is the probability of getting this sample data right over here. And if that probability is below some threshold, we will reject the null hypothesis in favor of the alternative hypothesis. Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have to say, what we're going to do is, we're going to assume that our null hypothesis is correct. And then given, with that assumption that the null hypothesis is correct, we're going to see what is the probability of getting this sample data right over here. And if that probability is below some threshold, we will reject the null hypothesis in favor of the alternative hypothesis. Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha. And here, we're going to decide for a significance level of 95%. Or another way to think about it, we want there to be, assuming that the null hypothesis is correct, we want there to be no more than a 5% chance of getting this result here, or no more than a 5% chance of incorrectly rejecting the null hypothesis when it is actually true, or that would be a type 1 error. So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha. And here, we're going to decide for a significance level of 95%. Or another way to think about it, we want there to be, assuming that the null hypothesis is correct, we want there to be no more than a 5% chance of getting this result here, or no more than a 5% chance of incorrectly rejecting the null hypothesis when it is actually true, or that would be a type 1 error. So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis. And go, less than a 5% probability, given the null hypothesis is true, then we're going to reject the null hypothesis in favor of the alternative. So let's think about this. So we have the null hypothesis."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis. And go, less than a 5% probability, given the null hypothesis is true, then we're going to reject the null hypothesis in favor of the alternative. So let's think about this. So we have the null hypothesis. Let me draw a distribution over here. The null hypothesis says that the mean of our differences, so the mean of the differences of the sampling distributions should be equal to 0. Now, in that situation, what is going to be our critical region here?"}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have the null hypothesis. Let me draw a distribution over here. The null hypothesis says that the mean of our differences, so the mean of the differences of the sampling distributions should be equal to 0. Now, in that situation, what is going to be our critical region here? Well, we need a result, so we need some critical z-value here, some critical z-score, or some critical, I should actually say some critical value here, because this isn't a normalized standard. This isn't a normalized normal distribution. But so there's some critical value here."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, in that situation, what is going to be our critical region here? Well, we need a result, so we need some critical z-value here, some critical z-score, or some critical, I should actually say some critical value here, because this isn't a normalized standard. This isn't a normalized normal distribution. But so there's some critical value here. The hardest thing in statistics is getting the wording right. There's some critical value here that the probability of getting a sample from this distribution above that value is only 5%. So we just need to figure out what this critical value is."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "But so there's some critical value here. The hardest thing in statistics is getting the wording right. There's some critical value here that the probability of getting a sample from this distribution above that value is only 5%. So we just need to figure out what this critical value is. And if our value is larger than that critical value, then we can reject the null hypothesis, because that means the probability of getting this is less than 5%. We could reject the null hypothesis and go with the alternative hypothesis. And to figure out that critical value, and remember, once again, we can use z-scores, and we can assume this is a normal distribution, because our sample size is large for either of those samples."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we just need to figure out what this critical value is. And if our value is larger than that critical value, then we can reject the null hypothesis, because that means the probability of getting this is less than 5%. We could reject the null hypothesis and go with the alternative hypothesis. And to figure out that critical value, and remember, once again, we can use z-scores, and we can assume this is a normal distribution, because our sample size is large for either of those samples. We have a sample size of 100. And to figure that out, we just have to figure out, the first step is to say, well, if we just look at a normalized normal distribution like this, what is your critical z-value We're getting a result above that z-value only has a 5% chance, and to do that, so this is actually cumulative, so this whole area right over here is going to be 95% chance. We can just look at the z-table."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to figure out that critical value, and remember, once again, we can use z-scores, and we can assume this is a normal distribution, because our sample size is large for either of those samples. We have a sample size of 100. And to figure that out, we just have to figure out, the first step is to say, well, if we just look at a normalized normal distribution like this, what is your critical z-value We're getting a result above that z-value only has a 5% chance, and to do that, so this is actually cumulative, so this whole area right over here is going to be 95% chance. We can just look at the z-table. We're going to look for 95%, because this is a one-tailed case, so let's look for 95%. This is the closest thing. We want to err on the side of being a little bit maybe to the right of this."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "We can just look at the z-table. We're going to look for 95%, because this is a one-tailed case, so let's look for 95%. This is the closest thing. We want to err on the side of being a little bit maybe to the right of this. So let's say 95.05 is pretty good. So that's 1.65. So this critical z-value is equal to 1.65."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to err on the side of being a little bit maybe to the right of this. So let's say 95.05 is pretty good. So that's 1.65. So this critical z-value is equal to 1.65. Or another way to view it is, this distance right here is going to be 1.65 standard deviations. I know my writing is really small. I'm just saying the standard deviation of that distribution."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this critical z-value is equal to 1.65. Or another way to view it is, this distance right here is going to be 1.65 standard deviations. I know my writing is really small. I'm just saying the standard deviation of that distribution. So what is the standard deviation of that distribution? We actually calculated it in the last video, and I'll recalculate it here. The standard deviation of our distribution, of the difference of the mean, the distribution of the difference of the sample means, is going to be equal to the square root of the variance of our first population."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just saying the standard deviation of that distribution. So what is the standard deviation of that distribution? We actually calculated it in the last video, and I'll recalculate it here. The standard deviation of our distribution, of the difference of the mean, the distribution of the difference of the sample means, is going to be equal to the square root of the variance of our first population. Now the variance of our first population, we don't know it, but we can estimate it with our sample standard deviation. If you take your sample standard deviation, 4.67, and you square it, you get your sample variance. So this is the variance, this is our best estimate, of the variance of the population."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "The standard deviation of our distribution, of the difference of the mean, the distribution of the difference of the sample means, is going to be equal to the square root of the variance of our first population. Now the variance of our first population, we don't know it, but we can estimate it with our sample standard deviation. If you take your sample standard deviation, 4.67, and you square it, you get your sample variance. So this is the variance, this is our best estimate, of the variance of the population. And we want to divide that by the sample size. And then plus our best estimate of the variance of the population of group 2, which is 4.04 squared, the sample standard deviation of group 2 squared, that gives us the variance, divided by 100. And this gives us, I did it before in the last, maybe it's still sitting on my calculator."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the variance, this is our best estimate, of the variance of the population. And we want to divide that by the sample size. And then plus our best estimate of the variance of the population of group 2, which is 4.04 squared, the sample standard deviation of group 2 squared, that gives us the variance, divided by 100. And this gives us, I did it before in the last, maybe it's still sitting on my calculator. So, yep, it's still sitting on the calculator. It's this quantity right up here, 4.67 squared divided by 100, plus 4.04 squared divided by 100, so it's 0.617. So this right here is going to be, this right here is 0.617."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this gives us, I did it before in the last, maybe it's still sitting on my calculator. So, yep, it's still sitting on the calculator. It's this quantity right up here, 4.67 squared divided by 100, plus 4.04 squared divided by 100, so it's 0.617. So this right here is going to be, this right here is 0.617. So this distance right here, this distance right here is going to be 1.65 times 0.617. So let's figure out what that is. So let's take 0.617, and we can even add, well, I'll just leave it there, times 1.65."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right here is going to be, this right here is 0.617. So this distance right here, this distance right here is going to be 1.65 times 0.617. So let's figure out what that is. So let's take 0.617, and we can even add, well, I'll just leave it there, times 1.65. So it's 1.02. I'll go with 1.02. This distance right here is 1.02."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's take 0.617, and we can even add, well, I'll just leave it there, times 1.65. So it's 1.02. I'll go with 1.02. This distance right here is 1.02. So what this tells us is that there is only a 5% chance that the difference, if we assume that the diet actually does nothing, there's only a 5% chance of having a difference between these two means, the means of these two samples, to have a difference of more than 1.02. There's only a 5% chance of that. Well, the mean that we actually got is 1.91."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "This distance right here is 1.02. So what this tells us is that there is only a 5% chance that the difference, if we assume that the diet actually does nothing, there's only a 5% chance of having a difference between these two means, the means of these two samples, to have a difference of more than 1.02. There's only a 5% chance of that. Well, the mean that we actually got is 1.91. The mean that we actually got is 1.91. So that's sitting out here someplace. So it definitely falls in this critical region."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the mean that we actually got is 1.91. The mean that we actually got is 1.91. So that's sitting out here someplace. So it definitely falls in this critical region. The probability of getting this, assuming that the null hypothesis is correct, is less than 5%. So that is, it's smaller probability than our significance level. Actually, let me be very clear."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it definitely falls in this critical region. The probability of getting this, assuming that the null hypothesis is correct, is less than 5%. So that is, it's smaller probability than our significance level. Actually, let me be very clear. The significance level, this alpha right here, the significance level, I don't want to give you the wrong, the significance level needs to be 5%, not the 95%. I think I might have said it here, but I wrote down the wrong number there. I subtracted it from 1 by accident, probably in my head."}, {"video_title": "Hypothesis test for difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me be very clear. The significance level, this alpha right here, the significance level, I don't want to give you the wrong, the significance level needs to be 5%, not the 95%. I think I might have said it here, but I wrote down the wrong number there. I subtracted it from 1 by accident, probably in my head. But anyway, the significance level is 5%. The probability, given that the null hypothesis is true, the probability of getting the result that we got, the probability of getting that difference, is less than our significance level. It is less than 5%."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Mom is always grouchy when it rains, Adam's brother said to him. So Adam decided to figure out if the statement was actually true. For the next year, he charted every time it rained and every time his mom was grouchy. What he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. Some of his data are shown in the table below. Fill in the missing values from the frequency table."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "What he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. Some of his data are shown in the table below. Fill in the missing values from the frequency table. And let's see, we have this raining days, not raining days, and the total days that he kept data for. And then he tabulated on, say, the raining day, whether his mom was grouchy or not grouchy, and on a not raining day, whether his mom was grouchy or not grouchy. And there was a total of 35 days it rained, 330 days that it didn't rain."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Fill in the missing values from the frequency table. And let's see, we have this raining days, not raining days, and the total days that he kept data for. And then he tabulated on, say, the raining day, whether his mom was grouchy or not grouchy, and on a not raining day, whether his mom was grouchy or not grouchy. And there was a total of 35 days it rained, 330 days that it didn't rain. And then 73 times his mom was grouchy and 292 times his mom was not grouchy. So the first thing that we said, well, how do we figure this out? We have these four boxes here."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And there was a total of 35 days it rained, 330 days that it didn't rain. And then 73 times his mom was grouchy and 292 times his mom was not grouchy. So the first thing that we said, well, how do we figure this out? We have these four boxes here. It's not clear that we have enough information to fill it out just with this table. But we have to remember what they told us. They told us that his mom being grouchy and it raining were entirely independent events."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have these four boxes here. It's not clear that we have enough information to fill it out just with this table. But we have to remember what they told us. They told us that his mom being grouchy and it raining were entirely independent events. Another way of saying that is the probability of his, let me do this in a color that you're more likely to see. Another way of saying that, so independent events, that means that the probability, my pen is acting up a little bit, probability that mom is grouchy, so let me write that. Mom, my pen is really, mom is grouchy given it is raining, it shouldn't really matter whether it's raining."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "They told us that his mom being grouchy and it raining were entirely independent events. Another way of saying that is the probability of his, let me do this in a color that you're more likely to see. Another way of saying that, so independent events, that means that the probability, my pen is acting up a little bit, probability that mom is grouchy, so let me write that. Mom, my pen is really, mom is grouchy given it is raining, it shouldn't really matter whether it's raining. It should just be the same thing as the probability of mom being grouchy in general. So what does that tell us? Well, we can figure out the probability that mom is grouchy in general."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Mom, my pen is really, mom is grouchy given it is raining, it shouldn't really matter whether it's raining. It should just be the same thing as the probability of mom being grouchy in general. So what does that tell us? Well, we can figure out the probability that mom is grouchy in general. She's grouchy 73 out of 365 days. So the probability that mom is grouchy in general is going to be 73 divided by 365. Or at least just based on the data we have, that's the best estimate that mom is grouchy, of the probability that mom is grouchy."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we can figure out the probability that mom is grouchy in general. She's grouchy 73 out of 365 days. So the probability that mom is grouchy in general is going to be 73 divided by 365. Or at least just based on the data we have, that's the best estimate that mom is grouchy, of the probability that mom is grouchy. It's the percentage of days that she's been grouchy. So that is.2. So based on the data, the best estimate of the probability of mom being grouchy is.2 or 20%."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or at least just based on the data we have, that's the best estimate that mom is grouchy, of the probability that mom is grouchy. It's the percentage of days that she's been grouchy. So that is.2. So based on the data, the best estimate of the probability of mom being grouchy is.2 or 20%. And so we should have the probability of mom being grouchy given that it's raining, should be 20% as well. So this number, so given that it's raining, we should also have 20% of the time mom is grouchy because these are independent events. It shouldn't matter whether it's raining or not."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So based on the data, the best estimate of the probability of mom being grouchy is.2 or 20%. And so we should have the probability of mom being grouchy given that it's raining, should be 20% as well. So this number, so given that it's raining, we should also have 20% of the time mom is grouchy because these are independent events. It shouldn't matter whether it's raining or not. This should be 20%, this should be, she should be grouchy 20% of the time that it's raining and she should be grouchy 20% of the time that it's not raining. That would be consistent with the data saying that these were entirely independent events. So what is 20% of 35?"}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "It shouldn't matter whether it's raining or not. This should be 20%, this should be, she should be grouchy 20% of the time that it's raining and she should be grouchy 20% of the time that it's not raining. That would be consistent with the data saying that these were entirely independent events. So what is 20% of 35? Well, 20% is 1 5th. 1 5th of 35 is 7. And once again, all I did is I said 20% of 35 is 7."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is 20% of 35? Well, 20% is 1 5th. 1 5th of 35 is 7. And once again, all I did is I said 20% of 35 is 7. And if that's 7, then 35 minus 7, that's gonna be 28 right over there. And then if this is 7, then 73 minus 7 is going to be 66. And 333, I guess there's a couple of ways we could do it."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And once again, all I did is I said 20% of 35 is 7. And if that's 7, then 35 minus 7, that's gonna be 28 right over there. And then if this is 7, then 73 minus 7 is going to be 66. And 333, I guess there's a couple of ways we could do it. We could take, actually we could just take 292 minus 28 is going to be, let's see, 292 minus 8 would be 284 minus another 264, 264. Do the numbers all add up? Yes, 66 plus 264 is 330."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And 333, I guess there's a couple of ways we could do it. We could take, actually we could just take 292 minus 28 is going to be, let's see, 292 minus 8 would be 284 minus another 264, 264. Do the numbers all add up? Yes, 66 plus 264 is 330. So the key realization here is, what he's saying he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. That means that the probability of his mom being grouchy, it shouldn't matter whether it's raining or not."}, {"video_title": "Filling out frequency table for independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yes, 66 plus 264 is 330. So the key realization here is, what he's saying he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. That means that the probability of his mom being grouchy, it shouldn't matter whether it's raining or not. It should just be, it should be the same probability of whether it's raining or not. And our best estimate of the probability of his mom being grouchy is on the total days is 20%. And so if the data is backing up that it's independent events, then the best way to fill this out would be the probability of his mom being grouchy on a rainy day or a not rainy day should be the same."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "The monthly data on ticket sales is shown below. What are the best and worst months for cruise sales? So what they've given us this diagram, this is usually called a pie chart or pie graph because it looks like a pie that's sliced up into a bunch of pieces. Sometimes this is called a circle graph, but pie graph is much more common. And then they say it's monthly ticket sales. So each of these slices represent the sales in a given month. So for example, this blue slice over here represents the sales in January."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Sometimes this is called a circle graph, but pie graph is much more common. And then they say it's monthly ticket sales. So each of these slices represent the sales in a given month. So for example, this blue slice over here represents the sales in January. And the way that a pie chart is set up, each slice is bigger or smaller depending on what fraction of the whole it represents. So for example, they're telling us in January, they sold 18% of the total year's ticket sales in January. So if you add up all of these percentages, it should add up to 100%."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So for example, this blue slice over here represents the sales in January. And the way that a pie chart is set up, each slice is bigger or smaller depending on what fraction of the whole it represents. So for example, they're telling us in January, they sold 18% of the total year's ticket sales in January. So if you add up all of these percentages, it should add up to 100%. And not only do they tell us that they sold 18%, but the slice of this pie should be 18% of the area of the entire pie. It is literally 18% of the pie. If you were to eat this slice, you would have eaten 18% of the pie."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So if you add up all of these percentages, it should add up to 100%. And not only do they tell us that they sold 18%, but the slice of this pie should be 18% of the area of the entire pie. It is literally 18% of the pie. If you were to eat this slice, you would have eaten 18% of the pie. Now with that out of the way, let's think about their questions. What are the best and the worst months for cruise sales? So the best month is obviously the month where they sell, where they have the largest or the largest percentage of their tickets were sold."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "If you were to eat this slice, you would have eaten 18% of the pie. Now with that out of the way, let's think about their questions. What are the best and the worst months for cruise sales? So the best month is obviously the month where they sell, where they have the largest or the largest percentage of their tickets were sold. And actually I started with January, and this is what's neat about pie graphs. You wouldn't even have to look at the numbers. January just jumps out as the biggest slice of pie."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So the best month is obviously the month where they sell, where they have the largest or the largest percentage of their tickets were sold. And actually I started with January, and this is what's neat about pie graphs. You wouldn't even have to look at the numbers. January just jumps out as the biggest slice of pie. If you didn't even see the numbers, if you couldn't even read, and you just looked at this and someone said, what is the largest slice of pie? You would immediately say, hey, this is clearly the largest slice of pie right over there. And so that is actually the best month for cruise sales because they sold 18%."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "January just jumps out as the biggest slice of pie. If you didn't even see the numbers, if you couldn't even read, and you just looked at this and someone said, what is the largest slice of pie? You would immediately say, hey, this is clearly the largest slice of pie right over there. And so that is actually the best month for cruise sales because they sold 18%. You see this 18% is larger than all of the other percentages over here. But it's clear just by looking at the graph. This is the largest slice."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And so that is actually the best month for cruise sales because they sold 18%. You see this 18% is larger than all of the other percentages over here. But it's clear just by looking at the graph. This is the largest slice. Now what's the worst month for cruise ticket sales? The worst month? Well then we just have to find the thinnest slice of pie."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "This is the largest slice. Now what's the worst month for cruise ticket sales? The worst month? Well then we just have to find the thinnest slice of pie. And if we look over here, the slices of pie get pretty thin out down here. This is in the summer in June and July and in May. But the smallest are actually July and June."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Well then we just have to find the thinnest slice of pie. And if we look over here, the slices of pie get pretty thin out down here. This is in the summer in June and July and in May. But the smallest are actually July and June. And this is where the numbers become useful because when you just look at it by, you know, when you just eyeball it, you're not sure, hey, are these exactly the same or do they just look exactly the same? And that's where these numbers are valuable. And based on the data they've given us, it looks like these are tied for the worst in terms of ticket sales."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "But the smallest are actually July and June. And this is where the numbers become useful because when you just look at it by, you know, when you just eyeball it, you're not sure, hey, are these exactly the same or do they just look exactly the same? And that's where these numbers are valuable. And based on the data they've given us, it looks like these are tied for the worst in terms of ticket sales. In both of these months, they sell only 3% in each month. So the worst months for cruise sales are July and June. July and June are tied for the worst."}, {"video_title": "Reading pie graphs (circle graphs)   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And based on the data they've given us, it looks like these are tied for the worst in terms of ticket sales. In both of these months, they sell only 3% in each month. So the worst months for cruise sales are July and June. July and June are tied for the worst. The best is clearly January. And then after January, the next best, they're not really asking us that, but since we have the pie chart in front of us, might as well ask ourselves that. What's the next best?"}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "We've already been introduced to the chi-squared statistic in other videos. Now we're going to use it for a test for homogeneity. And homogeneity or homogeneity, in everyday language, this means how similar things are. And that's what we're essentially going to test here. We're gonna look at two different groups and see whether the distributions of those groups for a certain variable are similar or not. And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains?"}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And that's what we're essentially going to test here. We're gonna look at two different groups and see whether the distributions of those groups for a certain variable are similar or not. And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, humanities, or neither? And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, humanities, or neither? And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed folks. And then the alternative hypothesis, well, no, there is a difference. So there is a difference."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed folks. And then the alternative hypothesis, well, no, there is a difference. So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent. They liked them equally."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent. They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the expected data would be if we are assuming that the null hypothesis is true, that there's no difference in preference between right and left-handed folks. This is the right-handed column. This is the left-handed column."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the expected data would be if we are assuming that the null hypothesis is true, that there's no difference in preference between right and left-handed folks. This is the right-handed column. This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from this total column. Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from this total column. Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60?"}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. And these, once again, they add up to 60."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM. 40% of 40, that is 16."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM. 40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. And then all of these add up to 40."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition. And so these need to be truly random samples."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition. And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well. And now we're ready to calculate our chi-squared statistic. We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "So let's assume that that is the case as well. And now we're ready to calculate our chi-squared statistic. We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24. I'm running out of colors. And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected. And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24. I'm running out of colors. And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected. And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12. And then last but not least, let me find a color I haven't used, we would look at that and that, and we would say plus five minus eight squared over expected, over eight. Now once you get that value for the chi-square statistic, the next question is what are the degrees of freedom? Now a simple rule of thumb is to just look at your data and think about the number of rows and the number of columns."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12. And then last but not least, let me find a color I haven't used, we would look at that and that, and we would say plus five minus eight squared over expected, over eight. Now once you get that value for the chi-square statistic, the next question is what are the degrees of freedom? Now a simple rule of thumb is to just look at your data and think about the number of rows and the number of columns. And we have three rows and two columns. And so your degrees of freedom are going to be the number of rows minus one, three minus one, times the number of columns minus one, two minus one. And so this is going to be equal to two times one, which is equal to two."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "Now a simple rule of thumb is to just look at your data and think about the number of rows and the number of columns. And we have three rows and two columns. And so your degrees of freedom are going to be the number of rows minus one, three minus one, times the number of columns minus one, two minus one. And so this is going to be equal to two times one, which is equal to two. Now the reason why that makes intuitive sense is think about it. If you knew two of these data points, and if you knew all of the totals, then you could figure out the other data points. If you knew these two data points, you could figure out that."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is going to be equal to two times one, which is equal to two. Now the reason why that makes intuitive sense is think about it. If you knew two of these data points, and if you knew all of the totals, then you could figure out the other data points. If you knew these two data points, you could figure out that. If you knew this data point, you knew the total. You could figure out that. If you knew this data point and you knew the total, you could figure out that."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "If you knew these two data points, you could figure out that. If you knew this data point, you knew the total. You could figure out that. If you knew this data point and you knew the total, you could figure out that. And if you figured out that and that, then you could figure out this right over here. And so that's why this rule of thumb works. The number of rows minus one times the number of columns minus one gives you your degrees of freedom."}, {"video_title": "Introduction to the chi-square test for homogeneity   AP Statistics   Khan Academy.mp3", "Sentence": "If you knew this data point and you knew the total, you could figure out that. And if you figured out that and that, then you could figure out this right over here. And so that's why this rule of thumb works. The number of rows minus one times the number of columns minus one gives you your degrees of freedom. Now, given this chi-squared statistic that I haven't calculated, but you could type this into a calculator and figure it out, and this degrees of freedom, we could then figure out the p-value. We could figure out the probability of getting a chi-squared statistic this extreme or more extreme. And if this is less than our significance level, which we should have set ahead of time, then we would reject the null hypothesis and it would suggest the alternative."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "For warmup, Jeremiah likes to shoot three-point shots until he successfully makes one. Alright, this is the telltale signs of a geometric random variable. How many trials do I have to take until I get a success? Let m be the number of shots it takes Jeremiah to successfully make his first three-point shot. Okay, so they're defining the random variable here, the number of shots it takes, the number of trials it takes until we get a successful three-point shot. Assume that the results of each shot are independent. Alright, the probability that he makes a given shot is not dependent on whether he made or missed the previous shots."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let m be the number of shots it takes Jeremiah to successfully make his first three-point shot. Okay, so they're defining the random variable here, the number of shots it takes, the number of trials it takes until we get a successful three-point shot. Assume that the results of each shot are independent. Alright, the probability that he makes a given shot is not dependent on whether he made or missed the previous shots. Find the probability that Jeremiah's first successful shot occurs on his third attempt. So like always, pause this video and see if you can have a go at it. Alright, now let's work through this together."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Alright, the probability that he makes a given shot is not dependent on whether he made or missed the previous shots. Find the probability that Jeremiah's first successful shot occurs on his third attempt. So like always, pause this video and see if you can have a go at it. Alright, now let's work through this together. So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successful one. And so what they're really asking is find the probability that m is equal to three, that his first successful shot occurs on his third attempt. So m is equal to three."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Alright, now let's work through this together. So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successful one. And so what they're really asking is find the probability that m is equal to three, that his first successful shot occurs on his third attempt. So m is equal to three. So that the number of shots it takes Jeremiah to make his first successful shot is three. So how do we do this? Well, what's just the probability of that happening?"}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So m is equal to three. So that the number of shots it takes Jeremiah to make his first successful shot is three. So how do we do this? Well, what's just the probability of that happening? Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has a 1 4th chance of making his shots, he has a 3 4th probability of missing his shots."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, what's just the probability of that happening? Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has a 1 4th chance of making his shots, he has a 3 4th probability of missing his shots. So this will be 3 4ths, so he misses the first shot, times he has to miss the second shot, and then he has to make his third shot. So there you have it, that's the probability. Miss, miss, make."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if he has a 1 4th chance of making his shots, he has a 3 4th probability of missing his shots. So this will be 3 4ths, so he misses the first shot, times he has to miss the second shot, and then he has to make his third shot. So there you have it, that's the probability. Miss, miss, make. And so what is this going to be? This is equal to nine over 60 4ths. So there you have it."}, {"video_title": "Probability for a geometric random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Miss, miss, make. And so what is this going to be? This is equal to nine over 60 4ths. So there you have it. If you wanted to have this as a decimal, we could get a calculator out real fast. So this is nine, whoops, nine divided by 64 is equal to roughly 0.14, approximately 0.14. Or another way to think about it is, a roughly 14% chance or 14% probability that it takes him, that his first successful shot occurs in his third attempt."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This is just going to be a ton of algebraic manipulation, but I'll try to color-code it well so we don't get lost in the math. Let me just rewrite this expression over here. This whole video is just going to be rewriting this over and over again, just simplifying it a bit with algebra. This first term right over here, y1 minus mx1 plus b squared, that's going to be, and we could write this as all going to be the squared error of the line. This first term over here, I'll keep it in blue, is going to be, if we just expand it, y1 squared minus 2 times y1 times mx1 plus b plus mx1 plus b squared. All I did is I just squared this binomial right here. You can imagine this, if this was a minus b, it would be a squared minus 2ab plus b squared."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This first term right over here, y1 minus mx1 plus b squared, that's going to be, and we could write this as all going to be the squared error of the line. This first term over here, I'll keep it in blue, is going to be, if we just expand it, y1 squared minus 2 times y1 times mx1 plus b plus mx1 plus b squared. All I did is I just squared this binomial right here. You can imagine this, if this was a minus b, it would be a squared minus 2ab plus b squared. That's all I did. Now I'll just have to do that for each of the terms. Each term is only different by the x and the y coordinates right over here."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "You can imagine this, if this was a minus b, it would be a squared minus 2ab plus b squared. That's all I did. Now I'll just have to do that for each of the terms. Each term is only different by the x and the y coordinates right over here. The next term, and I'll write it, I'll go down so that we can combine like terms. This term over here squared is going to be y2 squared minus 2 times y2 times mx2 plus b plus mx2 plus b squared. Same exact thing up here, except now it was with x2 and y2 as opposed to x1 and y1."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Each term is only different by the x and the y coordinates right over here. The next term, and I'll write it, I'll go down so that we can combine like terms. This term over here squared is going to be y2 squared minus 2 times y2 times mx2 plus b plus mx2 plus b squared. Same exact thing up here, except now it was with x2 and y2 as opposed to x1 and y1. Then we're just going to keep doing that n times. We're just going to keep doing it n times. We're going to do it for the third, x3, y3, keep going, keep going, all the way until we get to this nth term over here."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Same exact thing up here, except now it was with x2 and y2 as opposed to x1 and y1. Then we're just going to keep doing that n times. We're just going to keep doing it n times. We're going to do it for the third, x3, y3, keep going, keep going, all the way until we get to this nth term over here. This nth term over here when we square it is going to be yn squared minus 2yn times mxn plus b plus mxn plus b squared. Now the next thing I want to do is actually expand these out a little bit more. Let's expand these out a little more."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We're going to do it for the third, x3, y3, keep going, keep going, all the way until we get to this nth term over here. This nth term over here when we square it is going to be yn squared minus 2yn times mxn plus b plus mxn plus b squared. Now the next thing I want to do is actually expand these out a little bit more. Let's expand these out a little more. Let's actually scroll down. This whole expression, I'm just going to rewrite it, is the same thing as, and remember this is just a squared error of the line, so let me rewrite this top line over here. This top line over here is y1 squared."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Let's expand these out a little more. Let's actually scroll down. This whole expression, I'm just going to rewrite it, is the same thing as, and remember this is just a squared error of the line, so let me rewrite this top line over here. This top line over here is y1 squared. Then I'm going to distribute this 2y1. This is going to be minus 2y1mx1, that's just that times that, minus 2y1b and then plus, and now let's expand mx1 plus b squared. That's going to be m squared x1 squared plus 2 times mx1 times b plus b squared."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This top line over here is y1 squared. Then I'm going to distribute this 2y1. This is going to be minus 2y1mx1, that's just that times that, minus 2y1b and then plus, and now let's expand mx1 plus b squared. That's going to be m squared x1 squared plus 2 times mx1 times b plus b squared. All I did, if this was a plus b squared, this is a squared plus 2ab plus b squared. We're just going to do that for each of these terms, or each of these colors I guess you could say. Now let's move to the second term."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "That's going to be m squared x1 squared plus 2 times mx1 times b plus b squared. All I did, if this was a plus b squared, this is a squared plus 2ab plus b squared. We're just going to do that for each of these terms, or each of these colors I guess you could say. Now let's move to the second term. Plus, it's going to be the same thing, but instead of y1s and x1s, it's going to be y2s and x2s. So it is y2 squared minus 2y2mx2 minus 2y2b plus m squared x2 squared plus 2 times mx2b plus b squared. We're going to keep doing this all the way until we get the nth term."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Now let's move to the second term. Plus, it's going to be the same thing, but instead of y1s and x1s, it's going to be y2s and x2s. So it is y2 squared minus 2y2mx2 minus 2y2b plus m squared x2 squared plus 2 times mx2b plus b squared. We're going to keep doing this all the way until we get the nth term. All the way until we get to the nth color we should say. So this is going to be yn squared minus 2ynmxn. You don't even have to think, you just have to substitute these with n's now."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We're going to keep doing this all the way until we get the nth term. All the way until we get to the nth color we should say. So this is going to be yn squared minus 2ynmxn. You don't even have to think, you just have to substitute these with n's now. We could actually look at this, but it's going to be the exact same thing. mxn minus 2ynb plus m squared xn squared plus 2mxnb plus b squared. So once again, this is just the squared error of that line with n points, between those n points and the line y equals mx plus b."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "You don't even have to think, you just have to substitute these with n's now. We could actually look at this, but it's going to be the exact same thing. mxn minus 2ynb plus m squared xn squared plus 2mxnb plus b squared. So once again, this is just the squared error of that line with n points, between those n points and the line y equals mx plus b. So let's see if we can simplify this somehow. And to do that, what I'm going to do is I'm going to kind of try to add up a bunch of these terms here. So if I were to add up all of these terms right here, if I were to add up this column right over there, what do I get?"}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So once again, this is just the squared error of that line with n points, between those n points and the line y equals mx plus b. So let's see if we can simplify this somehow. And to do that, what I'm going to do is I'm going to kind of try to add up a bunch of these terms here. So if I were to add up all of these terms right here, if I were to add up this column right over there, what do I get? Well it's going to be y1 squared plus y2 squared plus y all the way to yn squared. That's those terms right over there. So I'm going to have that."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So if I were to add up all of these terms right here, if I were to add up this column right over there, what do I get? Well it's going to be y1 squared plus y2 squared plus y all the way to yn squared. That's those terms right over there. So I'm going to have that. And then I'm going to have minus, you have this common 2m amongst all of these terms over here. So let me write that down. 2m here, 2m here, 2m here."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So I'm going to have that. And then I'm going to have minus, you have this common 2m amongst all of these terms over here. So let me write that down. 2m here, 2m here, 2m here. So then you're going to have, let me put parentheses around here. So you have these terms all added up. Then you have minus 2m times all of these terms."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "2m here, 2m here, 2m here. So then you're going to have, let me put parentheses around here. So you have these terms all added up. Then you have minus 2m times all of these terms. So you have, actually let me color code it just so you see what we're doing. I want to be very careful with this math so that nothing seems too confusing. Although this is really just algebraic manipulation."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Then you have minus 2m times all of these terms. So you have, actually let me color code it just so you see what we're doing. I want to be very careful with this math so that nothing seems too confusing. Although this is really just algebraic manipulation. So if I add all of these up, I get y1 squared plus y2 squared all the way to yn squared. I'll put some parentheses around that. And then to that, we have these common terms."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Although this is really just algebraic manipulation. So if I add all of these up, I get y1 squared plus y2 squared all the way to yn squared. I'll put some parentheses around that. And then to that, we have these common terms. We have this minus 2m, minus 2m, minus 2m. So we can distribute those out. And so this actually, I should actually write it like this."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then to that, we have these common terms. We have this minus 2m, minus 2m, minus 2m. So we can distribute those out. And so this actually, I should actually write it like this. So we have a minus 2m times, once we distribute it out, here we're just going to be left with a y1 x1. Maybe I could call it an x1 y1. x1 y1, that's that over there with the 2m factored out."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And so this actually, I should actually write it like this. So we have a minus 2m times, once we distribute it out, here we're just going to be left with a y1 x1. Maybe I could call it an x1 y1. x1 y1, that's that over there with the 2m factored out. Plus x2, let me do that in another color. I want to make this easy to read. Plus x2 y2 plus xn yn."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "x1 y1, that's that over there with the 2m factored out. Plus x2, let me do that in another color. I want to make this easy to read. Plus x2 y2 plus xn yn. Plus x, and well we're going to keep adding up. We're going to do this n times, all the way to plus xn yn. This last term over here, yn xn, same thing."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Plus x2 y2 plus xn yn. Plus x, and well we're going to keep adding up. We're going to do this n times, all the way to plus xn yn. This last term over here, yn xn, same thing. So that's the sum. So this stuff over here, let me just add a new color. The sum of all of this stuff right over here is the same thing as this term right over here."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This last term over here, yn xn, same thing. So that's the sum. So this stuff over here, let me just add a new color. The sum of all of this stuff right over here is the same thing as this term right over here. And then we have to sum this right over here. And you see again, we can factor out. We can factor out here a minus 2b out of all of these terms."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "The sum of all of this stuff right over here is the same thing as this term right over here. And then we have to sum this right over here. And you see again, we can factor out. We can factor out here a minus 2b out of all of these terms. So we have minus 2b times y1 plus y2 plus all the way to yn. So this business, so these terms right over here, these terms right over here when you add them up, give you these terms or this term right over there. And let's just keep going."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We can factor out here a minus 2b out of all of these terms. So we have minus 2b times y1 plus y2 plus all the way to yn. So this business, so these terms right over here, these terms right over here when you add them up, give you these terms or this term right over there. And let's just keep going. And then in the next video, we're probably going to run out of time in this one. In the next video I'll simplify this more and I'll actually clean up the algebra a good bit. So then the next term, what is this going to be?"}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And let's just keep going. And then in the next video, we're probably going to run out of time in this one. In the next video I'll simplify this more and I'll actually clean up the algebra a good bit. So then the next term, what is this going to be? Same drill. We can factor out an m squared. So we have m squared times x1 squared plus x2 squared plus all the way."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So then the next term, what is this going to be? Same drill. We can factor out an m squared. So we have m squared times x1 squared plus x2 squared plus all the way. Actually, I want to color code them. I forgot to color code these over here. Plus x2 squared plus all the way to xn squared."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So we have m squared times x1 squared plus x2 squared plus all the way. Actually, I want to color code them. I forgot to color code these over here. Plus x2 squared plus all the way to xn squared. Let me color code these. This was a yn squared and this over here was a y2 squared. So this is exactly this."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Plus x2 squared plus all the way to xn squared. Let me color code these. This was a yn squared and this over here was a y2 squared. So this is exactly this. So we've written, so in this last step we just did, this thing over here is this thing right over here. And of course we have to add it, so I'll put a plus out front. We're almost done with this stage of the simplification."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So this is exactly this. So we've written, so in this last step we just did, this thing over here is this thing right over here. And of course we have to add it, so I'll put a plus out front. We're almost done with this stage of the simplification. So over here we have a common 2mb. So let's put a plus 2mb times, once again, x1 plus x2 plus all the way to xn. So this term right over here is the exact same thing as this term over here."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We're almost done with this stage of the simplification. So over here we have a common 2mb. So let's put a plus 2mb times, once again, x1 plus x2 plus all the way to xn. So this term right over here is the exact same thing as this term over here. And then finally we have a b squared in each of these. And how many of these b squares do we have? Well, we have n of these lines, right?"}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So this term right over here is the exact same thing as this term over here. And then finally we have a b squared in each of these. And how many of these b squares do we have? Well, we have n of these lines, right? This is the first line, second line, then a bunch, bunch, bunch, all the way to the nth line. So we have b squared added to itself n times. So this right over here is just b squared n times."}, {"video_title": "Proof (part 1) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Well, we have n of these lines, right? This is the first line, second line, then a bunch, bunch, bunch, all the way to the nth line. So we have b squared added to itself n times. So this right over here is just b squared n times. So we'll just write that as plus n times b squared. Now it doesn't look like, let me remind ourselves what this is all about. This is all just algebraic manipulation of the squared error between those n points and the line y equals mx plus b."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the last video, we were able to find the equation for the regression line for these four data points. What I want to do in this video is figure out the r squared for these data points. Figure out how good this line fits the data, or even better, figure out the percentage, which is really the same thing, of the variation of these data points, especially the variation in y, that can be explained by a variation in x. So to do that, I'm actually going to get a spreadsheet out. I actually have tried to do this with a calculator, and it's much harder. So hopefully this doesn't confuse you too much to use a spreadsheet. And I'm going to make a couple of columns here."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So to do that, I'm actually going to get a spreadsheet out. I actually have tried to do this with a calculator, and it's much harder. So hopefully this doesn't confuse you too much to use a spreadsheet. And I'm going to make a couple of columns here. And spreadsheets actually have functions that will do all of this automatically, but I really want to do it so that you could do it by hand if you had to. So I'm going to make a couple of columns here. This is going to be my x column."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm going to make a couple of columns here. And spreadsheets actually have functions that will do all of this automatically, but I really want to do it so that you could do it by hand if you had to. So I'm going to make a couple of columns here. This is going to be my x column. This is going to be my y column. This is going to be the column, I'll call this y star. This will be the y value that our line predicts based on our x value."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be my x column. This is going to be my y column. This is going to be the column, I'll call this y star. This will be the y value that our line predicts based on our x value. This is going to be the error with the line. So it's going to be the difference, and we call it the squared error with line. Actually, let me just do the error with line."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This will be the y value that our line predicts based on our x value. This is going to be the error with the line. So it's going to be the difference, and we call it the squared error with line. Actually, let me just do the error with line. I'll do the squared error. I don't want this to take up too much space. Squared error with line."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me just do the error with line. I'll do the squared error. I don't want this to take up too much space. Squared error with line. And then the next one I want to do the squared error. Actually, no, I already had the squared error. And then the next one I am going to have the squared variation for that y value, squared from the mean y. I think these columns by themselves will be enough for us to do everything."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Squared error with line. And then the next one I want to do the squared error. Actually, no, I already had the squared error. And then the next one I am going to have the squared variation for that y value, squared from the mean y. I think these columns by themselves will be enough for us to do everything. So let's first put all the data points in. So we had negative 2 comma negative 3. That was one data point."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the next one I am going to have the squared variation for that y value, squared from the mean y. I think these columns by themselves will be enough for us to do everything. So let's first put all the data points in. So we had negative 2 comma negative 3. That was one data point. Negative 1 comma negative 1. Then we had 1 comma 2. Then we have 4 comma 3."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "That was one data point. Negative 1 comma negative 1. Then we had 1 comma 2. Then we have 4 comma 3. Now, what does our line predict? Well, our line says, look, you give me an x value, and I'm going to tell you what y value I'll predict. So when x is equal to negative 2, the y value on the line is going to be the slope."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we have 4 comma 3. Now, what does our line predict? Well, our line says, look, you give me an x value, and I'm going to tell you what y value I'll predict. So when x is equal to negative 2, the y value on the line is going to be the slope. So this is going to be equal to 41 divided by 42 times our x value, and I just selected that cell. And just a little bit of a primer on spreadsheets, I'm selecting the cell D2. I was able to just move my cursor over and select that."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when x is equal to negative 2, the y value on the line is going to be the slope. So this is going to be equal to 41 divided by 42 times our x value, and I just selected that cell. And just a little bit of a primer on spreadsheets, I'm selecting the cell D2. I was able to just move my cursor over and select that. That tells me the x value minus 5 over 21. Minus 5 divided by 21. Just like that."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I was able to just move my cursor over and select that. That tells me the x value minus 5 over 21. Minus 5 divided by 21. Just like that. So just to be clear of what we're even doing, this y star here, I got negative 2.19. That tells us that this point right over here is negative 2.19 right over here. So when we figure out the error, we're going to figure out the distance between negative 3, that's our y value, between negative 3 and negative 2.19."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just like that. So just to be clear of what we're even doing, this y star here, I got negative 2.19. That tells us that this point right over here is negative 2.19 right over here. So when we figure out the error, we're going to figure out the distance between negative 3, that's our y value, between negative 3 and negative 2.19. So let's do that. So the error is just going to be equal to our y value, that cell E2, minus the value that our line would predict. And we want the square."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when we figure out the error, we're going to figure out the distance between negative 3, that's our y value, between negative 3 and negative 2.19. So let's do that. So the error is just going to be equal to our y value, that cell E2, minus the value that our line would predict. And we want the square. So just that value is the actual error, but we want to square it. So we want to square it just like that. So we will square it."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we want the square. So just that value is the actual error, but we want to square it. So we want to square it just like that. So we will square it. And then, let me make sure I did the right thing. Yep. And then the next thing we want to do is the squared distance, so this is equal to the squared distance of our y value from the y's mean."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we will square it. And then, let me make sure I did the right thing. Yep. And then the next thing we want to do is the squared distance, so this is equal to the squared distance of our y value from the y's mean. So what's the mean of the y's? Mean of the y's is 1 4th, so minus 0.25. It's the same thing as 1 4th."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the next thing we want to do is the squared distance, so this is equal to the squared distance of our y value from the y's mean. So what's the mean of the y's? Mean of the y's is 1 4th, so minus 0.25. It's the same thing as 1 4th. And we also want to square that. Now, this is what's fun about spreadsheets. I can apply those formulas to every row now."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's the same thing as 1 4th. And we also want to square that. Now, this is what's fun about spreadsheets. I can apply those formulas to every row now. And notice what it did when I did that. Now all of a sudden, this is the y value that my line would predict, it's now using this x value and sticking it over here. It's now figuring out the squared distance from the line using what the line would predict and using the y value, this one."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I can apply those formulas to every row now. And notice what it did when I did that. Now all of a sudden, this is the y value that my line would predict, it's now using this x value and sticking it over here. It's now figuring out the squared distance from the line using what the line would predict and using the y value, this one. And then it does the same thing over here. It figures out the squared distance of this y value from the mean. So what is the total squared error with the line?"}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's now figuring out the squared distance from the line using what the line would predict and using the y value, this one. And then it does the same thing over here. It figures out the squared distance of this y value from the mean. So what is the total squared error with the line? So let me just sum this up. The total squared error with the line is 2.73. And then the total variation from the mean, the squared distances from the mean of the y are 22.75."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the total squared error with the line? So let me just sum this up. The total squared error with the line is 2.73. And then the total variation from the mean, the squared distances from the mean of the y are 22.75. So let me be very clear what this is. So let me write these numbers down. So our squared, I'll write it up here so we can keep looking at this actual graph."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the total variation from the mean, the squared distances from the mean of the y are 22.75. So let me be very clear what this is. So let me write these numbers down. So our squared, I'll write it up here so we can keep looking at this actual graph. So our squared error versus our line, our total squared error, we just computed to be 2.74. I rounded it a little bit. And what that is, is you take each of these data points, vertical distance to the line."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our squared, I'll write it up here so we can keep looking at this actual graph. So our squared error versus our line, our total squared error, we just computed to be 2.74. I rounded it a little bit. And what that is, is you take each of these data points, vertical distance to the line. So this distance squared plus this distance squared plus this distance squared plus this distance squared. That's all we just calculated on Excel. And that total squared variation to the line is 2.74, our total squared error with the line."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what that is, is you take each of these data points, vertical distance to the line. So this distance squared plus this distance squared plus this distance squared plus this distance squared. That's all we just calculated on Excel. And that total squared variation to the line is 2.74, our total squared error with the line. And then the other number we figured out was the total distance from the mean. So the mean here is y is equal to 1 4th. So that's going to be right over here."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that total squared variation to the line is 2.74, our total squared error with the line. And then the other number we figured out was the total distance from the mean. So the mean here is y is equal to 1 4th. So that's going to be right over here. So y is equal to 1 4th is going to be right over, this is 1 half, so right over here. So this is our mean y, let me draw it a little bit neater than that, this is our mean y value. This is our mean y value, or the central tendency for our y values."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's going to be right over here. So y is equal to 1 4th is going to be right over, this is 1 half, so right over here. So this is our mean y, let me draw it a little bit neater than that, this is our mean y value. This is our mean y value, or the central tendency for our y values. And so what we calculated next was the total error, the squared error from the means of our y values. That's what we calculated over here. This is what we calculated over here in the spreadsheet."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our mean y value, or the central tendency for our y values. And so what we calculated next was the total error, the squared error from the means of our y values. That's what we calculated over here. This is what we calculated over here in the spreadsheet. You see it in the formula. It is this number, e2 minus 0.25, which is the mean of our y's, squared. That's exactly what we calculated."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is what we calculated over here in the spreadsheet. You see it in the formula. It is this number, e2 minus 0.25, which is the mean of our y's, squared. That's exactly what we calculated. We calculated for each of the y values and then we summed them all up. It's 22.75. It is equal to 22.75."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's exactly what we calculated. We calculated for each of the y values and then we summed them all up. It's 22.75. It is equal to 22.75. So if you wanted to know, so this is essentially the error that the line does not explain. This is the total error, this is the total variation of the numbers. So if you wanted to know the percentage of the total variation that is not explained by the line, you could take this number divided by this number."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It is equal to 22.75. So if you wanted to know, so this is essentially the error that the line does not explain. This is the total error, this is the total variation of the numbers. So if you wanted to know the percentage of the total variation that is not explained by the line, you could take this number divided by this number. So 2.74 over 22.75. This tells us the percentage of total variation not explained by the line or by the variation in x. By variation in x."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you wanted to know the percentage of the total variation that is not explained by the line, you could take this number divided by this number. So 2.74 over 22.75. This tells us the percentage of total variation not explained by the line or by the variation in x. By variation in x. And so what is this number going to be? I could just use Excel for this. So if I'm just going to divide this number divided by this number right over there, I get 0.12."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "By variation in x. And so what is this number going to be? I could just use Excel for this. So if I'm just going to divide this number divided by this number right over there, I get 0.12. So this is equal to 0.12. So this is equal right over here. This is equal to 0.12."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I'm just going to divide this number divided by this number right over there, I get 0.12. So this is equal to 0.12. So this is equal right over here. This is equal to 0.12. Or another way to think about it is 12% of the total variation is not explained by the variation in x. The total squared distance between each of the points or their kind of spread, their variation, is not explained by the variation in x. So if you want the amount that is explained by the variance in x, you just subtract that from 1."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is equal to 0.12. Or another way to think about it is 12% of the total variation is not explained by the variation in x. The total squared distance between each of the points or their kind of spread, their variation, is not explained by the variation in x. So if you want the amount that is explained by the variance in x, you just subtract that from 1. So let me write it right over here. So we have our r squared, which is the percent of the total variation that is explained by x is going to be 1 minus that 0.12 that we just calculated, which is going to be 0.88. So our r squared here is 0.88."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you want the amount that is explained by the variance in x, you just subtract that from 1. So let me write it right over here. So we have our r squared, which is the percent of the total variation that is explained by x is going to be 1 minus that 0.12 that we just calculated, which is going to be 0.88. So our r squared here is 0.88. It's very, very close to 1. The highest number it can be is 1. So what this tells us, or a way to interpret this, is 88% of the total variation of these y values is explained by the line or by the variation in x."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our r squared here is 0.88. It's very, very close to 1. The highest number it can be is 1. So what this tells us, or a way to interpret this, is 88% of the total variation of these y values is explained by the line or by the variation in x. And you can see that. It looks like a pretty good fit. Each of these aren't too far."}, {"video_title": "Calculating R-squared   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what this tells us, or a way to interpret this, is 88% of the total variation of these y values is explained by the line or by the variation in x. And you can see that. It looks like a pretty good fit. Each of these aren't too far. They're definitely much closer to this line. Each of these points are definitely much closer to the line than they are to the mean line. In fact, all of them are closer to our actual line than to the mean."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So a good place to start is just to define a random variable that essentially represents what you care about. So let's just say the number of cars that pass in some amount of time, let's say in an hour. And your goal is to figure out the probability distribution of this random variable. And then once you know the probability distribution, then you can figure out what's the probability that 100 cars pass in an hour, or the probability that no cars pass in an hour, and you'd be unstoppable. And just a little aside, just to move forward with this video, there's two assumptions we need to make, because we're going to study the Poisson distribution. In order to study it, there's two assumptions we have to make, that any hour at this point on the street is no different than any other hour. And we know that's probably false."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then once you know the probability distribution, then you can figure out what's the probability that 100 cars pass in an hour, or the probability that no cars pass in an hour, and you'd be unstoppable. And just a little aside, just to move forward with this video, there's two assumptions we need to make, because we're going to study the Poisson distribution. In order to study it, there's two assumptions we have to make, that any hour at this point on the street is no different than any other hour. And we know that's probably false. During rush hour in a real situation, you probably would have more cars than in another rush hour. And if you wanted to be more realistic, maybe we do it in a day. Because in a day, any period of time, actually, no, I shouldn't do a day."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we know that's probably false. During rush hour in a real situation, you probably would have more cars than in another rush hour. And if you wanted to be more realistic, maybe we do it in a day. Because in a day, any period of time, actually, no, I shouldn't do a day. We have to assume that every hour is completely just like any other hour. And actually, even within the hour, there's really no differentiation from one second to the other in terms of the probabilities that a car arrives. So that's a little bit of a simplifying assumption that might not truly apply to traffic, but I think we can make that assumption."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because in a day, any period of time, actually, no, I shouldn't do a day. We have to assume that every hour is completely just like any other hour. And actually, even within the hour, there's really no differentiation from one second to the other in terms of the probabilities that a car arrives. So that's a little bit of a simplifying assumption that might not truly apply to traffic, but I think we can make that assumption. And then the other assumption we need to make is that if a bunch of cars pass in one hour, that doesn't mean that fewer cars will pass in the next. That in no way does the number of cars that pass in one period affect or correlate or somehow influence the number of cars that pass in the next. That they're really independent."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's a little bit of a simplifying assumption that might not truly apply to traffic, but I think we can make that assumption. And then the other assumption we need to make is that if a bunch of cars pass in one hour, that doesn't mean that fewer cars will pass in the next. That in no way does the number of cars that pass in one period affect or correlate or somehow influence the number of cars that pass in the next. That they're really independent. Given that, we can then at least try using the skills we have to model out some type of a distribution. The first thing you do, and I'd recommend doing this for any distribution, is maybe we can estimate the mean. Let's sit out on that curb and measure what this variable is over a bunch of hours and then average it up."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "That they're really independent. Given that, we can then at least try using the skills we have to model out some type of a distribution. The first thing you do, and I'd recommend doing this for any distribution, is maybe we can estimate the mean. Let's sit out on that curb and measure what this variable is over a bunch of hours and then average it up. And that's going to be a pretty good estimator for the actual mean of our population, or since it's a random variable, the expected value of this random variable. Let's say you do that and you get your best estimate of the expected value of this random variable is, I'll use the letter lambda. So this could be 9 cars per hour."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's sit out on that curb and measure what this variable is over a bunch of hours and then average it up. And that's going to be a pretty good estimator for the actual mean of our population, or since it's a random variable, the expected value of this random variable. Let's say you do that and you get your best estimate of the expected value of this random variable is, I'll use the letter lambda. So this could be 9 cars per hour. You sat out there, it could be 9.3 cars per hour. You sat out there over hundreds of hours and you just counted the number of cars each hour and you averaged them all up. And you said on average there are 9.3 cars per hour and you feel that's a pretty good estimate."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this could be 9 cars per hour. You sat out there, it could be 9.3 cars per hour. You sat out there over hundreds of hours and you just counted the number of cars each hour and you averaged them all up. And you said on average there are 9.3 cars per hour and you feel that's a pretty good estimate. So that's what you have there. And let's see what we could do. We know the binomial distribution."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you said on average there are 9.3 cars per hour and you feel that's a pretty good estimate. So that's what you have there. And let's see what we could do. We know the binomial distribution. The binomial distribution tells us that the expected value of a random variable is equal to the number of trials that that random variable is kind of composed of. Before in the previous videos we were counting the number of heads in a coin toss. So this would be the number of coin tosses times the probability of success over each toss."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know the binomial distribution. The binomial distribution tells us that the expected value of a random variable is equal to the number of trials that that random variable is kind of composed of. Before in the previous videos we were counting the number of heads in a coin toss. So this would be the number of coin tosses times the probability of success over each toss. This is what we did with the binomial distribution. So maybe we can model our traffic situation something similar. This is the number of cars that pass in an hour."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this would be the number of coin tosses times the probability of success over each toss. This is what we did with the binomial distribution. So maybe we can model our traffic situation something similar. This is the number of cars that pass in an hour. So maybe we could say lambda cars per hour is equal to, I don't know, let's make each experiment or each toss of the coin equal to whether a car passes in a given minute. So there's 60 minutes per hour. And then so there would be 60 trials."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the number of cars that pass in an hour. So maybe we could say lambda cars per hour is equal to, I don't know, let's make each experiment or each toss of the coin equal to whether a car passes in a given minute. So there's 60 minutes per hour. And then so there would be 60 trials. And then the probability that we have success in each of those trials, if we model this as a binomial distribution, would be lambda over 60 cars per minute. And this would be a probability. This would be n. And this would be the probability."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then so there would be 60 trials. And then the probability that we have success in each of those trials, if we model this as a binomial distribution, would be lambda over 60 cars per minute. And this would be a probability. This would be n. And this would be the probability. If we said that this is a binomial distribution. And this probably wouldn't be that bad of an approximation. If you actually then said, oh, this is a binomial distribution, so the probability that our random variable equals some given value k, the probability that exactly three cars pass in a given hour, it would then be equal to n. So n would be 60."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would be n. And this would be the probability. If we said that this is a binomial distribution. And this probably wouldn't be that bad of an approximation. If you actually then said, oh, this is a binomial distribution, so the probability that our random variable equals some given value k, the probability that exactly three cars pass in a given hour, it would then be equal to n. So n would be 60. Choose k, and three cars, times the probability of success, so the probability that a car passes in any minute, so it would be lambda over 60 to the number of successes we need, so to the kth power, times the probability of no success, or that no cars pass, to the n minus k. If we have k successes, we have to have 60 minus k failures. There are 60 minus k minutes where no car passed. And this actually wouldn't be that bad of an approximation, where you have 60 intervals and you say this is a binomial distribution, and you'd probably get reasonable results."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you actually then said, oh, this is a binomial distribution, so the probability that our random variable equals some given value k, the probability that exactly three cars pass in a given hour, it would then be equal to n. So n would be 60. Choose k, and three cars, times the probability of success, so the probability that a car passes in any minute, so it would be lambda over 60 to the number of successes we need, so to the kth power, times the probability of no success, or that no cars pass, to the n minus k. If we have k successes, we have to have 60 minus k failures. There are 60 minus k minutes where no car passed. And this actually wouldn't be that bad of an approximation, where you have 60 intervals and you say this is a binomial distribution, and you'd probably get reasonable results. But there's a core issue here. In this model, where we model it as a binomial distribution, what happens if more than one car passes in an hour? Or more than one car passes in a minute?"}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this actually wouldn't be that bad of an approximation, where you have 60 intervals and you say this is a binomial distribution, and you'd probably get reasonable results. But there's a core issue here. In this model, where we model it as a binomial distribution, what happens if more than one car passes in an hour? Or more than one car passes in a minute? The way we have it right now, we call it a success if one car passes in a minute. And if you're kind of counting, it counts as one success, even if five cars pass in that minute. And so you say, oh, OK, Sal."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or more than one car passes in a minute? The way we have it right now, we call it a success if one car passes in a minute. And if you're kind of counting, it counts as one success, even if five cars pass in that minute. And so you say, oh, OK, Sal. I know the solution there. I just have to get more granular. Instead of dividing it into minutes, why don't I divide it into seconds?"}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so you say, oh, OK, Sal. I know the solution there. I just have to get more granular. Instead of dividing it into minutes, why don't I divide it into seconds? So the probability that I have k successes, instead of 60 intervals, I'll do 3,600 intervals. And so the probability of k successful seconds, so a second where a car is passing at that moment, out of 3,600 possible seconds, so that's 3,600, choose k, times the probability that a car passes in any given second. Well, that's the probability, that's the expected number of cars in an hour divided by number of seconds in an hour."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Instead of dividing it into minutes, why don't I divide it into seconds? So the probability that I have k successes, instead of 60 intervals, I'll do 3,600 intervals. And so the probability of k successful seconds, so a second where a car is passing at that moment, out of 3,600 possible seconds, so that's 3,600, choose k, times the probability that a car passes in any given second. Well, that's the probability, that's the expected number of cars in an hour divided by number of seconds in an hour. And we're going to have k successes, and then we're going to have, and these are the failures, the probability of failure, and you're going to have 3,600 minus k failures. And this would be even a better approximation. This actually would not be so bad, but still you have the situation where two cars can come within a half a second of each other, and you say, oh, OK, Sal, I see the pattern here."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that's the probability, that's the expected number of cars in an hour divided by number of seconds in an hour. And we're going to have k successes, and then we're going to have, and these are the failures, the probability of failure, and you're going to have 3,600 minus k failures. And this would be even a better approximation. This actually would not be so bad, but still you have the situation where two cars can come within a half a second of each other, and you say, oh, OK, Sal, I see the pattern here. We just have to get more and more granular. We have to just make this number larger and larger and larger, and your intuition is correct. And if you do that, you'll end up getting the Poisson distribution."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This actually would not be so bad, but still you have the situation where two cars can come within a half a second of each other, and you say, oh, OK, Sal, I see the pattern here. We just have to get more and more granular. We have to just make this number larger and larger and larger, and your intuition is correct. And if you do that, you'll end up getting the Poisson distribution. And this is really interesting, because a lot of times people give you the formula for the Poisson distribution, and you can kind of just plug in the numbers and use it, but it's neat to know that it really is just the binomial distribution, and the binomial distribution really did come from kind of the common sense of flipping coins. That's where everything is coming from. But before we kind of prove that this, if we take the limit as, let me change colors, before we prove that as we take the limit as this number right here, the number of intervals approaches infinity, that this becomes the Poisson distribution, I'm going to make sure we have a couple of mathematical tools in our belt."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you do that, you'll end up getting the Poisson distribution. And this is really interesting, because a lot of times people give you the formula for the Poisson distribution, and you can kind of just plug in the numbers and use it, but it's neat to know that it really is just the binomial distribution, and the binomial distribution really did come from kind of the common sense of flipping coins. That's where everything is coming from. But before we kind of prove that this, if we take the limit as, let me change colors, before we prove that as we take the limit as this number right here, the number of intervals approaches infinity, that this becomes the Poisson distribution, I'm going to make sure we have a couple of mathematical tools in our belt. So the first is something that you're probably reasonably familiar with by now, but I just want to make sure that the limit as x approaches infinity of 1 plus a over x to the x power is equal to e to the a x. No, sorry, is equal to e to the a. And just to prove this to you, let's make a little substitution here."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "But before we kind of prove that this, if we take the limit as, let me change colors, before we prove that as we take the limit as this number right here, the number of intervals approaches infinity, that this becomes the Poisson distribution, I'm going to make sure we have a couple of mathematical tools in our belt. So the first is something that you're probably reasonably familiar with by now, but I just want to make sure that the limit as x approaches infinity of 1 plus a over x to the x power is equal to e to the a x. No, sorry, is equal to e to the a. And just to prove this to you, let's make a little substitution here. Let's say that n is equal to, let me say 1 over n is equal to a over x, and then what would be x would be equal to n a, right, x times 1 is equal to n times a. And so the limit as x approaches infinity, what does a approach? a is, sorry, as x approaches infinity, what does n approach?"}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just to prove this to you, let's make a little substitution here. Let's say that n is equal to, let me say 1 over n is equal to a over x, and then what would be x would be equal to n a, right, x times 1 is equal to n times a. And so the limit as x approaches infinity, what does a approach? a is, sorry, as x approaches infinity, what does n approach? Well, n is x divided by a, right? So n would also approach infinity. So this thing would be the same thing as just making our substitution."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "a is, sorry, as x approaches infinity, what does n approach? Well, n is x divided by a, right? So n would also approach infinity. So this thing would be the same thing as just making our substitution. The limit as n approaches infinity of 1 plus a over x, I mean the substitution is 1 over n, and x is, by this substitution, n times a. And this is going to be the same thing as the limit as n approaches infinity of 1 plus 1 over n to the n, all of that, to the a. And since there's no n out here, we could just take the limit of this and then take that to the a power, so that's going to be equal to the limit as n approaches infinity of 1 plus 1 over n to the nth power, all of that, to the a."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this thing would be the same thing as just making our substitution. The limit as n approaches infinity of 1 plus a over x, I mean the substitution is 1 over n, and x is, by this substitution, n times a. And this is going to be the same thing as the limit as n approaches infinity of 1 plus 1 over n to the n, all of that, to the a. And since there's no n out here, we could just take the limit of this and then take that to the a power, so that's going to be equal to the limit as n approaches infinity of 1 plus 1 over n to the nth power, all of that, to the a. And this is our definition, or one of the ways to get to e, if you watch the videos on compound interests and all of that. This is how we got to e. And if you try it on your calculator, just try larger and larger n's here, and you'll get e. So this is equal to, this inner part is equal to e. So, and we raised it to the a power, so it's equal to e to the a. So hopefully you're pretty satisfied that this limit is equal to e to the a."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And since there's no n out here, we could just take the limit of this and then take that to the a power, so that's going to be equal to the limit as n approaches infinity of 1 plus 1 over n to the nth power, all of that, to the a. And this is our definition, or one of the ways to get to e, if you watch the videos on compound interests and all of that. This is how we got to e. And if you try it on your calculator, just try larger and larger n's here, and you'll get e. So this is equal to, this inner part is equal to e. So, and we raised it to the a power, so it's equal to e to the a. So hopefully you're pretty satisfied that this limit is equal to e to the a. And then one other toolkit I want on our belt, and I'll probably actually do the proof in the next video, the other toolkit is to recognize that x factorial, let me do this, x factorial over x minus k factorial is equal to x times x minus 1 times x minus 2, all the way down to times x minus k plus 1. And we've done this a lot of times, but this is the most abstract we've ever written it. I can give you a couple of examples."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So hopefully you're pretty satisfied that this limit is equal to e to the a. And then one other toolkit I want on our belt, and I'll probably actually do the proof in the next video, the other toolkit is to recognize that x factorial, let me do this, x factorial over x minus k factorial is equal to x times x minus 1 times x minus 2, all the way down to times x minus k plus 1. And we've done this a lot of times, but this is the most abstract we've ever written it. I can give you a couple of examples. And they'll be exactly, and just so you know, they'll be exactly k terms here. 1, 2, 3, this is the first term, second term, third term, all the way, and this is the kth term. And this is important to our derivation of the Poisson distribution."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "I can give you a couple of examples. And they'll be exactly, and just so you know, they'll be exactly k terms here. 1, 2, 3, this is the first term, second term, third term, all the way, and this is the kth term. And this is important to our derivation of the Poisson distribution. But just to make this in real numbers, if I had 7 factorial over 7 minus 2 factorial, that's equal to 7 times 6 times 5 times 4 times 3 times 2 times 1 over 2 times, no sorry, 7 minus 2, this is 5. So it's over 5 times 4 times 3 times 2 times 1. These cancel out, and you just have 7 times 6."}, {"video_title": "Poisson process 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is important to our derivation of the Poisson distribution. But just to make this in real numbers, if I had 7 factorial over 7 minus 2 factorial, that's equal to 7 times 6 times 5 times 4 times 3 times 2 times 1 over 2 times, no sorry, 7 minus 2, this is 5. So it's over 5 times 4 times 3 times 2 times 1. These cancel out, and you just have 7 times 6. And so it's 7, and then the last term is 7 minus 2 plus 1, which is 6. 7 minus 2 plus 1. And you had, in this example, k was 2, and you had exactly 2 terms."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We will now begin our journey into the world of statistics, which is really a way to understand or get our head around data. So statistics is all about data. And as we begin our journey into the world of statistics, we will be doing a lot of what we can call descriptive statistics. So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six. And we could write this as a mixed number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six. And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median, I'm running out of colors."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median, I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I will write median, I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the three and the four."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set. Kind of a crazy data set."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. We only have one three. But we have two ones. We have one six and one seven."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, we only have one four. We only have one three. But we have two ones. We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently."}, {"video_title": "Statistics intro Mean, median, and mode   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. The median is really good if you have some kind of crazy number out here that could have otherwise skewed the arithmetic mean. The mode could also be useful in situations like that, especially if you do have one number that's showing up a lot more frequently. Anyway, I'll leave you there."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And maybe, well, I mean, if you go into almost any scientific field, you might even argue it's the single most important concept. And I've actually told people that it's kind of sad that they don't cover this in the core curriculum. Everyone should know about this, because it touches on every single aspect of our lives. And that's the normal distribution, or the Gaussian distribution, or the bell curve. And just to kind of give you a preview of what it is, and my preview will actually make it seem pretty strange, but as we go through this video, hopefully you'll get a little bit more intuition of what it's all about. But the Gaussian distribution, or the normal distribution, they're just two words for the same thing. It was actually Gauss who came up with it."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's the normal distribution, or the Gaussian distribution, or the bell curve. And just to kind of give you a preview of what it is, and my preview will actually make it seem pretty strange, but as we go through this video, hopefully you'll get a little bit more intuition of what it's all about. But the Gaussian distribution, or the normal distribution, they're just two words for the same thing. It was actually Gauss who came up with it. I think he was studying astronomical phenomenon when he did. But it's a probability density function, just like we studied the Poisson distribution. It's just like that."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "It was actually Gauss who came up with it. I think he was studying astronomical phenomenon when he did. But it's a probability density function, just like we studied the Poisson distribution. It's just like that. And just to give you the preview, it looks like this. The probability of getting any x, and it's a class of probability distribution functions. Just like the binomial distribution is and the Poisson distribution is, based on a bunch of parameters."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's just like that. And just to give you the preview, it looks like this. The probability of getting any x, and it's a class of probability distribution functions. Just like the binomial distribution is and the Poisson distribution is, based on a bunch of parameters. But it's equal to, and this is how you would traditionally see it written in a lot of textbooks. And if we have time, I'd like to rearrange the algebra, just so you get a little bit more intuition of how it all works out. Or maybe we could get some insights on where it all came from."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just like the binomial distribution is and the Poisson distribution is, based on a bunch of parameters. But it's equal to, and this is how you would traditionally see it written in a lot of textbooks. And if we have time, I'd like to rearrange the algebra, just so you get a little bit more intuition of how it all works out. Or maybe we could get some insights on where it all came from. I'm not going to prove it in this video. That's a little bit beyond our scope. Although I do want to do it."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or maybe we could get some insights on where it all came from. I'm not going to prove it in this video. That's a little bit beyond our scope. Although I do want to do it. And there's actually some really neat mathematics that might show up in, if you're a math lead, there's something called Stirling's formula, which you might want to do a Wikipedia search on, which is really fascinating. It approximates factorials with essentially a continuous function, but I won't go into that right now. But the normal distribution is 1 over, this is how it's normally written, the standard deviation times the square root of 2 pi times e to the minus 1 half."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Although I do want to do it. And there's actually some really neat mathematics that might show up in, if you're a math lead, there's something called Stirling's formula, which you might want to do a Wikipedia search on, which is really fascinating. It approximates factorials with essentially a continuous function, but I won't go into that right now. But the normal distribution is 1 over, this is how it's normally written, the standard deviation times the square root of 2 pi times e to the minus 1 half. Well, I like to write it this way. It's easier to remember. Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution squared."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the normal distribution is 1 over, this is how it's normally written, the standard deviation times the square root of 2 pi times e to the minus 1 half. Well, I like to write it this way. It's easier to remember. Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution squared. And so if you think about it, actually, this is a good thing to just notice right now. This is how far I'm from the mean, and we're dividing that by the standard deviation of whatever our distribution is. And this is a preview of actually a normal distribution that I've plotted."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution squared. And so if you think about it, actually, this is a good thing to just notice right now. This is how far I'm from the mean, and we're dividing that by the standard deviation of whatever our distribution is. And this is a preview of actually a normal distribution that I've plotted. The purple line right here is the normal distribution. And just so you know, the whole exercise here, I know I jump around a little bit, is to show you that the normal distribution is a good approximation for the binomial distribution and vice versa. If the binomial distribution, if you take enough trials in your binomial distribution, we'll touch on that in a second, but the intuition of this term right here I think is interesting."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is a preview of actually a normal distribution that I've plotted. The purple line right here is the normal distribution. And just so you know, the whole exercise here, I know I jump around a little bit, is to show you that the normal distribution is a good approximation for the binomial distribution and vice versa. If the binomial distribution, if you take enough trials in your binomial distribution, we'll touch on that in a second, but the intuition of this term right here I think is interesting. Because we're saying how far are we away from the mean? We're dividing by the standard deviation. We're saying, so this whole term right here is how many standard deviations we are away from the mean."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If the binomial distribution, if you take enough trials in your binomial distribution, we'll touch on that in a second, but the intuition of this term right here I think is interesting. Because we're saying how far are we away from the mean? We're dividing by the standard deviation. We're saying, so this whole term right here is how many standard deviations we are away from the mean. And this is actually called a standard z-score. One thing I've found in statistics is there's a lot of words and a lot of definitions and they all sound very fancy. The standard z-score."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're saying, so this whole term right here is how many standard deviations we are away from the mean. And this is actually called a standard z-score. One thing I've found in statistics is there's a lot of words and a lot of definitions and they all sound very fancy. The standard z-score. But the underlying concept is pretty straightforward. Let's say I had a probability distribution and I get some x value that's out here and it's 3 and 1 half standard deviations away from the mean. Then it's standard z-score is 3 and 1 half."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "The standard z-score. But the underlying concept is pretty straightforward. Let's say I had a probability distribution and I get some x value that's out here and it's 3 and 1 half standard deviations away from the mean. Then it's standard z-score is 3 and 1 half. But anyway, let's focus on the purpose of this video. So that's what the normal distribution, I guess the probability density function for the normal distribution looks like. But how did it get there?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then it's standard z-score is 3 and 1 half. But anyway, let's focus on the purpose of this video. So that's what the normal distribution, I guess the probability density function for the normal distribution looks like. But how did it get there? I guess even more importantly, by the end of this video you should at least feel comfortable that this is a good approximation for the binomial distribution if you take enough trials. And that's what's fascinating about the normal distribution is that if you have the sum, and I'll do a whole other video on the central limit theorem, but if you have the sum of many, many independent trials approaching infinity, that the distribution of those, even though the distribution of each of those trials might have been non-normal, but the distribution of the sum of all of those trials approaches the normal distribution. And I'll talk more about that later."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But how did it get there? I guess even more importantly, by the end of this video you should at least feel comfortable that this is a good approximation for the binomial distribution if you take enough trials. And that's what's fascinating about the normal distribution is that if you have the sum, and I'll do a whole other video on the central limit theorem, but if you have the sum of many, many independent trials approaching infinity, that the distribution of those, even though the distribution of each of those trials might have been non-normal, but the distribution of the sum of all of those trials approaches the normal distribution. And I'll talk more about that later. But that's why it's such a good distribution to kind of assume for a lot of underlying phenomenon if you're kind of modeling weather patterns or drug interactions. And we'll talk about where it might work well and where it might not work so well. Like sometimes people might assume things like a normal distribution in finance and we see in the financial crisis that's led to a lot of things blowing up."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll talk more about that later. But that's why it's such a good distribution to kind of assume for a lot of underlying phenomenon if you're kind of modeling weather patterns or drug interactions. And we'll talk about where it might work well and where it might not work so well. Like sometimes people might assume things like a normal distribution in finance and we see in the financial crisis that's led to a lot of things blowing up. But anyway, let's get back to this. And this is a spreadsheet right here. I just made a black background."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Like sometimes people might assume things like a normal distribution in finance and we see in the financial crisis that's led to a lot of things blowing up. But anyway, let's get back to this. And this is a spreadsheet right here. I just made a black background. And you can download it at, let me write it right here, at khanacademy.org slash downloads. And actually if you just do that, you'll see all of the downloads, I haven't put it there yet, I'm going to do it right after I record the video. This downloads slash normal distribution."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I just made a black background. And you can download it at, let me write it right here, at khanacademy.org slash downloads. And actually if you just do that, you'll see all of the downloads, I haven't put it there yet, I'm going to do it right after I record the video. This downloads slash normal distribution. Normal distribution. That's distribution.xls. If you just go up to khanacademy.org slash downloads slash, you'll see all of the things there."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "This downloads slash normal distribution. Normal distribution. That's distribution.xls. If you just go up to khanacademy.org slash downloads slash, you'll see all of the things there. And you'll see that this spreadsheet. And I encourage you to play with it. And maybe do other spreadsheets where you experiment with it."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you just go up to khanacademy.org slash downloads slash, you'll see all of the things there. And you'll see that this spreadsheet. And I encourage you to play with it. And maybe do other spreadsheets where you experiment with it. So this spreadsheet, what we do is we're doing a game where let's say I'm sitting, I'm on a street, and I flip a coin. I flip a completely fair coin. If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And maybe do other spreadsheets where you experiment with it. So this spreadsheet, what we do is we're doing a game where let's say I'm sitting, I'm on a street, and I flip a coin. I flip a completely fair coin. If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left. And if I get a tails, I take a step to the right. So in general, I always have a 50, this is a completely fair coin. I have a 50% chance of taking a step to the left."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left. And if I get a tails, I take a step to the right. So in general, I always have a 50, this is a completely fair coin. I have a 50% chance of taking a step to the left. And I have a 50% chance of taking a step to the right. So your intuition there is, if I told you I took 1,000 flips of the coin, you're going to keep going left and right. If by chance you get a bunch of heads, you might end up really kind of moving over to the left."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I have a 50% chance of taking a step to the left. And I have a 50% chance of taking a step to the right. So your intuition there is, if I told you I took 1,000 flips of the coin, you're going to keep going left and right. If by chance you get a bunch of heads, you might end up really kind of moving over to the left. If you get a bunch of tails, you might move over to the right. And we learned already that the odds of getting a bunch of tails, or many, many, many, many more tails than heads, is a lot lower than things kind of being equal or close to equal. And right here, what I've done, this is the mean number of left steps."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If by chance you get a bunch of heads, you might end up really kind of moving over to the left. If you get a bunch of tails, you might move over to the right. And we learned already that the odds of getting a bunch of tails, or many, many, many, many more tails than heads, is a lot lower than things kind of being equal or close to equal. And right here, what I've done, this is the mean number of left steps. And all I did is I got the probability, and we figured out the mean of the binomial distribution. The mean of the binomial distribution is essentially the probability of taking a left step times the total number of trials. And so that's equal to 5."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And right here, what I've done, this is the mean number of left steps. And all I did is I got the probability, and we figured out the mean of the binomial distribution. The mean of the binomial distribution is essentially the probability of taking a left step times the total number of trials. And so that's equal to 5. That's where that number comes from. And then the variance, and I'm not sure if I went over this. I need to prove this to you if I have."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so that's equal to 5. That's where that number comes from. And then the variance, and I'm not sure if I went over this. I need to prove this to you if I have. And I'll make a whole other video on the variance of the binomial distribution. But the variance is essentially equal to the number of trials, 10, times the probability of taking the left step, or kind of a successful trial. That's what I'm defining left as a successful trial, but could be right as well."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I need to prove this to you if I have. And I'll make a whole other video on the variance of the binomial distribution. But the variance is essentially equal to the number of trials, 10, times the probability of taking the left step, or kind of a successful trial. That's what I'm defining left as a successful trial, but could be right as well. Times the probability of 1 minus a successful trial, or non-successful trial. In this case, they're equally probable, and that's where I got the 2.5 from. And that's all in the spreadsheet."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what I'm defining left as a successful trial, but could be right as well. Times the probability of 1 minus a successful trial, or non-successful trial. In this case, they're equally probable, and that's where I got the 2.5 from. And that's all in the spreadsheet. If you actually click on the cell and look at the actual formula, I did that. Although sometimes when you see it in Excel, it's a little bit confusing. And this is just the square root of that number, right?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's all in the spreadsheet. If you actually click on the cell and look at the actual formula, I did that. Although sometimes when you see it in Excel, it's a little bit confusing. And this is just the square root of that number, right? The standard deviation is just the square root of the variance. So that's just the square root of 2.5. And so if you look here, this says, OK, what is the probability that I take 0 steps?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is just the square root of that number, right? The standard deviation is just the square root of the variance. So that's just the square root of 2.5. And so if you look here, this says, OK, what is the probability that I take 0 steps? So I take a total of 10 steps, just to understand the spreadsheet. So if I take a total of 10 steps, what is the probability that I take 0 left steps? Well, and just to clarify, if I take 0 left steps, that means I must have taken 10 right steps."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if you look here, this says, OK, what is the probability that I take 0 steps? So I take a total of 10 steps, just to understand the spreadsheet. So if I take a total of 10 steps, what is the probability that I take 0 left steps? Well, and just to clarify, if I take 0 left steps, that means I must have taken 10 right steps. And I calculate this probability, and I should have drawn maybe a line here. I calculate this using the binomial distribution. And how do I do that?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, and just to clarify, if I take 0 left steps, that means I must have taken 10 right steps. And I calculate this probability, and I should have drawn maybe a line here. I calculate this using the binomial distribution. And how do I do that? I say, what is the probability that I take a total of 10 steps, so let me actually switch colors just to make things interesting. Let's see, do they have a purple here? I'll do a blue."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And how do I do that? I say, what is the probability that I take a total of 10 steps, so let me actually switch colors just to make things interesting. Let's see, do they have a purple here? I'll do a blue. So blue for binomial. So what I have here is, how many total steps? So there's a total of 10 steps, so 10 factorial."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll do a blue. So blue for binomial. So what I have here is, how many total steps? So there's a total of 10 steps, so 10 factorial. That's kind of the number of trials I have. Of that, I'm choosing 0 to go left. So 0 factorial divided by 10 minus 0 factorial, right?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a total of 10 steps, so 10 factorial. That's kind of the number of trials I have. Of that, I'm choosing 0 to go left. So 0 factorial divided by 10 minus 0 factorial, right? This is 10 choose 0. I'm choosing 0 left steps of the total 10 steps I'm taking times the probability of 0 left steps. So it's the probability of a left step, I'm only taking 0 of them, times the probability of a right step, and I'm taking 10 of those."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 0 factorial divided by 10 minus 0 factorial, right? This is 10 choose 0. I'm choosing 0 left steps of the total 10 steps I'm taking times the probability of 0 left steps. So it's the probability of a left step, I'm only taking 0 of them, times the probability of a right step, and I'm taking 10 of those. So that's where this number came from, this 0.001. That's what the binomial distribution tells us. And then this one, similarly, is 10 factorial over 1 factorial over 10 minus 1 factorial, and that's how I get that one."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's the probability of a left step, I'm only taking 0 of them, times the probability of a right step, and I'm taking 10 of those. So that's where this number came from, this 0.001. That's what the binomial distribution tells us. And then this one, similarly, is 10 factorial over 1 factorial over 10 minus 1 factorial, and that's how I get that one. And once again, if you click on the actual cell, you'll see that explained. So we've done this multiple times, it's just the binomial calculation. And then right here, after this line right here, you can almost ignore it."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then this one, similarly, is 10 factorial over 1 factorial over 10 minus 1 factorial, and that's how I get that one. And once again, if you click on the actual cell, you'll see that explained. So we've done this multiple times, it's just the binomial calculation. And then right here, after this line right here, you can almost ignore it. And I did that so that I can do a bunch of different scenarios. So for example, if I were to go to my spreadsheet, and instead of doing 10 steps, I wanted to do 20 steps, then everything changes. And that's why down here, after you get to a certain point, the whole thing just repeats."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then right here, after this line right here, you can almost ignore it. And I did that so that I can do a bunch of different scenarios. So for example, if I were to go to my spreadsheet, and instead of doing 10 steps, I wanted to do 20 steps, then everything changes. And that's why down here, after you get to a certain point, the whole thing just repeats. And I'll let you think about why I do that. Maybe I should have made a cleaner spreadsheet. But it doesn't affect the scatter plot chart that I did."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's why down here, after you get to a certain point, the whole thing just repeats. And I'll let you think about why I do that. Maybe I should have made a cleaner spreadsheet. But it doesn't affect the scatter plot chart that I did. And so this plot in blue, and you can't see it because the purple is almost right over it. Actually, let me make it smaller so that you can see. So if I only took 6 steps, well, it's still hard to see the difference between the two."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it doesn't affect the scatter plot chart that I did. And so this plot in blue, and you can't see it because the purple is almost right over it. Actually, let me make it smaller so that you can see. So if I only took 6 steps, well, it's still hard to see the difference between the two. And then once again, the whole point of this is to see that the normal distribution is a good approximation, but they're so close that you can't even see the difference on mine. If you only took 4 steps, OK, I think you can see here, the blue here is definitely, let me get my screen drawer on. So let me draw this."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I only took 6 steps, well, it's still hard to see the difference between the two. And then once again, the whole point of this is to see that the normal distribution is a good approximation, but they're so close that you can't even see the difference on mine. If you only took 4 steps, OK, I think you can see here, the blue here is definitely, let me get my screen drawer on. So let me draw this. The blue curve is right around there. So this is the binomial. There's only a few points here."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me draw this. The blue curve is right around there. So this is the binomial. There's only a few points here. The points only go up to here. This is if I take 0 steps left, 1 step left, 2 steps left, 3 steps left, 4 steps left, and then I plot it. And then I say, what's the probability using the binomial distribution?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's only a few points here. The points only go up to here. This is if I take 0 steps left, 1 step left, 2 steps left, 3 steps left, 4 steps left, and then I plot it. And then I say, what's the probability using the binomial distribution? And then this is my final position. If I take 0 steps to the left, then I take 4 steps to the right, so my final position is at 4. So that's this scenario right here."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then I say, what's the probability using the binomial distribution? And then this is my final position. If I take 0 steps to the left, then I take 4 steps to the right, so my final position is at 4. So that's this scenario right here. Let me switch my color back to yellow. It's easier to see. If I take 4 steps to the left, I take 0 steps to the right, and so my final position is going to be at minus 4."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's this scenario right here. Let me switch my color back to yellow. It's easier to see. If I take 4 steps to the left, I take 0 steps to the right, and so my final position is going to be at minus 4. It's going to be here. If I take an equal amount of both, so that's this scenario, then I'm neutral. I'm just stuck in the middle right here."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I take 4 steps to the left, I take 0 steps to the right, and so my final position is going to be at minus 4. It's going to be here. If I take an equal amount of both, so that's this scenario, then I'm neutral. I'm just stuck in the middle right here. I take 2 steps to the right, and then I take 2 steps to the left, or vice versa. I take 2 steps to the left, and then I take 2 steps right, and I end up right there. Hopefully that makes a little sense of how this is going to seem."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just stuck in the middle right here. I take 2 steps to the right, and then I take 2 steps to the left, or vice versa. I take 2 steps to the left, and then I take 2 steps right, and I end up right there. Hopefully that makes a little sense of how this is going to seem. My phone is ringing. No, I'll ignore that. Because the normal distribution is so important."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Hopefully that makes a little sense of how this is going to seem. My phone is ringing. No, I'll ignore that. Because the normal distribution is so important. And actually, my 9-week-old son is watching, so this is the first time I have a live audience. But I thought he might pick up something about the normal distribution. But anyway, the blue line right here, and I'll trace it maybe in yellow just so you can see it, is the plot of the binomial distribution."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because the normal distribution is so important. And actually, my 9-week-old son is watching, so this is the first time I have a live audience. But I thought he might pick up something about the normal distribution. But anyway, the blue line right here, and I'll trace it maybe in yellow just so you can see it, is the plot of the binomial distribution. And I connected the lines, but you see the binomial distribution look something more like this. Where this is the probability of getting to minus 4. This is the probability of going to minus 2."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But anyway, the blue line right here, and I'll trace it maybe in yellow just so you can see it, is the plot of the binomial distribution. And I connected the lines, but you see the binomial distribution look something more like this. Where this is the probability of getting to minus 4. This is the probability of going to minus 2. This right here is the probability of ending up nowhere. And then this is the probability. Actually, no, the point is right here."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the probability of going to minus 2. This right here is the probability of ending up nowhere. And then this is the probability. Actually, no, the point is right here. This is the probability of ending up 2 to the right. And this is the probability of ending 4 to the right. This is the binomial distribution."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, no, the point is right here. This is the probability of ending up 2 to the right. And this is the probability of ending 4 to the right. This is the binomial distribution. I just plotted these points right here. This is 0.375. This is 0.375."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the binomial distribution. I just plotted these points right here. This is 0.375. This is 0.375. That's the height of that. Now, what I wanted to show you is that the normal distribution approximates the binomial distribution. So this right here, I wanted to say, what does the normal distribution tell me is the probability of ending up with exactly 0 left steps."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 0.375. That's the height of that. Now, what I wanted to show you is that the normal distribution approximates the binomial distribution. So this right here, I wanted to say, what does the normal distribution tell me is the probability of ending up with exactly 0 left steps. And then this is a little bit tricky because the binomial distribution is a discrete probability distribution. You could just look at this chart or look here and you say, what is the probability of having exactly, let's say, 1 left step and 3 right steps, which puts me right here. Well, you just look at this chart and you say, oh, that puts me right there."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right here, I wanted to say, what does the normal distribution tell me is the probability of ending up with exactly 0 left steps. And then this is a little bit tricky because the binomial distribution is a discrete probability distribution. You could just look at this chart or look here and you say, what is the probability of having exactly, let's say, 1 left step and 3 right steps, which puts me right here. Well, you just look at this chart and you say, oh, that puts me right there. I just read that probability. It's actually 0.25. And I say, oh, I have a 25% chance of ending up 2 steps to the right."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, you just look at this chart and you say, oh, that puts me right there. I just read that probability. It's actually 0.25. And I say, oh, I have a 25% chance of ending up 2 steps to the right. There's a 25% chance. The normal distribution function is a continuous probability distribution. So it's a continuous curve."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I say, oh, I have a 25% chance of ending up 2 steps to the right. There's a 25% chance. The normal distribution function is a continuous probability distribution. So it's a continuous curve. It looks like that. It's the bell curve. And it goes off to infinity and starts approaching 0 on both sides."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's a continuous curve. It looks like that. It's the bell curve. And it goes off to infinity and starts approaching 0 on both sides. It looks something like that. But this is a continuous probability distribution. You can't just take a point here and you say, what's the probability that I end up 2 feet to the right?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it goes off to infinity and starts approaching 0 on both sides. It looks something like that. But this is a continuous probability distribution. You can't just take a point here and you say, what's the probability that I end up 2 feet to the right? Because if you just say that, the actual probability of being exactly, and you should watch my video on probability density functions, but the probability of being exactly 2 feet to the right, exactly, I mean, I'm talking to the atom, is close to 0. So you actually have to specify a range around this. And what I assume in this is I assume that within, essentially, a half a foot in either direction, if we're talking about feet."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can't just take a point here and you say, what's the probability that I end up 2 feet to the right? Because if you just say that, the actual probability of being exactly, and you should watch my video on probability density functions, but the probability of being exactly 2 feet to the right, exactly, I mean, I'm talking to the atom, is close to 0. So you actually have to specify a range around this. And what I assume in this is I assume that within, essentially, a half a foot in either direction, if we're talking about feet. So to figure that out, what I did here is I took the value of the probability density function there, and I'll show you how I evaluated that. And then I multiplied that by 1. So it gives me this area."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what I assume in this is I assume that within, essentially, a half a foot in either direction, if we're talking about feet. So to figure that out, what I did here is I took the value of the probability density function there, and I'll show you how I evaluated that. And then I multiplied that by 1. So it gives me this area. And I use that as an approximation for this area. If you really wanted to be particular about it, what you would do is you would take the integral of this curve between this point and this point as a better approximation. We'll do that in the future."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it gives me this area. And I use that as an approximation for this area. If you really wanted to be particular about it, what you would do is you would take the integral of this curve between this point and this point as a better approximation. We'll do that in the future. But right now, I just want to give you the intuition that the binomial distribution really does converge to the normal distribution. So what is this number right here? Well, I said, what is the probability?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "We'll do that in the future. But right now, I just want to give you the intuition that the binomial distribution really does converge to the normal distribution. So what is this number right here? Well, I said, what is the probability? Well, let me do something like that. Let's say this one right here, because I don't want to use the 0. So what is the probability that I take one left step?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, I said, what is the probability? Well, let me do something like that. Let's say this one right here, because I don't want to use the 0. So what is the probability that I take one left step? I kind of used left steps as a success. So what is the probability of 1? And that equaled 1 over the standard deviation."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the probability that I take one left step? I kind of used left steps as a success. So what is the probability of 1? And that equaled 1 over the standard deviation. When I only took 4 steps, the standard deviation was 1. So 1 over 1. Actually, let me change this, because I think it might be let me change it to a higher number."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that equaled 1 over the standard deviation. When I only took 4 steps, the standard deviation was 1. So 1 over 1. Actually, let me change this, because I think it might be let me change it to a higher number. Let's see, if I make this, I don't know. Let me go back to the example where I'm at 10. All right, so if this is at 10, let me go back to my drawing tool."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me change this, because I think it might be let me change it to a higher number. Let's see, if I make this, I don't know. Let me go back to the example where I'm at 10. All right, so if this is at 10, let me go back to my drawing tool. So this calculation right here, let me do this calculation. Actually, even better, let me do this calculation right here. All right, so what's the probability that I have 2 left steps, and if I have 2 left steps, I took a total of 10 steps, so I'm going to have 8 right steps, and that puts me 6 to the right."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, so if this is at 10, let me go back to my drawing tool. So this calculation right here, let me do this calculation. Actually, even better, let me do this calculation right here. All right, so what's the probability that I have 2 left steps, and if I have 2 left steps, I took a total of 10 steps, so I'm going to have 8 right steps, and that puts me 6 to the right. So that's this point right here. So what's that probability? How do I figure this out using the probability density function?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, so what's the probability that I have 2 left steps, and if I have 2 left steps, I took a total of 10 steps, so I'm going to have 8 right steps, and that puts me 6 to the right. So that's this point right here. So what's that probability? How do I figure this out using the probability density function? How do I figure this out? Well, I say the probability of taking 2 left steps, that's how I calculated it. If you actually click on the cell, you'll see that, is equal to 1 over the standard deviation, 1.581."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "How do I figure this out using the probability density function? How do I figure this out? Well, I say the probability of taking 2 left steps, that's how I calculated it. If you actually click on the cell, you'll see that, is equal to 1 over the standard deviation, 1.581. And I just directly referenced the cell there. Divide it times the square root of 2 pi. And I always go on in awe of the whole notion of e to the i pi is equal to negative 1 and all of that."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you actually click on the cell, you'll see that, is equal to 1 over the standard deviation, 1.581. And I just directly referenced the cell there. Divide it times the square root of 2 pi. And I always go on in awe of the whole notion of e to the i pi is equal to negative 1 and all of that. But this is another amazing thing, that all of a sudden we have this, as we take many, many, many trials, we have this formula that has e and pi in it and square roots. But once again, these two numbers just keep showing up, and it tells you something about the order of the universe with a capital O. But let's see, so times e to the minus 1 half times x."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I always go on in awe of the whole notion of e to the i pi is equal to negative 1 and all of that. But this is another amazing thing, that all of a sudden we have this, as we take many, many, many trials, we have this formula that has e and pi in it and square roots. But once again, these two numbers just keep showing up, and it tells you something about the order of the universe with a capital O. But let's see, so times e to the minus 1 half times x. Well, the x is what we're trying to calculate. 2 successes. So to have exactly 2 left, so it's 2 minus the mean."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But let's see, so times e to the minus 1 half times x. Well, the x is what we're trying to calculate. 2 successes. So to have exactly 2 left, so it's 2 minus the mean. So the mean is 5. Divided by the standard deviation, divided by 1.581. All of that squared."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So to have exactly 2 left, so it's 2 minus the mean. So the mean is 5. Divided by the standard deviation, divided by 1.581. All of that squared. That's where this calculation came from. And so I told you in the last one, I mean, this just tells you, this right here just tells me this value up here. And I assume that the probability, if I want to know this exact probability, it's the area of this."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of that squared. That's where this calculation came from. And so I told you in the last one, I mean, this just tells you, this right here just tells me this value up here. And I assume that the probability, if I want to know this exact probability, it's the area of this. And if I just take one line, the area is 0. So to be exactly 2 feet away using the, remember, I mean, in this case, you can only be 2 feet away because we're taking very exact steps. But what the normal distribution is, it's a continuous probability density function."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I assume that the probability, if I want to know this exact probability, it's the area of this. And if I just take one line, the area is 0. So to be exactly 2 feet away using the, remember, I mean, in this case, you can only be 2 feet away because we're taking very exact steps. But what the normal distribution is, it's a continuous probability density function. So it can tell us, what's the probability of being 2.183 feet away? Which obviously can only happen if we're kind of taking infinitely small steps every time. But that's what its use is."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what the normal distribution is, it's a continuous probability density function. So it can tell us, what's the probability of being 2.183 feet away? Which obviously can only happen if we're kind of taking infinitely small steps every time. But that's what its use is. It happens kind of when you start taking an infinite number of steps. But it can approximate the discrete. And the way I approximate it is I say, oh, what's the probability of being within a foot of that?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But that's what its use is. It happens kind of when you start taking an infinite number of steps. But it can approximate the discrete. And the way I approximate it is I say, oh, what's the probability of being within a foot of that? And so I multiply this height, which I calculate here, times 1. So let's say this has a base of 1 to calculate this area, which I use as an approximation. So you just multiply that times 1, and that's what you get here."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the way I approximate it is I say, oh, what's the probability of being within a foot of that? And so I multiply this height, which I calculate here, times 1. So let's say this has a base of 1 to calculate this area, which I use as an approximation. So you just multiply that times 1, and that's what you get here. And I just want to show you, I mean, even with just 10 trials, the curves for the normal distribution here is in purple, and the binomial distribution is in blue. So they're almost right on top of each other. I mean, when the number of steps I took was a little bit smaller, and as you take many, many, many, many more steps, they almost converge right on top of each other."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you just multiply that times 1, and that's what you get here. And I just want to show you, I mean, even with just 10 trials, the curves for the normal distribution here is in purple, and the binomial distribution is in blue. So they're almost right on top of each other. I mean, when the number of steps I took was a little bit smaller, and as you take many, many, many, many more steps, they almost converge right on top of each other. And I encourage you to play with this spreadsheet. And actually, let me show you that they converge. So I made another, there's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I mean, when the number of steps I took was a little bit smaller, and as you take many, many, many, many more steps, they almost converge right on top of each other. And I encourage you to play with this spreadsheet. And actually, let me show you that they converge. So I made another, there's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence. And I did, this is the same thing, but I just wanted to show you what happens at any given point. So let's say that I wanted to, let me explain this part, this spreadsheet to you. So this is, what's the probability of moving left, right, this is, so this is just saying, I'm just fixing a point, what's the probability, and you could change this, of my final position being 10."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I made another, there's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence. And I did, this is the same thing, but I just wanted to show you what happens at any given point. So let's say that I wanted to, let me explain this part, this spreadsheet to you. So this is, what's the probability of moving left, right, this is, so this is just saying, I'm just fixing a point, what's the probability, and you could change this, of my final position being 10. And then this essentially tells you that if I take 10 moves, then for my final position to be 10 to the right, I have to take 10 right moves and 0 left moves. That's a typo right there, it should be moves, not moves-ed. If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is, what's the probability of moving left, right, this is, so this is just saying, I'm just fixing a point, what's the probability, and you could change this, of my final position being 10. And then this essentially tells you that if I take 10 moves, then for my final position to be 10 to the right, I have to take 10 right moves and 0 left moves. That's a typo right there, it should be moves, not moves-ed. If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves. And likewise, if I take a total of 80 moves, if I take 80 flips of my coin to make me go left to right, in order to end up 10 to the right, I have to take 45 right moves and 35 left moves, in any order. And it'll end up with 10 to the right. So what I want to figure out is, as I start taking a bunch of total moves, it's like my total moves, I mean here I max it out at 170, but you could kind of say, if I started flipping this coin an infinite number of times, I want to figure out what's the probability that my final position is 10 to the right."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves. And likewise, if I take a total of 80 moves, if I take 80 flips of my coin to make me go left to right, in order to end up 10 to the right, I have to take 45 right moves and 35 left moves, in any order. And it'll end up with 10 to the right. So what I want to figure out is, as I start taking a bunch of total moves, it's like my total moves, I mean here I max it out at 170, but you could kind of say, if I started flipping this coin an infinite number of times, I want to figure out what's the probability that my final position is 10 to the right. And what I want to show you is that as you take more and more moves, the normal distribution becomes a better and better approximation for the binomial distribution. So right here, this calculates the binomial probability, just the way we did it before. And you could look at the cell to figure it out."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what I want to figure out is, as I start taking a bunch of total moves, it's like my total moves, I mean here I max it out at 170, but you could kind of say, if I started flipping this coin an infinite number of times, I want to figure out what's the probability that my final position is 10 to the right. And what I want to show you is that as you take more and more moves, the normal distribution becomes a better and better approximation for the binomial distribution. So right here, this calculates the binomial probability, just the way we did it before. And you could look at the cell to figure it out. You know, this says 10, you know, I use left moves as a success. So this is 10 choose 0, and we know what that is, it's 10 factorial over 0 factorial over 10 minus 0 factorial, times 0.5 to the 0, times 0.5 to the 10. That's where that number comes from."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you could look at the cell to figure it out. You know, this says 10, you know, I use left moves as a success. So this is 10 choose 0, and we know what that is, it's 10 factorial over 0 factorial over 10 minus 0 factorial, times 0.5 to the 0, times 0.5 to the 10. That's where that number comes from. If I go to, let's say, this one right here, this one right here is calculated. Actually, let me write it out, because I think it's interesting. I have a total of 60 total moves, so it's 60 factorial over, I have to have 25 left moves, so 25 factorial, 60 minus 25 factorial, times the probability of a left move, and I have 25 of them, times the probability of a right move, and I have 35 of those."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's where that number comes from. If I go to, let's say, this one right here, this one right here is calculated. Actually, let me write it out, because I think it's interesting. I have a total of 60 total moves, so it's 60 factorial over, I have to have 25 left moves, so 25 factorial, 60 minus 25 factorial, times the probability of a left move, and I have 25 of them, times the probability of a right move, and I have 35 of those. So that's just what the binomial probability distribution will tell us. And then it figures out the mean and the variance for each of those circumstances, and you could look at the formula, but the mean is just the probability of having a left move times the total number of moves. The variance is probability of left times probability of right times total number of moves."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I have a total of 60 total moves, so it's 60 factorial over, I have to have 25 left moves, so 25 factorial, 60 minus 25 factorial, times the probability of a left move, and I have 25 of them, times the probability of a right move, and I have 35 of those. So that's just what the binomial probability distribution will tell us. And then it figures out the mean and the variance for each of those circumstances, and you could look at the formula, but the mean is just the probability of having a left move times the total number of moves. The variance is probability of left times probability of right times total number of moves. And then the normal probability, once again, I just use the normal probability, I just use the, so I approximate it the same way. So for example, in this situation right here, and Excel has a normal distribution function, but I actually typed in the formula because I wanted to kind of see what was under the covers for that function that Excel actually has. So I actually say, what's the probability of 45 left moves?"}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "The variance is probability of left times probability of right times total number of moves. And then the normal probability, once again, I just use the normal probability, I just use the, so I approximate it the same way. So for example, in this situation right here, and Excel has a normal distribution function, but I actually typed in the formula because I wanted to kind of see what was under the covers for that function that Excel actually has. So I actually say, what's the probability of 45 left moves? So I say the probability of 45 left moves is equal to 1 over the standard deviation. So in this situation, the standard deviation is the square root of 25, so it's 5 times 2 pi times e to the minus 1 half times 45 minus the mean, minus 50, over the standard deviation, which we figured out was 5, squared. So that tells me the value of what the normal distribution tells me for this situation with this standard deviation and this mean."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I actually say, what's the probability of 45 left moves? So I say the probability of 45 left moves is equal to 1 over the standard deviation. So in this situation, the standard deviation is the square root of 25, so it's 5 times 2 pi times e to the minus 1 half times 45 minus the mean, minus 50, over the standard deviation, which we figured out was 5, squared. So that tells me the value of what the normal distribution tells me for this situation with this standard deviation and this mean. And then I multiply that by 1, so you don't see that in formula, I don't actually write times 1, to actually figure out the area under the curve, right? Because remember, it's a continuous distribution function like that. This right here just gives me the value."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that tells me the value of what the normal distribution tells me for this situation with this standard deviation and this mean. And then I multiply that by 1, so you don't see that in formula, I don't actually write times 1, to actually figure out the area under the curve, right? Because remember, it's a continuous distribution function like that. This right here just gives me the value. But to figure out the probability of being within a foot of it, I have to multiply it by 1. Or I'm approximating, really. I really should take the integral from there to there, but this little rectangle is a pretty good approximation."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right here just gives me the value. But to figure out the probability of being within a foot of it, I have to multiply it by 1. Or I'm approximating, really. I really should take the integral from there to there, but this little rectangle is a pretty good approximation. And this chart, I show you that as the total number of moves gets larger and larger, the difference between what the normal probability distribution tells us and the binomial probability distribution tells us gets smaller and smaller in terms of the probability of you ending up 10 moves to the right. And you can change this number here. Let me change it just to show you."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "I really should take the integral from there to there, but this little rectangle is a pretty good approximation. And this chart, I show you that as the total number of moves gets larger and larger, the difference between what the normal probability distribution tells us and the binomial probability distribution tells us gets smaller and smaller in terms of the probability of you ending up 10 moves to the right. And you can change this number here. Let me change it just to show you. You could say, what's the probability of being 15 moves to the right? And actually, no, that one doesn't. 15 moves to the right."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me change it just to show you. You could say, what's the probability of being 15 moves to the right? And actually, no, that one doesn't. 15 moves to the right. It looks like it kind of, no, that's not. Let's see, 10. And if I go 12, it converges."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "15 moves to the right. It looks like it kind of, no, that's not. Let's see, 10. And if I go 12, it converges. And then if you go to 13, I think that something's happening with the floating point error. Because when you get to large factorials, I think something weird happens out here. But like if you do 3, something weird is happening."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if I go 12, it converges. And then if you go to 13, I think that something's happening with the floating point error. Because when you get to large factorials, I think something weird happens out here. But like if you do 3, something weird is happening. 5, 10. Yeah, you maybe have to just get even further out. So for 10, you can see clearly that it converges."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "But like if you do 3, something weird is happening. 5, 10. Yeah, you maybe have to just get even further out. So for 10, you can see clearly that it converges. And I'll try to figure out why I was getting those weird sawtooth patterns. If you get 11, no, everything is, maybe while I do screen capture, something weird is happening. But anyway, the whole point of this was to show you that if you wanted to figure out the probability of being 10 moves to the right, as you take more and more flips of your coin, the normal distribution becomes a much better approximation for the actual binomial distribution."}, {"video_title": "Normal distribution excel exercise   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for 10, you can see clearly that it converges. And I'll try to figure out why I was getting those weird sawtooth patterns. If you get 11, no, everything is, maybe while I do screen capture, something weird is happening. But anyway, the whole point of this was to show you that if you wanted to figure out the probability of being 10 moves to the right, as you take more and more flips of your coin, the normal distribution becomes a much better approximation for the actual binomial distribution. And as you approach infinity, they actually converge to each other. Anyway, that's all for this video. I'll actually do several more videos on the normal distribution, just because it is such an important concept."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Probability, a word that you've probably heard a lot of and you are probably a little bit familiar with it, but hopefully this will give you a little deeper understanding. So let's say that I have a fair coin over here. So when I talk about a fair coin, I mean that it has an equal chance of landing on one side or another. So you can maybe view it as the sides are equal, the weight is the same on either side. If I flip it in the air, it's not more likely to land on one side or the other. It's equally likely. And so you have one side of this coin, so this would be the heads, I guess."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you can maybe view it as the sides are equal, the weight is the same on either side. If I flip it in the air, it's not more likely to land on one side or the other. It's equally likely. And so you have one side of this coin, so this would be the heads, I guess. Try to draw George Washington. I'll assume it's a quarter of some kind. And then the other side, of course, is the tails."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so you have one side of this coin, so this would be the heads, I guess. Try to draw George Washington. I'll assume it's a quarter of some kind. And then the other side, of course, is the tails. So that is heads. The other side right over there is tails. And so if I were to ask you, what is the probability?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the other side, of course, is the tails. So that is heads. The other side right over there is tails. And so if I were to ask you, what is the probability? I'm going to flip a coin, and I want to know what is the probability of getting heads. And I could write that like this. The probability of getting heads."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if I were to ask you, what is the probability? I'm going to flip a coin, and I want to know what is the probability of getting heads. And I could write that like this. The probability of getting heads. And you probably, just based on that question, have a sense of what probability is asking. It's asking for some type of way of getting your hands around an event that's fundamentally random. We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability of getting heads. And you probably, just based on that question, have a sense of what probability is asking. It's asking for some type of way of getting your hands around an event that's fundamentally random. We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. And we'll talk about different ways of describing that. So one way to think about it, and this is the way that probability tends to be introduced in textbooks, is you say, well look, how many different equally likely possibilities are there? So how many equally likely possibilities?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. And we'll talk about different ways of describing that. So one way to think about it, and this is the way that probability tends to be introduced in textbooks, is you say, well look, how many different equally likely possibilities are there? So how many equally likely possibilities? So number of equally likely possibilities. And of the number of equally likely possibilities, I care about the number that contain my event right here. So the number of possibilities that meet my constraint."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many equally likely possibilities? So number of equally likely possibilities. And of the number of equally likely possibilities, I care about the number that contain my event right here. So the number of possibilities that meet my constraint. That meet my conditions. So in the case of the probability of figuring out heads, what is the number of equally likely possibilities? Well there's only two possibilities."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the number of possibilities that meet my constraint. That meet my conditions. So in the case of the probability of figuring out heads, what is the number of equally likely possibilities? Well there's only two possibilities. We're assuming that the coin can't land on its corner and just stand straight up. We're assuming that it lands flat. So there's two possibilities here."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well there's only two possibilities. We're assuming that the coin can't land on its corner and just stand straight up. We're assuming that it lands flat. So there's two possibilities here. Two equally likely possibilities. You could either get heads or you could get tails. And what's the number of possibilities that meet my conditions?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's two possibilities here. Two equally likely possibilities. You could either get heads or you could get tails. And what's the number of possibilities that meet my conditions? Well there's only one. The condition of heads. So it'll be 1 over 2."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what's the number of possibilities that meet my conditions? Well there's only one. The condition of heads. So it'll be 1 over 2. So the one way to think about it is the probability of getting heads is equal to 1 over 2. Is equal to 1 half. If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it'll be 1 over 2. So the one way to think about it is the probability of getting heads is equal to 1 over 2. Is equal to 1 half. If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. Now another way to think about or conceptualize probability that will give you this exact same answer, is to say, well if I were to run the experiment of flipping a coin. So this flip, you view this as an experiment. I know this isn't the kind of experiment that you're used to."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. Now another way to think about or conceptualize probability that will give you this exact same answer, is to say, well if I were to run the experiment of flipping a coin. So this flip, you view this as an experiment. I know this isn't the kind of experiment that you're used to. You know, you normally think an experiment is doing something in chemistry or physics or all the rest. But an experiment is every time you run this random event. So one way to think about probability is if I were to do this experiment, an experiment many, many, many times."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "I know this isn't the kind of experiment that you're used to. You know, you normally think an experiment is doing something in chemistry or physics or all the rest. But an experiment is every time you run this random event. So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. If I were to do it a thousand times or a million times or a billion times or a trillion times, and the more the better. What percentage of those would give me what I care about? What percentage of those would give me heads?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. If I were to do it a thousand times or a million times or a billion times or a trillion times, and the more the better. What percentage of those would give me what I care about? What percentage of those would give me heads? And so another way to think about this 50% probability of getting heads, is if I were to run this experiment tons of times. If I were to run this forever and closer or an infinite number of times, what percentage of those would be heads? You would get this 50%."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "What percentage of those would give me heads? And so another way to think about this 50% probability of getting heads, is if I were to run this experiment tons of times. If I were to run this forever and closer or an infinite number of times, what percentage of those would be heads? You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually a fun thing to do."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually a fun thing to do. I encourage you to do it. If you take 100 or 200 quarters or pennies, stick them in a big box, shake the box. So you're kind of simultaneously flipping all of the coins."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's actually a fun thing to do. I encourage you to do it. If you take 100 or 200 quarters or pennies, stick them in a big box, shake the box. So you're kind of simultaneously flipping all of the coins. And then count how many of those are going to be heads. And you're going to see that the larger the number that you are doing, the more likely you're going to get something really close to 50%. There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you're kind of simultaneously flipping all of the coins. And then count how many of those are going to be heads. And you're going to see that the larger the number that you are doing, the more likely you're going to get something really close to 50%. There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. But the more you do, the more likely that things are going to trend towards 50% of them are going to be heads. Now let's just apply these same ideas. And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. But the more you do, the more likely that things are going to trend towards 50% of them are going to be heads. Now let's just apply these same ideas. And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. But a lot of times this is actually a helpful one too, this idea that if you run the experiment many, many, many, many times, what percentage of those trials are going to give you what you're asking for? In this case, it was heads. Now let's do another very typical example when you first learn probability."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. But a lot of times this is actually a helpful one too, this idea that if you run the experiment many, many, many, many times, what percentage of those trials are going to give you what you're asking for? In this case, it was heads. Now let's do another very typical example when you first learn probability. And this is the idea of rolling a die. So here's my die right over here. And of course you have the different sides of the die."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's do another very typical example when you first learn probability. And this is the idea of rolling a die. So here's my die right over here. And of course you have the different sides of the die. So that's the 1, that's the 2, that's the 3. And what I want to do, and we know of course that there are, and I'm assuming this is a fair die, and so there are 6 equally likely possibilities. When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And of course you have the different sides of the die. So that's the 1, that's the 2, that's the 3. And what I want to do, and we know of course that there are, and I'm assuming this is a fair die, and so there are 6 equally likely possibilities. When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. And they are all equally likely. So if I were to ask you, what is the probability, given that I'm rolling a fair die, so the experiment is rolling this fair die, what is the probability of getting a 1? Well, what are the number of equally likely possibilities?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. And they are all equally likely. So if I were to ask you, what is the probability, given that I'm rolling a fair die, so the experiment is rolling this fair die, what is the probability of getting a 1? Well, what are the number of equally likely possibilities? Well, I have 6 equally likely possibilities. And how many of those meet my conditions? Well, only one of them meets my condition, that right there."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, what are the number of equally likely possibilities? Well, I have 6 equally likely possibilities. And how many of those meet my conditions? Well, only one of them meets my condition, that right there. So there is a 1 6 probability of rolling a 1. What is the probability of rolling a 1 or a 6? Well, once again, there are 6 equally likely possibilities for what I can get."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, only one of them meets my condition, that right there. So there is a 1 6 probability of rolling a 1. What is the probability of rolling a 1 or a 6? Well, once again, there are 6 equally likely possibilities for what I can get. And there are now 2 possibilities that meet my conditions. I could roll a 1 or I could roll a 6. So now there are 2 possibilities that meet my constraints, my conditions."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, once again, there are 6 equally likely possibilities for what I can get. And there are now 2 possibilities that meet my conditions. I could roll a 1 or I could roll a 6. So now there are 2 possibilities that meet my constraints, my conditions. So there is a 1 3 probability of rolling a 1 or a 6. Now what is the probability, this might seem a little silly to even ask this question, but I'll ask it just to make it clear. What is the probability of rolling a 2 and a 3?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So now there are 2 possibilities that meet my constraints, my conditions. So there is a 1 3 probability of rolling a 1 or a 6. Now what is the probability, this might seem a little silly to even ask this question, but I'll ask it just to make it clear. What is the probability of rolling a 2 and a 3? And I'm just talking about one roll of the die. Well, in any roll of the die, I can only get a 2 or a 3. I'm not talking about taking 2 rolls of this die."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of rolling a 2 and a 3? And I'm just talking about one roll of the die. Well, in any roll of the die, I can only get a 2 or a 3. I'm not talking about taking 2 rolls of this die. So in this situation, there are 6 possibilities, but none of these possibilities are 2 and a 3. 2 and a 3 cannot exist on one trial. You cannot get a 2 and a 3 in the same experiment."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm not talking about taking 2 rolls of this die. So in this situation, there are 6 possibilities, but none of these possibilities are 2 and a 3. 2 and a 3 cannot exist on one trial. You cannot get a 2 and a 3 in the same experiment. Getting a 2 and a 3 are mutually exclusive events. They cannot happen at the same time. So the probability of this is actually 0."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "You cannot get a 2 and a 3 in the same experiment. Getting a 2 and a 3 are mutually exclusive events. They cannot happen at the same time. So the probability of this is actually 0. There's no way to roll this normal die and all of a sudden you get a 2 and a 3. In fact, I don't want to confuse you with that because it's kind of abstract and impossible. So let's cross this out right over here."}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of this is actually 0. There's no way to roll this normal die and all of a sudden you get a 2 and a 3. In fact, I don't want to confuse you with that because it's kind of abstract and impossible. So let's cross this out right over here. Now what is the probability of getting an even number? So once again, you have 6 equally likely possibilities when I roll that die. And which of these possibilities meet my conditions, the condition of being even?"}, {"video_title": "Probability explained   Independent and dependent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's cross this out right over here. Now what is the probability of getting an even number? So once again, you have 6 equally likely possibilities when I roll that die. And which of these possibilities meet my conditions, the condition of being even? Well, 2 is even, 4 is even, and 6 is even. So 3 of the possibilities meet my conditions, meet my constraints. So this is 1 half."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say we have six people. We have person A, we have person B, we have person C, person D, person E, and we have person F. So we have six people, and for the sake of this video, we're going to say, oh, we want to figure out all the scenarios, all the possibilities, all the permutations, all the ways that we could put them into three chairs. So that's chair number one, chair number two, and chair number three. This is all a review, this is covered in the permutations video, but it'll be very instructive as we move into a new concept. So what are all of the permutations of putting six different people into three chairs? Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one? Well, there's six different people who could be in chair number one."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is all a review, this is covered in the permutations video, but it'll be very instructive as we move into a new concept. So what are all of the permutations of putting six different people into three chairs? Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one? Well, there's six different people who could be in chair number one. Let me do that in a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's six different people who could be in chair number one. Let me do that in a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. Now, for each of those six scenarios, how many people, how many different people could sit in chair number two? Well, in each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Six different scenarios for who sits in chair number one. Now, for each of those six scenarios, how many people, how many different people could sit in chair number two? Well, in each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now, if you want to say, well, what about for the three chairs? Well, for each of these 30 scenarios, how many different people could you put in chair number three?"}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now, if you want to say, well, what about for the three chairs? Well, for each of these 30 scenarios, how many different people could you put in chair number three? Well, you're still gonna have four people standing up, not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, for each of these 30 scenarios, how many different people could you put in chair number three? Well, you're still gonna have four people standing up, not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. Permutations. Now, permutations. Now, it's worth thinking about what permutations are counting."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So your total number of scenarios or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. Permutations. Now, permutations. Now, it's worth thinking about what permutations are counting. Now, remember, we care, when we're talking about permutations, we care about who's sitting in which chair. So, for example, this is one permutation, and this would be counted as another permutation. And this would be counted as another permutation."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, it's worth thinking about what permutations are counting. Now, remember, we care, when we're talking about permutations, we care about who's sitting in which chair. So, for example, this is one permutation, and this would be counted as another permutation. And this would be counted as another permutation. This would be counted as another permutation. So notice, these are all the same three people, but we're putting them in different chairs. And this counted that."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this would be counted as another permutation. This would be counted as another permutation. So notice, these are all the same three people, but we're putting them in different chairs. And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that. So when we're thinking in the permutation world, we would count all of these, or we would count this as six different permutations. These are going towards this 120. And of course, we have other permutations where we involve other people, where we have, it could be FBC, FCB, FAC, F, F, actually, let me do it this way."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could have that, or we could have that. So when we're thinking in the permutation world, we would count all of these, or we would count this as six different permutations. These are going towards this 120. And of course, we have other permutations where we involve other people, where we have, it could be FBC, FCB, FAC, F, F, actually, let me do it this way. That would be a little bit more systematic. F, let me do it, BBFC, BCF, and obviously, I could keep going. I could do 120 of these."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And of course, we have other permutations where we involve other people, where we have, it could be FBC, FCB, FAC, F, F, actually, let me do it this way. That would be a little bit more systematic. F, let me do it, BBFC, BCF, and obviously, I could keep going. I could do 120 of these. I'll do two more. You could have CFB, and then you could have CBF. So in the permutation world, these are literally 12 of the 120 permutations."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could do 120 of these. I'll do two more. You could have CFB, and then you could have CBF. So in the permutation world, these are literally 12 of the 120 permutations. But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting? So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in the permutation world, these are literally 12 of the 120 permutations. But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting? So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question, if I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video and try to think of what that number would actually be."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question, if I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video and try to think of what that number would actually be. Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people. And you pick a certain group of three people that turned into six permutations. And so if all you want to do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I encourage you to pause the video and try to think of what that number would actually be. Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people. And you pick a certain group of three people that turned into six permutations. And so if all you want to do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. Number of ways to arrange three people. And we see that you can arrange three people, or even three letters, you can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if all you want to do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. Number of ways to arrange three people. And we see that you can arrange three people, or even three letters, you can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. So there were 120 permutations here. If you said, how many different arrangements are there of taking six people and putting them into three chairs, that's 120. But now we're asking another thing."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this would be equal to 120 divided by six, or this would be equal to 20. So there were 120 permutations here. If you said, how many different arrangements are there of taking six people and putting them into three chairs, that's 120. But now we're asking another thing. We're saying, if we start with 120 people, and we want to choose, and we want to, sorry, if we're starting, if we're starting with six people, and we want to figure out how many ways, how many combinations, how many ways are there for us to choose three of them, then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "But now we're asking another thing. We're saying, if we start with 120 people, and we want to choose, and we want to, sorry, if we're starting, if we're starting with six people, and we want to figure out how many ways, how many combinations, how many ways are there for us to choose three of them, then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination. It's the combination A, B, C. I don't care what order they sit in. I've chosen them. I've chosen these three of the six."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right over here, once again, this right over here is just one combination. It's the combination A, B, C. I don't care what order they sit in. I've chosen them. I've chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I've chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination. It is F, C, and B. Once again, I don't care about the order. I just care that I've chosen these three people."}, {"video_title": "Introduction to combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right over here is another combination. It is F, C, and B. Once again, I don't care about the order. I just care that I've chosen these three people. So how many ways are there to choose three people out of six? It is 20. It's the total number of permutations, it's 120, divided by the number of ways you can arrange three people."}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "So the probability of rolling even numbers. So even roll on six-sided die. So let's think about that probability. Well, how many total outcomes are there? How many possible rolls could we get? Well, you could get 1, 2, 3, 4, 5, 6. And how many of them satisfy these conditions, that it's an even number?"}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "Well, how many total outcomes are there? How many possible rolls could we get? Well, you could get 1, 2, 3, 4, 5, 6. And how many of them satisfy these conditions, that it's an even number? Well, it could be a 2, it could be a 4, or it could be a 6. So the probability is the events that match what you need, your condition for right here. So three of the possible events are an even roll."}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "And how many of them satisfy these conditions, that it's an even number? Well, it could be a 2, it could be a 4, or it could be a 6. So the probability is the events that match what you need, your condition for right here. So three of the possible events are an even roll. And it's out of a total of six possible events. So there is a 3 over 6, the same thing as 1 half probability of rolling even on each roll. Now, they want to roll even three times."}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "So three of the possible events are an even roll. And it's out of a total of six possible events. So there is a 3 over 6, the same thing as 1 half probability of rolling even on each roll. Now, they want to roll even three times. And these are all going to be independent events. Every time you roll, it's not going to affect what happens in the next roll, despite what some gamblers might think. It has no impact on what happens on the next roll."}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "Now, they want to roll even three times. And these are all going to be independent events. Every time you roll, it's not going to affect what happens in the next roll, despite what some gamblers might think. It has no impact on what happens on the next roll. So the probability of rolling even three times is equal to the probability of an even roll one time, or even roll on six-sided die, this thing over here, is equal to that thing times that thing again. That's our first roll. Let me copy and paste it."}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "It has no impact on what happens on the next roll. So the probability of rolling even three times is equal to the probability of an even roll one time, or even roll on six-sided die, this thing over here, is equal to that thing times that thing again. That's our first roll. Let me copy and paste it. Times that thing, and then times that thing again. That's our first roll, which is that. That's our second roll."}, {"video_title": "Die rolling probability with independent events   Precalculus   Khan Academy.mp3", "Sentence": "Let me copy and paste it. Times that thing, and then times that thing again. That's our first roll, which is that. That's our second roll. That's our third roll. They're independent events. So this is going to be equal to 1 half, that's the same 1 half right there, times 1 half, times 1 half, which is equal to 1 over 8."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Administrators at Riverview High School surveyed a random sample of 100 of their seniors to see how they felt about the lunch offerings at the school's cafeteria. So you have all of the seniors, I'm assuming there's more than 100 of them, and then they sampled 100 of them. So this is the sample. So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings."}, {"video_title": "Identifying a sample and population   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings. We probably think it's impractical, or the administrators feel it's impractical to talk to everyone so they get exactly what the population thinks. So instead, they're gonna do a random sample of 100 of them. So the sample is 100 seniors who are actually surveyed."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So they decide to ask a simple random sample of 160 students if they have experienced extreme levels of stress during the past month. Subsequently, they find that 10% of the sample replied yes to the question. Assuming the true proportion is 15%, which they tell us up here, they say 15% of the population of the 1,750 students actually have experienced extreme levels of stress during the past month. So that is the true proportion. So let me just write that. The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month?"}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So that is the true proportion. So let me just write that. The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal. So in this case, the newspaper is asking 160 students."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal. So in this case, the newspaper is asking 160 students. That's the sample size. So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So in this case, the newspaper is asking 160 students. That's the sample size. So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is going to be 85 hundredths. And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is going to be 85 hundredths. And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out. So I'm gonna take the square root of 0.15 times 0.85 divided by 160, and let me close those parentheses. And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So now let's get our calculator out. So I'm gonna take the square root of 0.15 times 0.85 divided by 160, and let me close those parentheses. And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to be equal to the same thing as our population proportion, 0.15."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to be equal to the same thing as our population proportion, 0.15. And we also know that our standard deviation here is going to be approximately equal to 0.028. And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So we could say that 10% would be right over here."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to be equal to the same thing as our population proportion, 0.15. And we also know that our standard deviation here is going to be approximately equal to 0.028. And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So we could say that 10% would be right over here. I'll say 0.10. And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here. So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So we could say that 10% would be right over here. I'll say 0.10. And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here. So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. I could write one zero just like that. And then to calculate it, I can get out our calculator again. So here I'm gonna go to my distribution menu right over there."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. I could write one zero just like that. And then to calculate it, I can get out our calculator again. So here I'm gonna go to my distribution menu right over there. And then I'm gonna do a normal cumulative distribution function. So let me click Enter there. And so what is my lower bound?"}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So here I'm gonna go to my distribution menu right over there. And then I'm gonna do a normal cumulative distribution function. So let me click Enter there. And so what is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound? Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And so what is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound? Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15. What is the standard deviation of our sampling distribution?"}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15. What is the standard deviation of our sampling distribution? Well, it's approximately 0.028. And then I can click Enter. And if you're taking an AP exam, you actually should write this."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "What is the standard deviation of our sampling distribution? Well, it's approximately 0.028. And then I can click Enter. And if you're taking an AP exam, you actually should write this. You should say, you should tell the graders what you're actually typing in in your normal CDF function. But if we click Enter right over here, and then Enter, there we have it. It's approximately 96%."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And if you're taking an AP exam, you actually should write this. You should say, you should tell the graders what you're actually typing in in your normal CDF function. But if we click Enter right over here, and then Enter, there we have it. It's approximately 96%. So this is approximately 0.96. And then out of our choices, it would be this one right over here. If you're taking this on the AP exam, you would say that called, called normal, normal CDF, where you have your lower bound, lower bound, and you would put in your 0.10."}, {"video_title": "Probability of sample proportions example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "It's approximately 96%. So this is approximately 0.96. And then out of our choices, it would be this one right over here. If you're taking this on the AP exam, you would say that called, called normal, normal CDF, where you have your lower bound, lower bound, and you would put in your 0.10. You would say that you use an upper bound, upper bound of one. You would say that you gave a mean of 0.15, and then you gave a standard deviation of 0.028, just so people know that you knew what you were doing. But hopefully this is helpful."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Here is computer output on the sample data. So we have some statistics calculated on the reaction time, on the memory time, and then he had his computer do a regression for the data that he collected, and then we're told assume that all conditions for inference have been met, calculate the test statistic that should be used for testing a null hypothesis that the population slope is actually zero. So pause this video and have a go at it. All right, so let's just make sure we understand what is going on. So let's first think about the population. So I'll do that right over here. So in the population, there might be some true linear relationship."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "All right, so let's just make sure we understand what is going on. So let's first think about the population. So I'll do that right over here. So in the population, there might be some true linear relationship. So in theory, on our x-axis, we would have our reaction time, and on our y-axis, you have your memory time. If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x. Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So in the population, there might be some true linear relationship. So in theory, on our x-axis, we would have our reaction time, and on our y-axis, you have your memory time. If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x. Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. So he's taking a sample of 24, so samples, samples, 24 data, data points, and that's much easier to then, you could even visualize it on a scatter plot like this, so you'd have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic. This b is just a statistic that is trying to estimate the true parameter beta."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. So he's taking a sample of 24, so samples, samples, 24 data, data points, and that's much easier to then, you could even visualize it on a scatter plot like this, so you'd have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic. This b is just a statistic that is trying to estimate the true parameter beta. Now, when we went and inputted these data points into a computer, we got values for a and b right over here. A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line. Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "This b is just a statistic that is trying to estimate the true parameter beta. Now, when we went and inputted these data points into a computer, we got values for a and b right over here. A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line. Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. Now, ideally, you would take your b, you would take your b, and from that, subtract the slope assumed in the null hypothesis, so the slope of the regression line you get, minus the slope that's assumed from the null hypothesis, and then divide by the standard deviation of the sampling distribution of the slope of the regression line, and if you did this, you would get a, it would be appropriate to use a z statistic over here. Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it. We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. Now, ideally, you would take your b, you would take your b, and from that, subtract the slope assumed in the null hypothesis, so the slope of the regression line you get, minus the slope that's assumed from the null hypothesis, and then divide by the standard deviation of the sampling distribution of the slope of the regression line, and if you did this, you would get a, it would be appropriate to use a z statistic over here. Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it. We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. In fact, our computer has already done it for us, and this is an estimate of this, and we know what number that is, so we know what all of these numbers are, but if you're using an estimate of the standard deviation of the sampling distribution, and we've seen this before when we've done inferential statistics using for means, it is appropriate to use a t statistic, but with that said, pause the video. What is this going to be equal to? Well, this is going to be equal to the slope for our sample regression line."}, {"video_title": "Calculating t statistic for slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. In fact, our computer has already done it for us, and this is an estimate of this, and we know what number that is, so we know what all of these numbers are, but if you're using an estimate of the standard deviation of the sampling distribution, and we've seen this before when we've done inferential statistics using for means, it is appropriate to use a t statistic, but with that said, pause the video. What is this going to be equal to? Well, this is going to be equal to the slope for our sample regression line. We know it's 14.686, minus our assumed true population parameter, the slope of the true regression line. Well, we're assuming that is zero, so minus zero, and then we divide that by the standard error, which is going to be, we could view this as a standard error for B, and so this is divided by 13.329, so it's just gonna be 14.686 divided by 13.329, and if we assume, if we're doing a one-sided test here, what we would then do is take this t statistic and think about the degrees of freedom, and then say, and then calculate a p value. What is the probability of getting a result at least this far above t is equal to zero, or what is the probability of getting a t statistic this high or higher, and that would be our p value, and if that's below some threshold, let's say, hey, that's pretty unlikely, then we would reject the null, and that which would suggest the alternative, but they're not asking us to do all of that."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "He decides to test his null hypothesis is that the mean number of years of experience is five years and his alternative hypothesis is that the true mean years of experience is less than five years using a sample of 25 teachers. His sample mean was four years and his sample standard deviation was two years. Rory wants to use these sample data to conduct a t-test on the mean. Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years. And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower?"}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "The sample mean was four years. And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test. If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Sometimes we use a z-test. If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value."}, {"video_title": "Example calculating t statistic for a test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value. And then if that p-value is below some preset significance level that Rory should have set, maybe 5% or 1%, then he'll reject the null hypothesis, which would suggest his suspicion that the true mean of years of experience for the teachers in his district is less than five. Now another really important thing to keep in mind is, they told us that assume all conditions for inference have been met. And so that's the, assuming that this was truly a random sample, that each, the individual observations are either truly independent or roughly independent, that maybe he observed either with replacement or it's less than 10% of the population, and he feels good that the sampling distribution is going to be roughly normal."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "You are planning a full day nature trip for 50 men, and will bring 110 liters of water. What is the probability that you will run out of water? So let's think about what's happening here. So there's some distribution of how many liters the average man needs when they're active outdoors. And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's some distribution of how many liters the average man needs when they're active outdoors. And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters. So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they're all going to need at least more than 0 liters. So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 2 liters would be right over here. So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water. So maybe this is the actual distribution. And then one standard deviation is going to be 0.7 liters away."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water. So maybe this is the actual distribution. And then one standard deviation is going to be 0.7 liters away. So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters. So one standard deviation is going to be about that far away from the mean. If you go above it, it'll be about that far if you go below it."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then one standard deviation is going to be 0.7 liters away. So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters. So one standard deviation is going to be about that far away from the mean. If you go above it, it'll be about that far if you go below it. So let me draw. This is the standard deviation. That right there is the standard deviation to the right."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you go above it, it'll be about that far if you go below it. So let me draw. This is the standard deviation. That right there is the standard deviation to the right. That's the standard deviation to the left. And we know that the standard deviation is equal to, I'll write the 0 in front, 0.7 liters. So that's the actual distribution of how much water the average man needs when active."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "That right there is the standard deviation to the right. That's the standard deviation to the left. And we know that the standard deviation is equal to, I'll write the 0 in front, 0.7 liters. So that's the actual distribution of how much water the average man needs when active. Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water. What is the probability that you will run out? So the probability that you will run out, let me write this down."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's the actual distribution of how much water the average man needs when active. Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water. What is the probability that you will run out? So the probability that you will run out, let me write this down. The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability. If we use more than 110 liters, that means that on average, because we have 50 men, so 110 divided by 50 is what? That's 2."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability that you will run out, let me write this down. The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability. If we use more than 110 liters, that means that on average, because we have 50 men, so 110 divided by 50 is what? That's 2. Let me get the calculator out just so we don't make any mistakes here. So this is going to be the calculator out. So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's 2. Let me get the calculator out just so we don't make any mistakes here. So this is going to be the calculator out. So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. So we would run out if on average more than 2.2 liters is used per man. So this is the same thing as the probability of the average, or maybe we should say the sample mean, or let me write it this way, that the average water use per man of our 50 men is greater than, or we could say greater than or equal to, let me say greater than, well, I'll say greater than, because if we write on the money, then we won't run out of water, is greater than 2.2 liters per man. So let's think about this."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. So we would run out if on average more than 2.2 liters is used per man. So this is the same thing as the probability of the average, or maybe we should say the sample mean, or let me write it this way, that the average water use per man of our 50 men is greater than, or we could say greater than or equal to, let me say greater than, well, I'll say greater than, because if we write on the money, then we won't run out of water, is greater than 2.2 liters per man. So let's think about this. We are essentially taking 50 men out of a universal sample. And we got this data. Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about this. We are essentially taking 50 men out of a universal sample. And we got this data. Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation. Now we're sampling 50 men. And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation. Now we're sampling 50 men. And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. And to do that, we have to figure out the distribution of the sampling mean. And we know what that's called. It's the sampling distribution of the sample means."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. And to do that, we have to figure out the distribution of the sampling mean. And we know what that's called. It's the sampling distribution of the sample means. And we know that that is going to be a normal distribution. We know a few of the properties of that normal distribution. So this is the distribution of just all men."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's the sampling distribution of the sample means. And we know that that is going to be a normal distribution. We know a few of the properties of that normal distribution. So this is the distribution of just all men. And then if you take samples of, say, 50 men. So this will be, let me write this down. So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the distribution of just all men. And then if you take samples of, say, 50 men. So this will be, let me write this down. So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use. So let me draw that. So let's say this is the frequency, and then here are the different values."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use. So let me draw that. So let's say this is the frequency, and then here are the different values. Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population. So it's going to be the same value. I want to do it in that same blue."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say this is the frequency, and then here are the different values. Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population. So it's going to be the same value. I want to do it in that same blue. It's going to be the same value as this population over here. So that is going to be 2 liters. So we're still centered at 2 liters."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "I want to do it in that same blue. It's going to be the same value as this population over here. So that is going to be 2 liters. So we're still centered at 2 liters. But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency. 50 people, find the mean. This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're still centered at 2 liters. But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency. 50 people, find the mean. This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. It's not normal. Even though this one isn't normal, this one over here will be. And we've seen it in multiple videos already."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. It's not normal. Even though this one isn't normal, this one over here will be. And we've seen it in multiple videos already. So this is going to be a normal distribution. And the standard deviation, and we saw this in the last video, and hopefully we've got a little bit of intuition for why this is true. The standard deviation, actually maybe put it a better way."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we've seen it in multiple videos already. So this is going to be a normal distribution. And the standard deviation, and we saw this in the last video, and hopefully we've got a little bit of intuition for why this is true. The standard deviation, actually maybe put it a better way. The variance of the sample mean is going to be the variance. So remember, this is standard deviation. So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "The standard deviation, actually maybe put it a better way. The variance of the sample mean is going to be the variance. So remember, this is standard deviation. So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. You take the square root of both sides of that. We have the standard deviation of the sample mean is going to be equal to, the square root of this side over here, is going to be equal to the standard deviation of the population divided by the square root of n. And what's this going to be in our case? We know what the standard deviation of the population is."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. You take the square root of both sides of that. We have the standard deviation of the sample mean is going to be equal to, the square root of this side over here, is going to be equal to the standard deviation of the population divided by the square root of n. And what's this going to be in our case? We know what the standard deviation of the population is. It is 0.7. And what is n? We have 50 men."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know what the standard deviation of the population is. It is 0.7. And what is n? We have 50 men. So 0.7 over the square root of 50. Now let's figure out what that is with the calculator. So we have 0.7 divided by the square root of 50."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 50 men. So 0.7 over the square root of 50. Now let's figure out what that is with the calculator. So we have 0.7 divided by the square root of 50. And we have 0.098. It was pretty close to 0.99. So I'll just write that down."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have 0.7 divided by the square root of 50. And we have 0.098. It was pretty close to 0.99. So I'll just write that down. So this is equal to 0.099. That's going to be the standard deviation of this. So it's going to have a lower standard deviation."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll just write that down. So this is equal to 0.099. That's going to be the standard deviation of this. So it's going to have a lower standard deviation. So it's going to look, the distribution is going to be normal. It's going to look something like this. So this is 3 liters over here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to have a lower standard deviation. So it's going to look, the distribution is going to be normal. It's going to look something like this. So this is 3 liters over here. This is 1 liter. The standard deviation is almost a tenth. So it's going to be a much narrower distribution."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is 3 liters over here. This is 1 liter. The standard deviation is almost a tenth. So it's going to be a much narrower distribution. It's going to look something. I'm trying my best to draw it. It's going to look something like this."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be a much narrower distribution. It's going to look something. I'm trying my best to draw it. It's going to look something like this. You get the idea. Where the standard deviation right now is almost 0.1. So it's 0.09, almost a tenth."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to look something like this. You get the idea. Where the standard deviation right now is almost 0.1. So it's 0.09, almost a tenth. So it's going to be one standard deviation away. It's going to look something like that. So we have our distribution."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 0.09, almost a tenth. So it's going to be one standard deviation away. It's going to look something like that. So we have our distribution. It's a normal distribution. And now let's go back to our question that we're asking. We want to know the probability that our sample will have an average greater than 2.2."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have our distribution. It's a normal distribution. And now let's go back to our question that we're asking. We want to know the probability that our sample will have an average greater than 2.2. So this is the distribution of all of the possible samples, the means of all of the possible samples. Now to be greater than 2.2, 2.2 is going to be right around here. 2.2 is going to be right around here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to know the probability that our sample will have an average greater than 2.2. So this is the distribution of all of the possible samples, the means of all of the possible samples. Now to be greater than 2.2, 2.2 is going to be right around here. 2.2 is going to be right around here. So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "2.2 is going to be right around here. So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here. That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's right over here. That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is. The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we figured out what that is. The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there. And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm taking 0.2 divided by this value over there. And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way. More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe I should write it this way. More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going to be?"}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going to be? And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is that going to be? And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right?"}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. 2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "We go to 2.0, and it was 2.02. 2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 2.02 is right over there. So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217."}, {"video_title": "Sampling distribution example problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217. Or another way you could say it, it is a 2.17% probability that we will run out of water. And we are done. Let me make sure I got that number right."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that you have a cholesterol test, and you know, you somehow magically know that the probability that it is accurate, that it gives the correct results, is 99, 99%. You have a 99 out of 100 chance that any time you apply this test, that it is going to be accurate. Now let's say that you, and you just magically know that, we're just assuming that. Now let's just say that you get 100 folks into this room, and you apply this test to all 100 of them. So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate?"}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's just say that you get 100 folks into this room, and you apply this test to all 100 of them. So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability and combinatorics of it, but if you're curious about it, there's a lot of good videos on probability and combinatorics on Khan Academy. But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability and combinatorics of it, but if you're curious about it, there's a lot of good videos on probability and combinatorics on Khan Academy. But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going. The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now we could keep going. The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect. Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is what you would expect. Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going. The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we can of course keep going. The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you get 99 out of 100 accurate, and so on and so forth. So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you get 99 out of 100 accurate, and so on and so forth. So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is. You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have a new test and you don't know how accurate it is. You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new test is accurate is this is your hypothesis because you want that to be your hypothesis because if you feel good about it, then you're like, okay, maybe I'll get approved by the appropriate governing body. So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new test is accurate is this is your hypothesis because you want that to be your hypothesis because if you feel good about it, then you're like, okay, maybe I'll get approved by the appropriate governing body. So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate. You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So the question you have is, well, does the hypothesis make sense to you?"}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You don't know the actual probability of it being accurate. You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So the question you have is, well, does the hypothesis make sense to you? Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the question you have is, well, does the hypothesis make sense to you? Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out. If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we figured that out. If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, you'll see that there are thresholds that people often set for, you know, if the probability of something happening or not happening is above or below some threshold, then we might reject a certain hypothesis. But in this world, you could see that, look, if my test really was accurate 99% of the time, for me to get, when I apply it to 100 people, it's only accurate 95 out of 100."}, {"video_title": "Idea behind hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, you'll see that there are thresholds that people often set for, you know, if the probability of something happening or not happening is above or below some threshold, then we might reject a certain hypothesis. But in this world, you could see that, look, if my test really was accurate 99% of the time, for me to get, when I apply it to 100 people, it's only accurate 95 out of 100. If my hypothesis is true, that would have only, there's only a 0.3% chance that I would have seen this observation. So based on that, it might be completely reasonable to say, you know what, I might reject my hypothesis. Look for a new test."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "I actually recorded this video earlier today, but then I realized my microphone wasn't plugged in, and I won't name names in terms of who unplugged it. But anyway, back to probability. My wife is giggling mischievously. Anyway, so let's do a slightly harder problem than we did before. We were dealing with fair coins. Let's deal with a slightly unfair coin. Let's say I have a coin, and it's a, actually instead of unfair coin, let's do basketball."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Anyway, so let's do a slightly harder problem than we did before. We were dealing with fair coins. Let's deal with a slightly unfair coin. Let's say I have a coin, and it's a, actually instead of unfair coin, let's do basketball. Let's say I'm shooting free throws, and I have a free throw percentage of 80%. So when I shoot a free throw 8 out of 10 times, or 80% of the time, I will make it. But that also says that 20% of the time, I will miss it."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I have a coin, and it's a, actually instead of unfair coin, let's do basketball. Let's say I'm shooting free throws, and I have a free throw percentage of 80%. So when I shoot a free throw 8 out of 10 times, or 80% of the time, I will make it. But that also says that 20% of the time, I will miss it. So given that, if I were to take, I don't know, 5 free throws, what is the probability that I make at least 3 of the 5 free throws? Let's think of it this way. What is the probability of any particular combination of making 3 out of the 5?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But that also says that 20% of the time, I will miss it. So given that, if I were to take, I don't know, 5 free throws, what is the probability that I make at least 3 of the 5 free throws? Let's think of it this way. What is the probability of any particular combination of making 3 out of the 5? So what do I mean by that? Let me pick a particular combination. Let's say it's a basket, basket, basket, and then I miss, miss."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of any particular combination of making 3 out of the 5? So what do I mean by that? Let me pick a particular combination. Let's say it's a basket, basket, basket, and then I miss, miss. So that would be, I made 3 out of the 5. It could be, I don't know, basket, miss, basket, miss, basket, and there's a bunch of them, and we'll actually try to figure out how many of them there are. But what is the probability of this particular combination?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say it's a basket, basket, basket, and then I miss, miss. So that would be, I made 3 out of the 5. It could be, I don't know, basket, miss, basket, miss, basket, and there's a bunch of them, and we'll actually try to figure out how many of them there are. But what is the probability of this particular combination? Well, I have an 80% chance of making this first basket times 80%. That's a times right there. Times 80%."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what is the probability of this particular combination? Well, I have an 80% chance of making this first basket times 80%. That's a times right there. Times 80%. And then what's my probability of missing? Well, that's 20%, right? Times 0.2, times 0.2."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Times 80%. And then what's my probability of missing? Well, that's 20%, right? Times 0.2, times 0.2. So this equals 0.8 to the third power times 0.2 squared. Well, what's the probability of getting this exact combination? Well, it's 0.8 times, and then I miss."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Times 0.2, times 0.2. So this equals 0.8 to the third power times 0.2 squared. Well, what's the probability of getting this exact combination? Well, it's 0.8 times, and then I miss. There's a 20% chance of that. So times 0.2, times 0.8, times 0.2, times 0.8, right? We could rearrange this, because when you multiply numbers, it doesn't matter what order you multiply them in."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's 0.8 times, and then I miss. There's a 20% chance of that. So times 0.2, times 0.8, times 0.2, times 0.8, right? We could rearrange this, because when you multiply numbers, it doesn't matter what order you multiply them in. So this is the same thing as 0.8 times 0.8 times 0.8 times 0.2 times 0.2. So this is also the same thing as 0.8 to the third times 0.2 squared. So pretty much any particular, the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could rearrange this, because when you multiply numbers, it doesn't matter what order you multiply them in. So this is the same thing as 0.8 times 0.8 times 0.8 times 0.2 times 0.2. So this is also the same thing as 0.8 to the third times 0.2 squared. So pretty much any particular, the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared. Now, what's the total probability of getting 3 out of 5? Well, it's going to be the sum of all of these combinations. You know, I could list them all, but we, hopefully now, are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking, or we have 5 shots, and we're picking 3 of them to be the ones that are basket shots."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So pretty much any particular, the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared. Now, what's the total probability of getting 3 out of 5? Well, it's going to be the sum of all of these combinations. You know, I could list them all, but we, hopefully now, are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking, or we have 5 shots, and we're picking 3 of them to be the ones that are basket shots. What do I mean? So let's say my 5 shots, so I have shot 1, 2, 3, 4, 5. And I'm going to, out of these 5, I'm going to choose 3."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "You know, I could list them all, but we, hopefully now, are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking, or we have 5 shots, and we're picking 3 of them to be the ones that are basket shots. What do I mean? So let's say my 5 shots, so I have shot 1, 2, 3, 4, 5. And I'm going to, out of these 5, I'm going to choose 3. So I'm, once again, I'm putting my hat on as the god of probability. And I will choose 3 of these shots to be the ones that happen to be the ones that get made. So essentially, out of 5, I am choosing 3."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm going to, out of these 5, I'm going to choose 3. So I'm, once again, I'm putting my hat on as the god of probability. And I will choose 3 of these shots to be the ones that happen to be the ones that get made. So essentially, out of 5, I am choosing 3. 5 choose 3. And what does that equal to? That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So essentially, out of 5, I am choosing 3. 5 choose 3. And what does that equal to? That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial. That equals 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1 times 2 times 1. We can ignore all the 1's. Let's see, we get 3 times 2 times 1, 3 times 2 times 1, we can cancel that."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial. That equals 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1 times 2 times 1. We can ignore all the 1's. Let's see, we get 3 times 2 times 1, 3 times 2 times 1, we can cancel that. This 1 we can ignore. And then this 2, and then this turns into 2. So there are 10 possible combinations."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see, we get 3 times 2 times 1, 3 times 2 times 1, we can cancel that. This 1 we can ignore. And then this 2, and then this turns into 2. So there are 10 possible combinations. These are 2 of them. Basket, basket, basket, miss, miss, basket, miss, basket, miss, basket. And it's a good exercise for you to list the other 8 of them."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there are 10 possible combinations. These are 2 of them. Basket, basket, basket, miss, miss, basket, miss, basket, miss, basket. And it's a good exercise for you to list the other 8 of them. But using just the binomial coefficient, and hopefully you have an intuition of why that works, and I'd be happy to make more videos if you feel that you need more explanation, but I made a couple. There are 10 combinations. So essentially, the probability of getting exactly 3 out of 5 baskets if I am an 80% free throw shot is going to be, let me switch colors, the probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared, right?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's a good exercise for you to list the other 8 of them. But using just the binomial coefficient, and hopefully you have an intuition of why that works, and I'd be happy to make more videos if you feel that you need more explanation, but I made a couple. There are 10 combinations. So essentially, the probability of getting exactly 3 out of 5 baskets if I am an 80% free throw shot is going to be, let me switch colors, the probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared, right? I make 3, miss 2, and then times the total number of combinations, right? Each of these has a probability of this much. And then there's 10 different arrangements that I could make."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So essentially, the probability of getting exactly 3 out of 5 baskets if I am an 80% free throw shot is going to be, let me switch colors, the probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared, right? I make 3, miss 2, and then times the total number of combinations, right? Each of these has a probability of this much. And then there's 10 different arrangements that I could make. There's 10 different ways of getting 3 baskets and 2 misses. So times 10. And what is that equal to?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then there's 10 different arrangements that I could make. There's 10 different ways of getting 3 baskets and 2 misses. So times 10. And what is that equal to? Let me get my high-end calculator here. So let's see what that is. That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20.48."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what is that equal to? Let me get my high-end calculator here. So let's see what that is. That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20.48. So it's essentially a 20.48% chance that I get exactly 3 out of 5 of the baskets. Now let's make it slightly more interesting. Let's say I don't want to know the probability of 3 out of 5."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20.48. So it's essentially a 20.48% chance that I get exactly 3 out of 5 of the baskets. Now let's make it slightly more interesting. Let's say I don't want to know the probability of 3 out of 5. And this is actually something that probably people are more likely to ask. What is the probability of getting at least 3 baskets? Well, if you think about it, this is the probability."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I don't want to know the probability of 3 out of 5. And this is actually something that probably people are more likely to ask. What is the probability of getting at least 3 baskets? Well, if you think about it, this is the probability. This is equal to the probability of getting 3 out of 5 baskets plus the probability of getting exactly 4 out of 5 baskets plus the probability of getting exactly 5 out of 5 baskets, right? Well, we already figured this one out. That's 20.48%."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, if you think about it, this is the probability. This is equal to the probability of getting 3 out of 5 baskets plus the probability of getting exactly 4 out of 5 baskets plus the probability of getting exactly 5 out of 5 baskets, right? Well, we already figured this one out. That's 20.48%. So what's the probability of getting 4 out of 5 baskets? Well, once again, if we want exactly 4 out of 5 baskets, so an example could be, I don't know, basket, basket, basket, basket, what's the probability of any one of the combinations where I make 4 baskets? Well, it's going to be 0.8 to the 4th times my, and then I have a 20% chance of that one miss, right?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's 20.48%. So what's the probability of getting 4 out of 5 baskets? Well, once again, if we want exactly 4 out of 5 baskets, so an example could be, I don't know, basket, basket, basket, basket, what's the probability of any one of the combinations where I make 4 baskets? Well, it's going to be 0.8 to the 4th times my, and then I have a 20% chance of that one miss, right? And it could have been basket, miss, basket, basket, basket. Right? That's the same."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's going to be 0.8 to the 4th times my, and then I have a 20% chance of that one miss, right? And it could have been basket, miss, basket, basket, basket. Right? That's the same. That's also exactly 4. But when you multiply them, the probability of getting any one of these particular combinations is exactly this, 0.8 to the 4th times 0.2. And so how many ways can you, if I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's the same. That's also exactly 4. But when you multiply them, the probability of getting any one of these particular combinations is exactly this, 0.8 to the 4th times 0.2. And so how many ways can you, if I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability? So this is going to be 0.8 to the 4th times 0.2 times out of 5 baskets, I'm choosing 4 that I'm going to make. So this is the number of combinations where I get 4 out of the 5. So what does 5 choose 4?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so how many ways can you, if I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability? So this is going to be 0.8 to the 4th times 0.2 times out of 5 baskets, I'm choosing 4 that I'm going to make. So this is the number of combinations where I get 4 out of the 5. So what does 5 choose 4? That's 5 factorial over 4 factorial times 1 factorial. Well, that equals just 5. You can work that out."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what does 5 choose 4? That's 5 factorial over 4 factorial times 1 factorial. Well, that equals just 5. You can work that out. So it's going to be, so let's just figure this out. So it's going to be 0.8 times 0.8 times 0.8. That's 3 times 0.8."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can work that out. So it's going to be, so let's just figure this out. So it's going to be 0.8 times 0.8 times 0.8. That's 3 times 0.8. That equals, did I do that right? Let's see, 0.1, 0.8, 0.8 times 0.8. Yeah, that's right, times 0.2 times 5."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's 3 times 0.8. That equals, did I do that right? Let's see, 0.1, 0.8, 0.8 times 0.8. Yeah, that's right, times 0.2 times 5. So 40.96%. So this is 40.96%. So roughly 41% chance that I get exactly 4 out of 5 baskets, which is interesting because that's kind of my free-throw percentage."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yeah, that's right, times 0.2 times 5. So 40.96%. So this is 40.96%. So roughly 41% chance that I get exactly 4 out of 5 baskets, which is interesting because that's kind of my free-throw percentage. So it's almost a little less, 2 3rd shot of kind of hitting my free-throw percentage on the mark on that time. And that's probably getting 5 out of 5. Well, there's only one way of getting 5 out of 5."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So roughly 41% chance that I get exactly 4 out of 5 baskets, which is interesting because that's kind of my free-throw percentage. So it's almost a little less, 2 3rd shot of kind of hitting my free-throw percentage on the mark on that time. And that's probably getting 5 out of 5. Well, there's only one way of getting 5 out of 5. You have to get all 5 of them. So this is 0.8 to the 5th power. Let me get the calculator back."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's only one way of getting 5 out of 5. You have to get all 5 of them. So this is 0.8 to the 5th power. Let me get the calculator back. So it's 0.8 times 0.8 times 0.8 times 0.8 equals 0.3276. So 32.77% shot. And then we can add them all up, right?"}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me get the calculator back. So it's 0.8 times 0.8 times 0.8 times 0.8 equals 0.3276. So 32.77% shot. And then we can add them all up, right? Because we want the probability of at least 3. So it's going to be that, the probability of getting 5 out of 5, plus the probability of getting a 4 out of 5, which is 0.4096, plus the probability of getting 3 out of 5. So that's 0.2048 equals 0.94208."}, {"video_title": "Probability and combinations (part 2)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we can add them all up, right? Because we want the probability of at least 3. So it's going to be that, the probability of getting 5 out of 5, plus the probability of getting a 4 out of 5, which is 0.4096, plus the probability of getting 3 out of 5. So that's 0.2048 equals 0.94208. So 94.21, roughly, rounding percent chance. Which makes sense. If I have an 80% free-throw percentage on any one shot, I have a very high probability of getting at least 3 out of 5 when I go to the free-throw line."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the last video, we figured out the mean, variance, and standard deviation for a Bernoulli distribution with specific numbers. What I want to do in this video is to generalize it, to figure out really the formulas for the mean and the variance of a Bernoulli distribution if we don't have the actual numbers. If we just know that the probability of success is p, and the probability of failure is 1 minus p. So let's look at this. Let's look at a population where the probability of success, and we'll define success as 1, as having a probability of p. And the probability of failure is 1 minus p, whatever this might be. And obviously, if you add these two up, if you view them as percentages, these are going to add up to 100%. Or if you add up these two values, you're going to add to 1. And that needs to be the case, because these are the only two possibilities that can occur."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's look at a population where the probability of success, and we'll define success as 1, as having a probability of p. And the probability of failure is 1 minus p, whatever this might be. And obviously, if you add these two up, if you view them as percentages, these are going to add up to 100%. Or if you add up these two values, you're going to add to 1. And that needs to be the case, because these are the only two possibilities that can occur. If this is 60% chance of success, there has to be a 40% chance of failure. 70% chance of success, 30% chance of failure. Now, with this definition of this, and this is the most general definition of a Bernoulli distribution, it's really exactly what we did in the last video."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that needs to be the case, because these are the only two possibilities that can occur. If this is 60% chance of success, there has to be a 40% chance of failure. 70% chance of success, 30% chance of failure. Now, with this definition of this, and this is the most general definition of a Bernoulli distribution, it's really exactly what we did in the last video. I now want to calculate the expected value, which is the same thing as the mean of this distribution. And I also want to calculate the variance, which is the same thing as the expected squared distance of a value from the mean. So let's do that."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, with this definition of this, and this is the most general definition of a Bernoulli distribution, it's really exactly what we did in the last video. I now want to calculate the expected value, which is the same thing as the mean of this distribution. And I also want to calculate the variance, which is the same thing as the expected squared distance of a value from the mean. So let's do that. So what is the mean over here? What is going to be the mean? Well, that's just the probability weighted sum of the values that this could take on."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's do that. So what is the mean over here? What is going to be the mean? Well, that's just the probability weighted sum of the values that this could take on. So there is a 1 minus p probability that we get failure, that we get 0. So there's 1 minus p probability of getting 0, so times 0. And then there is a p probability of getting 1, plus p times 1."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that's just the probability weighted sum of the values that this could take on. So there is a 1 minus p probability that we get failure, that we get 0. So there's 1 minus p probability of getting 0, so times 0. And then there is a p probability of getting 1, plus p times 1. Well, this is pretty easy to calculate. 0 times anything is 0, so that cancels out. And then p times 1 is just going to be p. So pretty straightforward."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then there is a p probability of getting 1, plus p times 1. Well, this is pretty easy to calculate. 0 times anything is 0, so that cancels out. And then p times 1 is just going to be p. So pretty straightforward. The mean, the expected value of this distribution is p. And p might be here or something. So once again, it's a value that you cannot actually take on in this distribution, which is interesting. But it is the expected value."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then p times 1 is just going to be p. So pretty straightforward. The mean, the expected value of this distribution is p. And p might be here or something. So once again, it's a value that you cannot actually take on in this distribution, which is interesting. But it is the expected value. Now, what is going to be the variance? What is the variance of this distribution? Remember, that is the weighted sum of the squared distances from the mean."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it is the expected value. Now, what is going to be the variance? What is the variance of this distribution? Remember, that is the weighted sum of the squared distances from the mean. Now, what's the probability that we get a 0? We already figured that out. There's a 1 minus p probability that we get a 0."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, that is the weighted sum of the squared distances from the mean. Now, what's the probability that we get a 0? We already figured that out. There's a 1 minus p probability that we get a 0. So that is the probability part. And what is the squared distance from 0 to our mean? Well, the squared distance from 0 to our mean, let me write it over here, is going to be 0."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a 1 minus p probability that we get a 0. So that is the probability part. And what is the squared distance from 0 to our mean? Well, the squared distance from 0 to our mean, let me write it over here, is going to be 0. That's the value we're taking on. Let me do that in blue, since I already wrote 0. 0 minus our mean."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the squared distance from 0 to our mean, let me write it over here, is going to be 0. That's the value we're taking on. Let me do that in blue, since I already wrote 0. 0 minus our mean. Let me do this in a new color. Minus our mean. That's too similar to that orange."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "0 minus our mean. Let me do this in a new color. Minus our mean. That's too similar to that orange. So I'm going to do the mean in white. 0 minus our mean, which is p, plus the probability that we get a 1, which is just p. This is the squared distance. Let me be very careful."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's too similar to that orange. So I'm going to do the mean in white. 0 minus our mean, which is p, plus the probability that we get a 1, which is just p. This is the squared distance. Let me be very careful. It's the probability weighted sum of the squared distances from the mean. Now, what's the distance? Now, we've got a 1."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me be very careful. It's the probability weighted sum of the squared distances from the mean. Now, what's the distance? Now, we've got a 1. And what's the distance between 1 and the mean? It's 1 minus our mean, which is going to be p over here. And we're going to want to square this as well."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, we've got a 1. And what's the distance between 1 and the mean? It's 1 minus our mean, which is going to be p over here. And we're going to want to square this as well. This right here is going to be the variance. Now, let's actually work this out. So this is going to be equal to 1 minus p. Now, 0 minus p is going to be negative p. If you square it, you're just going to get p squared."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to want to square this as well. This right here is going to be the variance. Now, let's actually work this out. So this is going to be equal to 1 minus p. Now, 0 minus p is going to be negative p. If you square it, you're just going to get p squared. So it's going to be p squared. Then plus p times. What's 1 minus p squared?"}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be equal to 1 minus p. Now, 0 minus p is going to be negative p. If you square it, you're just going to get p squared. So it's going to be p squared. Then plus p times. What's 1 minus p squared? 1 minus p squared is going to be 1 squared, which is just 1. Minus 2 times the product of this. So it's going to be minus 2p."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's 1 minus p squared? 1 minus p squared is going to be 1 squared, which is just 1. Minus 2 times the product of this. So it's going to be minus 2p. Let me write it over here. And then plus negative p squared. So plus p squared."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be minus 2p. Let me write it over here. And then plus negative p squared. So plus p squared. Just like that. And now let's multiply everything out. This term right over here is going to be p squared minus p to the third."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So plus p squared. Just like that. And now let's multiply everything out. This term right over here is going to be p squared minus p to the third. And then this term over here, this whole thing over here, is going to be plus. p times 1 is p. p times negative 2p is negative 2p squared. And then p times p squared is p to the third."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "This term right over here is going to be p squared minus p to the third. And then this term over here, this whole thing over here, is going to be plus. p times 1 is p. p times negative 2p is negative 2p squared. And then p times p squared is p to the third. Now, we can simplify these. p to the third cancels out with p to the third. And then we have p squared minus 2p squared."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then p times p squared is p to the third. Now, we can simplify these. p to the third cancels out with p to the third. And then we have p squared minus 2p squared. So this right here becomes, you have this p right over here, so this is equal to p. And then when you add p squared to negative 2p squared, you're left with negative p squared. So minus p squared. And if you want to factor a p out of this, this is going to be equal to p times, if you take p divided by p, you get a 1, p squared divided by p is p. So p times 1 minus p. Which is a pretty neat, clean formula."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have p squared minus 2p squared. So this right here becomes, you have this p right over here, so this is equal to p. And then when you add p squared to negative 2p squared, you're left with negative p squared. So minus p squared. And if you want to factor a p out of this, this is going to be equal to p times, if you take p divided by p, you get a 1, p squared divided by p is p. So p times 1 minus p. Which is a pretty neat, clean formula. So our variance is p times 1 minus p. And if we want to take it to the next level and figure out the standard deviation, the standard deviation is just the square root of the variance. Which is equal to the square root of p times 1 minus p. And we can even verify that this actually works for the example that we did up here. Our mean is p, the probability of success."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you want to factor a p out of this, this is going to be equal to p times, if you take p divided by p, you get a 1, p squared divided by p is p. So p times 1 minus p. Which is a pretty neat, clean formula. So our variance is p times 1 minus p. And if we want to take it to the next level and figure out the standard deviation, the standard deviation is just the square root of the variance. Which is equal to the square root of p times 1 minus p. And we can even verify that this actually works for the example that we did up here. Our mean is p, the probability of success. We see that it indeed was, it was 0.6. And we know that our variance is essentially the probability of success times the probability of failure. That's our variance right over there."}, {"video_title": "Bernoulli distribution mean and variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our mean is p, the probability of success. We see that it indeed was, it was 0.6. And we know that our variance is essentially the probability of success times the probability of failure. That's our variance right over there. Probability of success in this example was 0.4, or probability of success was 0.6, probability of failure was 0.4. You multiply the two, you get 0.24, which is exactly what we got in the last example. And if you take its square root for the standard deviation, which is what we do right here, it's 0.49."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in the last video, we actually estimated that using a 95% confidence interval for the difference in the proportion of men and the difference in the proportion of women. What I want to do in this video is just to ask the question more directly, or just do a straight up hypothesis test to see, is there a difference? So we're going to make our null hypothesis. Let me just get a clear screen here. Let's make our null hypothesis no difference between how the men and the women will vote. Or another way of viewing it is that the proportion of men who will vote for the candidate is going to be the same as the proportion of women who are going to vote for the candidate. Or another way you could say that is that the difference, P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate, is going to be 0."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me just get a clear screen here. Let's make our null hypothesis no difference between how the men and the women will vote. Or another way of viewing it is that the proportion of men who will vote for the candidate is going to be the same as the proportion of women who are going to vote for the candidate. Or another way you could say that is that the difference, P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate, is going to be 0. That's our null hypothesis. Our alternative hypothesis is that there is a difference. Our alternative hypothesis is that there is a difference."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way you could say that is that the difference, P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate, is going to be 0. That's our null hypothesis. Our alternative hypothesis is that there is a difference. Our alternative hypothesis is that there is a difference. Or that P1 does not equal P2. Or that P1 minus P2, the proportion of men voting minus the proportion of women voting, the true population proportions, do not equal 0. And we're going to test this."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our alternative hypothesis is that there is a difference. Or that P1 does not equal P2. Or that P1 minus P2, the proportion of men voting minus the proportion of women voting, the true population proportions, do not equal 0. And we're going to test this. We're going to do the hypothesis test with a significance level of 5%. And all that means, and we've done this multiple times, is we are going to assume the null hypothesis. And then assuming the null hypothesis is true, we're going to then figure out the probability of getting our actual difference of our sample proportions."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to test this. We're going to do the hypothesis test with a significance level of 5%. And all that means, and we've done this multiple times, is we are going to assume the null hypothesis. And then assuming the null hypothesis is true, we're going to then figure out the probability of getting our actual difference of our sample proportions. So we're going to figure out the probability of actually getting our actual difference between our male sample proportion and our female sample proportion, given the assumption that our null hypothesis is correct. And if this probability is less than 5%, if this probability is less than our significance level, so if the odds of getting these two samples and the difference between those two samples is less than 5%, then we are going to reject the null hypothesis. So how are we going to do this?"}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then assuming the null hypothesis is true, we're going to then figure out the probability of getting our actual difference of our sample proportions. So we're going to figure out the probability of actually getting our actual difference between our male sample proportion and our female sample proportion, given the assumption that our null hypothesis is correct. And if this probability is less than 5%, if this probability is less than our significance level, so if the odds of getting these two samples and the difference between those two samples is less than 5%, then we are going to reject the null hypothesis. So how are we going to do this? So if we assume the null hypothesis, what does the sampling distribution of this statistic start to look like? Well, the mean, if we assume that P1 and the true population proportions are actually the same between men and women, if P1 and P2 are actually the same, then this right here is going to be 0. This right here is going to be 0."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how are we going to do this? So if we assume the null hypothesis, what does the sampling distribution of this statistic start to look like? Well, the mean, if we assume that P1 and the true population proportions are actually the same between men and women, if P1 and P2 are actually the same, then this right here is going to be 0. This right here is going to be 0. So what we can do is we can figure out that we got, when we took the proportion of men and we subtracted from that the proportion of women, so this is our sample proportion of men who are going to vote for, at least in our poll, said they would vote for the candidate. This is the proportion of women who said they would vote for the candidate. The difference between the two was 0.051."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right here is going to be 0. So what we can do is we can figure out that we got, when we took the proportion of men and we subtracted from that the proportion of women, so this is our sample proportion of men who are going to vote for, at least in our poll, said they would vote for the candidate. This is the proportion of women who said they would vote for the candidate. The difference between the two was 0.051. So what we can do is figure out what's the probability, assuming that the true proportions are equal, that the mean of the sampling distribution of this statistic is actually 0, what's the probability that we get a difference of 0.051? So what's the likelihood that we get something that extreme? And what we're going to do here is just figure out a z score for this, essentially figure out how many standard deviations away from the mean this is."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The difference between the two was 0.051. So what we can do is figure out what's the probability, assuming that the true proportions are equal, that the mean of the sampling distribution of this statistic is actually 0, what's the probability that we get a difference of 0.051? So what's the likelihood that we get something that extreme? And what we're going to do here is just figure out a z score for this, essentially figure out how many standard deviations away from the mean this is. That would be our z score. And then figure out, is the likelihood of getting a standard deviation or that extreme of a result or that many standard deviations away from the mean, is that likelihood more or less than 5%? If it is less than 5%, we're going to reject the null hypothesis."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what we're going to do here is just figure out a z score for this, essentially figure out how many standard deviations away from the mean this is. That would be our z score. And then figure out, is the likelihood of getting a standard deviation or that extreme of a result or that many standard deviations away from the mean, is that likelihood more or less than 5%? If it is less than 5%, we're going to reject the null hypothesis. So let's first of all figure out our z score. So we're assuming the null hypothesis, p1 is equal to p2. Our z score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women, was 0.051."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it is less than 5%, we're going to reject the null hypothesis. So let's first of all figure out our z score. So we're assuming the null hypothesis, p1 is equal to p2. Our z score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women, was 0.051. And from that, we're going to subtract the assumed mean. Remember, we're assuming that these two things are equal. So the mean of this sampling distribution right here is 0."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our z score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women, was 0.051. And from that, we're going to subtract the assumed mean. Remember, we're assuming that these two things are equal. So the mean of this sampling distribution right here is 0. So we're just going to subtract 0. And then we have to divide this by the standard deviation of the sampling distribution of this statistic right here. So the sampling distribution of this statistic, p1 minus p2."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the mean of this sampling distribution right here is 0. So we're just going to subtract 0. And then we have to divide this by the standard deviation of the sampling distribution of this statistic right here. So the sampling distribution of this statistic, p1 minus p2. Now what's the standard deviation of the distribution going to be? In the last video, we figured out that we could represent it by this formula over here. But with our null hypothesis, we're assuming that p1 and p2 are the same value."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the sampling distribution of this statistic, p1 minus p2. Now what's the standard deviation of the distribution going to be? In the last video, we figured out that we could represent it by this formula over here. But with our null hypothesis, we're assuming that p1 and p2 are the same value. So let me rewrite it. So in our last video, and I don't want to confuse the issue, because in the last video I made this approximation over here. So let me write the clean version down here."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But with our null hypothesis, we're assuming that p1 and p2 are the same value. So let me rewrite it. So in our last video, and I don't want to confuse the issue, because in the last video I made this approximation over here. So let me write the clean version down here. We know that the standard deviation of our sampling distribution of this statistic, of the sample mean of p1 minus the sample proportion or sample mean of p2, is equal to the square root of p1 times 1 minus p1 over 1,000 plus p2 times 1 minus p2 over 1,000. We've seen this in several videos. But in the null hypothesis, we are going to assume that p1 is equal to p2."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write the clean version down here. We know that the standard deviation of our sampling distribution of this statistic, of the sample mean of p1 minus the sample proportion or sample mean of p2, is equal to the square root of p1 times 1 minus p1 over 1,000 plus p2 times 1 minus p2 over 1,000. We've seen this in several videos. But in the null hypothesis, we are going to assume that p1 is equal to p2. That's what we do. We assume the null hypothesis and see the probability of this occurring. So if p1 is equal to p2, we can just represent them as just some true population proportion."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But in the null hypothesis, we are going to assume that p1 is equal to p2. That's what we do. We assume the null hypothesis and see the probability of this occurring. So if p1 is equal to p2, we can just represent them as just some true population proportion. So we could say that this is going to be equal to, so we could write it like this. The square root of, we can literally just factor out, 1 over 1,000 times p times 1 minus p plus p times 1 minus p, because they're going to be the same value. That's what we're assuming in the null hypothesis."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if p1 is equal to p2, we can just represent them as just some true population proportion. So we could say that this is going to be equal to, so we could write it like this. The square root of, we can literally just factor out, 1 over 1,000 times p times 1 minus p plus p times 1 minus p, because they're going to be the same value. That's what we're assuming in the null hypothesis. And so this is just 2 of these over here. So this is going to be equal to 2p times 1 minus p, all of that over 1,000. And we're going to take the square root of that."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what we're assuming in the null hypothesis. And so this is just 2 of these over here. So this is going to be equal to 2p times 1 minus p, all of that over 1,000. And we're going to take the square root of that. Now this is the standard deviation, once again, of the distribution of this statistic right over here. The sample mean of the sample proportion for the men minus the sample proportion of the women. Now, we still don't know this."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to take the square root of that. Now this is the standard deviation, once again, of the distribution of this statistic right over here. The sample mean of the sample proportion for the men minus the sample proportion of the women. Now, we still don't know this. We still don't know the true proportion. We still don't know what that is. But we can estimate it."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, we still don't know this. We still don't know the true proportion. We still don't know what that is. But we can estimate it. We can estimate it using our samples. And since we're assuming that the men and the women, that there's no difference between them, we can actually view it as a sample size of 2,000 to figure out that true proportion. So we can actually substitute it."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we can estimate it. We can estimate it using our samples. And since we're assuming that the men and the women, that there's no difference between them, we can actually view it as a sample size of 2,000 to figure out that true proportion. So we can actually substitute it. We can actually substitute this with a sample proportion. And we can pretend like our survey of the men and women is just one huge survey. So you have your sample proportion is going to be equal to, we're surveying a total of 2,000 people."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can actually substitute it. We can actually substitute this with a sample proportion. And we can pretend like our survey of the men and women is just one huge survey. So you have your sample proportion is going to be equal to, we're surveying a total of 2,000 people. 1,000 men and 1,000 women. But we're assuming that they're no different. That's what our null hypothesis is all about."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have your sample proportion is going to be equal to, we're surveying a total of 2,000 people. 1,000 men and 1,000 women. But we're assuming that they're no different. That's what our null hypothesis is all about. Assuming there's no difference between men and women. And we got, let's go back to our original, we got 642 yeses amongst the men and 591 amongst the women. So we got a total of, I already forgot, 642, 591."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what our null hypothesis is all about. Assuming there's no difference between men and women. And we got, let's go back to our original, we got 642 yeses amongst the men and 591 amongst the women. So we got a total of, I already forgot, 642, 591. So it is 642 plus 591. If you viewed it as just one huge sample of 2,000 people, we got, get the calculator out, we got 642 plus 591 is equal to 1,233 divided by 2,000. Gives us 0.6165."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we got a total of, I already forgot, 642, 591. So it is 642 plus 591. If you viewed it as just one huge sample of 2,000 people, we got, get the calculator out, we got 642 plus 591 is equal to 1,233 divided by 2,000. Gives us 0.6165. And this is our best estimate of this consistent population proportion that is true of both men and women. Because we are assuming that they are no different. So we can substitute this value in for p to estimate the standard deviation of the sampling distribution of this statistic right over here."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Gives us 0.6165. And this is our best estimate of this consistent population proportion that is true of both men and women. Because we are assuming that they are no different. So we can substitute this value in for p to estimate the standard deviation of the sampling distribution of this statistic right over here. Assuming that the proportion of men and women are the same. Or the proportion that will vote for the candidate. So let's do that."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can substitute this value in for p to estimate the standard deviation of the sampling distribution of this statistic right over here. Assuming that the proportion of men and women are the same. Or the proportion that will vote for the candidate. So let's do that. It's going to be the square root of 2 times p, which is 0.6165 times 1 minus p. Divided by 1,000. Let me make sure I get it. 2 times 0.6165, that's that p right there, times 1 minus p divided by 1,000."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's do that. It's going to be the square root of 2 times p, which is 0.6165 times 1 minus p. Divided by 1,000. Let me make sure I get it. 2 times 0.6165, that's that p right there, times 1 minus p divided by 1,000. We're taking the square root of the whole thing. And so we get a standard deviation of 0.0217. Let me write this over here."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "2 times 0.6165, that's that p right there, times 1 minus p divided by 1,000. We're taking the square root of the whole thing. And so we get a standard deviation of 0.0217. Let me write this over here. So this thing right over here is 0.0217. So if we want to figure out our z-score, if we want to figure out how many standard deviations the actual sample that we got of this statistic right over here, if we want to figure out how many standard deviations that is away from our assumed mean, that the assumed mean is that there's no difference, then we just divide 0.051 by this standard deviation right over here. So let's do that."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write this over here. So this thing right over here is 0.0217. So if we want to figure out our z-score, if we want to figure out how many standard deviations the actual sample that we got of this statistic right over here, if we want to figure out how many standard deviations that is away from our assumed mean, that the assumed mean is that there's no difference, then we just divide 0.051 by this standard deviation right over here. So let's do that. So we have 0.051 divided by this standard deviation. That was our answer up here. So I'll just do divided by our answer."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's do that. So we have 0.051 divided by this standard deviation. That was our answer up here. So I'll just do divided by our answer. And we are 2.35 standard deviations away. So our z-score is equal to 2.35. So just to review what we're doing, we're assuming the null hypothesis."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll just do divided by our answer. And we are 2.35 standard deviations away. So our z-score is equal to 2.35. So just to review what we're doing, we're assuming the null hypothesis. There's no difference. If we assume there's no difference, then the sampling distribution of this statistic right here is going to have a mean of 0, and the result that we actually got for this statistic has a z-score of 2.34. Or this is equivalent to being 2.34 standard deviations away from this mean of 0."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just to review what we're doing, we're assuming the null hypothesis. There's no difference. If we assume there's no difference, then the sampling distribution of this statistic right here is going to have a mean of 0, and the result that we actually got for this statistic has a z-score of 2.34. Or this is equivalent to being 2.34 standard deviations away from this mean of 0. So in order to reject the null hypothesis, that has to be less probable than our significance level. And to see that, let's see what the minimum z-score we need to reject our hypothesis. So let's think about that a second."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or this is equivalent to being 2.34 standard deviations away from this mean of 0. So in order to reject the null hypothesis, that has to be less probable than our significance level. And to see that, let's see what the minimum z-score we need to reject our hypothesis. So let's think about that a second. I'll go back to my z-table. We want to have a significance level of 5%, which means the entire area of our rejection, the entire area in which we would reject the null hypothesis, is 5%. This is a two-tailed test."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about that a second. I'll go back to my z-table. We want to have a significance level of 5%, which means the entire area of our rejection, the entire area in which we would reject the null hypothesis, is 5%. This is a two-tailed test. An extreme event either far above the mean or far below the mean will allow us to reject the hypothesis. So we care about area over here. And over here we would put 2.5%."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is a two-tailed test. An extreme event either far above the mean or far below the mean will allow us to reject the hypothesis. So we care about area over here. And over here we would put 2.5%. And over here we would have 2.5%. And we would have 95% in the middle. So we need to find this critical z-score, critical z value, and if our z value is greater than the positive version of this critical z value, then that's less probable than the odds of getting something so extreme is less than 5%."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And over here we would put 2.5%. And over here we would have 2.5%. And we would have 95% in the middle. So we need to find this critical z-score, critical z value, and if our z value is greater than the positive version of this critical z value, then that's less probable than the odds of getting something so extreme is less than 5%. And we can, assuming the null hypothesis is correct, so then we can reject the null hypothesis. So let's see what this critical z value is. So essentially we want a z value where the entire percentage below it is going to be 97.5%, because then you're going to have 2.5% over here."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we need to find this critical z-score, critical z value, and if our z value is greater than the positive version of this critical z value, then that's less probable than the odds of getting something so extreme is less than 5%. And we can, assuming the null hypothesis is correct, so then we can reject the null hypothesis. So let's see what this critical z value is. So essentially we want a z value where the entire percentage below it is going to be 97.5%, because then you're going to have 2.5% over here. And we've actually already figured that out. This whole cumulative has to be 97.5%. We did that in the last video."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So essentially we want a z value where the entire percentage below it is going to be 97.5%, because then you're going to have 2.5% over here. And we've actually already figured that out. This whole cumulative has to be 97.5%. We did that in the last video. If you look for that, you get 97.975 right there. It's a z-score of 1.96. I even wrote it over there."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "We did that in the last video. If you look for that, you get 97.975 right there. It's a z-score of 1.96. I even wrote it over there. So this critical z value is 1.96. So what that tells you is there is a 5% chance. So this tells us that there is a 5% chance of sampling a z statistic greater than 1.96, assuming the null hypothesis is correct."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "I even wrote it over there. So this critical z value is 1.96. So what that tells you is there is a 5% chance. So this tells us that there is a 5% chance of sampling a z statistic greater than 1.96, assuming the null hypothesis is correct. Now, we just figured out that we just sampled a z statistic of 2.34, assuming the null hypothesis is correct. So the probability of sampling this, given the null hypothesis is correct, is going to be less than 5%. It is more extreme than this critical z value."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this tells us that there is a 5% chance of sampling a z statistic greater than 1.96, assuming the null hypothesis is correct. Now, we just figured out that we just sampled a z statistic of 2.34, assuming the null hypothesis is correct. So the probability of sampling this, given the null hypothesis is correct, is going to be less than 5%. It is more extreme than this critical z value. It's going to be out here someplace. And because of that, we can reject the null hypothesis. And sorry for jumping around so much in this video."}, {"video_title": "Hypothesis test comparing population proportions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It is more extreme than this critical z value. It's going to be out here someplace. And because of that, we can reject the null hypothesis. And sorry for jumping around so much in this video. I had already written a lot, so I just kind of leveraged what I had already written. But since the odds of getting that, assuming the null hypothesis, are less than 5%, and that was our significance level, we can reject the null hypothesis and say that there is a difference. We don't know 100% sure that there is, but statistically we are in favor of the idea that there is a difference between the proportion of men and the proportion of women who are going to vote for the candidate."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a 95% chance that the true mean, and let me put this here, this is also the same thing as the mean of the sampling distribution of the sampling mean, is in that interval. And to do that, let me just throw out a few ideas. What is the probability, that if I take a sample and I were to take a mean of that sample, so the probability that a random sample mean is within two standard deviations of the sampling mean of our sample mean. So what is this probability right over here? Let's just look at our actual distribution. So this is our distribution. This right here is our sampling mean."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is this probability right over here? Let's just look at our actual distribution. So this is our distribution. This right here is our sampling mean. Maybe I should do it in blue because that's the number up here. That's the color up here. This is our sampling mean."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right here is our sampling mean. Maybe I should do it in blue because that's the number up here. That's the color up here. This is our sampling mean. And so what is the probability that a random sampling mean is going to be in two standard deviations? Well, a random sampling mean is a sample from this distribution. It is a sample from the sampling distribution of the sample mean."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our sampling mean. And so what is the probability that a random sampling mean is going to be in two standard deviations? Well, a random sampling mean is a sample from this distribution. It is a sample from the sampling distribution of the sample mean. So it's literally, what is the probability of finding a sample within two standard deviations of the mean? That's one standard deviation. That's another standard deviation right over there."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It is a sample from the sampling distribution of the sample mean. So it's literally, what is the probability of finding a sample within two standard deviations of the mean? That's one standard deviation. That's another standard deviation right over there. And in general, if you haven't committed this to memory already, it's not a bad thing to commit to memory, is that if you have a normal distribution, the probability of taking a sample within two standard deviations is 95, and if you want to get a little bit more accurate, it's 95.4%. But you could say it's roughly 95%. And really that's all that matters because we have this little funny language here called reasonably confident, and we have to estimate the standard deviation anyway."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's another standard deviation right over there. And in general, if you haven't committed this to memory already, it's not a bad thing to commit to memory, is that if you have a normal distribution, the probability of taking a sample within two standard deviations is 95, and if you want to get a little bit more accurate, it's 95.4%. But you could say it's roughly 95%. And really that's all that matters because we have this little funny language here called reasonably confident, and we have to estimate the standard deviation anyway. In fact, we could say if we want, I could say it's going to be exactly equal to 95.4%. But in general, two standard deviations, 95%, that's what people equate with each other. Now, this statement is the exact same thing as the probability that the sampling mean, or the mean of the sample, not the sample mean, the probability of the mean of the sampling distribution is within two standard deviations of the sampling distribution of x is also going to be the same number, is also going to be equal to 95.4%."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And really that's all that matters because we have this little funny language here called reasonably confident, and we have to estimate the standard deviation anyway. In fact, we could say if we want, I could say it's going to be exactly equal to 95.4%. But in general, two standard deviations, 95%, that's what people equate with each other. Now, this statement is the exact same thing as the probability that the sampling mean, or the mean of the sample, not the sample mean, the probability of the mean of the sampling distribution is within two standard deviations of the sampling distribution of x is also going to be the same number, is also going to be equal to 95.4%. These are the exact same statement. If x is within two standard deviations of this, then this, then the mean is within two standard deviations of x. These are just two ways of phrasing the same thing."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, this statement is the exact same thing as the probability that the sampling mean, or the mean of the sample, not the sample mean, the probability of the mean of the sampling distribution is within two standard deviations of the sampling distribution of x is also going to be the same number, is also going to be equal to 95.4%. These are the exact same statement. If x is within two standard deviations of this, then this, then the mean is within two standard deviations of x. These are just two ways of phrasing the same thing. Now, we know that the mean of the sampling distribution is the same thing as the mean of the population distribution, which is the same thing as the parameter p, the proportion of people, or the proportion of the population that is a one. This right here is the same thing as the population mean. This statement right here, we can switch this with p. The probability that p is within two standard deviations of the sampling distribution of x is 95.4%."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are just two ways of phrasing the same thing. Now, we know that the mean of the sampling distribution is the same thing as the mean of the population distribution, which is the same thing as the parameter p, the proportion of people, or the proportion of the population that is a one. This right here is the same thing as the population mean. This statement right here, we can switch this with p. The probability that p is within two standard deviations of the sampling distribution of x is 95.4%. Now, we don't know what this number right here is, but we have estimated it. Remember, our best estimate of this is the true standard, or it is the true standard deviation of the population divided by 10. We can estimate the true standard deviation of the population with our sampling standard deviation, which was 0.5, 0.5 divided by 10."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This statement right here, we can switch this with p. The probability that p is within two standard deviations of the sampling distribution of x is 95.4%. Now, we don't know what this number right here is, but we have estimated it. Remember, our best estimate of this is the true standard, or it is the true standard deviation of the population divided by 10. We can estimate the true standard deviation of the population with our sampling standard deviation, which was 0.5, 0.5 divided by 10. Our best estimate of the standard deviation of the sampling distribution of the sample mean is 0.05. Now, we can say, and I'll switch colors, the probability that the parameter p, the proportion of the population saying one, is within two times, remember, our best estimate of this right here is 0.05, of a sample mean that we take is equal to 95.4%. And so, we could say the probability that p is within two times 0.05 is going to be equal to, 2.0 is going to be 0.10, of our mean is equal to 95."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We can estimate the true standard deviation of the population with our sampling standard deviation, which was 0.5, 0.5 divided by 10. Our best estimate of the standard deviation of the sampling distribution of the sample mean is 0.05. Now, we can say, and I'll switch colors, the probability that the parameter p, the proportion of the population saying one, is within two times, remember, our best estimate of this right here is 0.05, of a sample mean that we take is equal to 95.4%. And so, we could say the probability that p is within two times 0.05 is going to be equal to, 2.0 is going to be 0.10, of our mean is equal to 95. And actually, let me be a little careful here. I can't say the equal now, because over here, if we knew this, if we knew this parameter of the sampling distribution of the sample mean, we could say that it is 95.4%. We don't know it."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so, we could say the probability that p is within two times 0.05 is going to be equal to, 2.0 is going to be 0.10, of our mean is equal to 95. And actually, let me be a little careful here. I can't say the equal now, because over here, if we knew this, if we knew this parameter of the sampling distribution of the sample mean, we could say that it is 95.4%. We don't know it. We are just trying to find our best estimator for it. So actually, what I'm going to do here is actually just say is roughly, and just to show that we don't even have that level of accuracy, I'm going to say roughly 95%. We're reasonably confident that it's about 95% because we're using this estimator that came out of our sample."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't know it. We are just trying to find our best estimator for it. So actually, what I'm going to do here is actually just say is roughly, and just to show that we don't even have that level of accuracy, I'm going to say roughly 95%. We're reasonably confident that it's about 95% because we're using this estimator that came out of our sample. And if the sample is really skewed, this is going to be a really weird number. So this is why we just have to be a little bit more exact about what we're doing. But this is a tool for at least saying how good is our result."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're reasonably confident that it's about 95% because we're using this estimator that came out of our sample. And if the sample is really skewed, this is going to be a really weird number. So this is why we just have to be a little bit more exact about what we're doing. But this is a tool for at least saying how good is our result. And so, this is going to be about 95%. Or we could say that the probability that p is within 0.10 of our sample mean that we actually got. So what was the sample mean that we actually got?"}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But this is a tool for at least saying how good is our result. And so, this is going to be about 95%. Or we could say that the probability that p is within 0.10 of our sample mean that we actually got. So what was the sample mean that we actually got? It was 0.43. So if we're within 0.1 of 0.43, that means we are within 0.43 plus or minus 0.1 is also, roughly, we're reasonably confident, it's about 95%. And I want to be very clear."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what was the sample mean that we actually got? It was 0.43. So if we're within 0.1 of 0.43, that means we are within 0.43 plus or minus 0.1 is also, roughly, we're reasonably confident, it's about 95%. And I want to be very clear. Everything that I started, all the way from pure and brown to yellow to all this magenta, I'm just restating the same thing inside of this. It became a little bit more loosey-goosey once I went from the exact standard deviation of the sampling distribution to an estimator for it. And that's why this is just becoming, I kind of put the squiggly equal signs there to say, we're reasonably confident."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I want to be very clear. Everything that I started, all the way from pure and brown to yellow to all this magenta, I'm just restating the same thing inside of this. It became a little bit more loosey-goosey once I went from the exact standard deviation of the sampling distribution to an estimator for it. And that's why this is just becoming, I kind of put the squiggly equal signs there to say, we're reasonably confident. I even got rid of some of the precision. But we just found our interval. An interval that we can be reasonably confident that there's a 95% probability that p is within that is going to be 0.43 plus or minus 0.1."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's why this is just becoming, I kind of put the squiggly equal signs there to say, we're reasonably confident. I even got rid of some of the precision. But we just found our interval. An interval that we can be reasonably confident that there's a 95% probability that p is within that is going to be 0.43 plus or minus 0.1. Or an interval of, we have a confidence interval. We have a 95% confidence interval. And we could say 0.43 minus 0.1."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "An interval that we can be reasonably confident that there's a 95% probability that p is within that is going to be 0.43 plus or minus 0.1. Or an interval of, we have a confidence interval. We have a 95% confidence interval. And we could say 0.43 minus 0.1. Minus 0.1 is 0.33. If we write that as a percent, we could say 33% 2. And if we add the 0.1, 0.43 plus 0.1, we get 53%."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we could say 0.43 minus 0.1. Minus 0.1 is 0.33. If we write that as a percent, we could say 33% 2. And if we add the 0.1, 0.43 plus 0.1, we get 53%. 2, 53%. So we are 95% confident. So we're not saying precisely that the probability of the actual proportion is 95%."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we add the 0.1, 0.43 plus 0.1, we get 53%. 2, 53%. So we are 95% confident. So we're not saying precisely that the probability of the actual proportion is 95%. But we're 95% confident that the true proportion is between 33% and 55%. That p is in this range over here. Or another way, and you'll see this in a lot of surveys that have been done, people will say, we did a survey."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're not saying precisely that the probability of the actual proportion is 95%. But we're 95% confident that the true proportion is between 33% and 55%. That p is in this range over here. Or another way, and you'll see this in a lot of surveys that have been done, people will say, we did a survey. And we got 43% will vote for number one. And number one in this case is candidate B. For candidate B."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way, and you'll see this in a lot of surveys that have been done, people will say, we did a survey. And we got 43% will vote for number one. And number one in this case is candidate B. For candidate B. And then the other side, since everyone else voted for candidate A, 57% will vote for A. And then they're going to put a margin of error. And you'll see this in any survey that you see on TV."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "For candidate B. And then the other side, since everyone else voted for candidate A, 57% will vote for A. And then they're going to put a margin of error. And you'll see this in any survey that you see on TV. They'll put a margin of error. And the margin of error is just another way of describing this confidence interval. And they'll say that the margin of error in this case is 10%."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you'll see this in any survey that you see on TV. They'll put a margin of error. And the margin of error is just another way of describing this confidence interval. And they'll say that the margin of error in this case is 10%. Which means that there's a 95% confidence interval if you go plus or minus 10% from that value right over there. And I really want to emphasize, you can't say with certainty that there is a 95% chance that the true result will be within 10% of this. Because we had to estimate the standard deviation of the sampling mean."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And they'll say that the margin of error in this case is 10%. Which means that there's a 95% confidence interval if you go plus or minus 10% from that value right over there. And I really want to emphasize, you can't say with certainty that there is a 95% chance that the true result will be within 10% of this. Because we had to estimate the standard deviation of the sampling mean. But this is the best measure we can with the information you have. If you're going to do a survey of 100 people, this is the best kind of confidence that we can get. And this number is actually fairly big."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because we had to estimate the standard deviation of the sampling mean. But this is the best measure we can with the information you have. If you're going to do a survey of 100 people, this is the best kind of confidence that we can get. And this number is actually fairly big. So if you were to look at this, you would say, roughly, there's a 95% chance that the true value of this number is between 33% and 53%. So there's actually still a chance that candidate B can win, even though only 43% of your 100 are going to vote for him. If you wanted to make it a little bit more precise, you would want to take more samples."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this number is actually fairly big. So if you were to look at this, you would say, roughly, there's a 95% chance that the true value of this number is between 33% and 53%. So there's actually still a chance that candidate B can win, even though only 43% of your 100 are going to vote for him. If you wanted to make it a little bit more precise, you would want to take more samples. You can imagine, instead of taking 100 samples, instead of n being 100, if you made n equal 1,000, then you would take this number over here and divide by the square root of 1,000 instead of the square root of 100. So you'd be dividing by 33 or whatever. And so then your margin of the number, the size of the standard deviation of your sampling distribution, will go down."}, {"video_title": "Margin of error 2   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you wanted to make it a little bit more precise, you would want to take more samples. You can imagine, instead of taking 100 samples, instead of n being 100, if you made n equal 1,000, then you would take this number over here and divide by the square root of 1,000 instead of the square root of 100. So you'd be dividing by 33 or whatever. And so then your margin of the number, the size of the standard deviation of your sampling distribution, will go down. And so the distance of two standard deviations will be a smaller number. And so then you will have a smaller margin of error. And maybe you want to get the margin of error small enough so that you can figure out decisively who's going to win the election."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And these, well, I was going to say that they tend to be integers, but they don't always have to be integers. You have discrete random, and so finite meaning you can't have an infinite number of values for a discrete random variable. Well, then we have the continuous, which can take on an infinite number. And the example I gave for continuous is, let's say, random variable X. And people do tend to use, let me change it a little bit, just so you can see, it can be something other than an X. Let's say I have the random variable capital Y. They do tend to be capital letters."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the example I gave for continuous is, let's say, random variable X. And people do tend to use, let me change it a little bit, just so you can see, it can be something other than an X. Let's say I have the random variable capital Y. They do tend to be capital letters. Is equal to the exact amount of rain tomorrow. And I say rain because I'm in Northern California. It's actually raining quite hard right now."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "They do tend to be capital letters. Is equal to the exact amount of rain tomorrow. And I say rain because I'm in Northern California. It's actually raining quite hard right now. Which, we're short right now, so that's a positive. We've been having a drought, so it's a good thing. But the exact amount of rain tomorrow."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually raining quite hard right now. Which, we're short right now, so that's a positive. We've been having a drought, so it's a good thing. But the exact amount of rain tomorrow. And let's say, I don't know what the actual probability distribution function for this is, but I'll draw one. And then we'll interpret it. Just so you can kind of think about continuous random variables."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the exact amount of rain tomorrow. And let's say, I don't know what the actual probability distribution function for this is, but I'll draw one. And then we'll interpret it. Just so you can kind of think about continuous random variables. Let me draw its probability distribution, or they call it its probability density function. Let me draw it like this. And let's say that there is, let me think, it looks something like this."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just so you can kind of think about continuous random variables. Let me draw its probability distribution, or they call it its probability density function. Let me draw it like this. And let's say that there is, let me think, it looks something like this. All right. And then, I don't know what this height is. So this, the x-axis here is the amount of rain."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's say that there is, let me think, it looks something like this. All right. And then, I don't know what this height is. So this, the x-axis here is the amount of rain. Where this is 0 inches, this is 1 inch, this is 2 inches, this is 3 inches, 4 inches. And then this is some height. Let's say it peaks out here at, I don't know, let's say this is 0.5."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this, the x-axis here is the amount of rain. Where this is 0 inches, this is 1 inch, this is 2 inches, this is 3 inches, 4 inches. And then this is some height. Let's say it peaks out here at, I don't know, let's say this is 0.5. So the way to think about it, if you were to look at this and I were to ask you, what is the probability that y, because that's our random variable now, that y is exactly equal to 2 inches? That y is exactly equal to 2 inches. What's the probability of that happening?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say it peaks out here at, I don't know, let's say this is 0.5. So the way to think about it, if you were to look at this and I were to ask you, what is the probability that y, because that's our random variable now, that y is exactly equal to 2 inches? That y is exactly equal to 2 inches. What's the probability of that happening? Well, based on how we thought about the probability distribution functions for the discrete random variable, you'd say, OK, let's see, 2 inches, that's the case we care about right now. Let me go up here. Say, OK, it looks like it's about 0.5."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's the probability of that happening? Well, based on how we thought about the probability distribution functions for the discrete random variable, you'd say, OK, let's see, 2 inches, that's the case we care about right now. Let me go up here. Say, OK, it looks like it's about 0.5. And you say, well, I don't know, is it a 0.5 chance? And I would say, no, it is not a 0.5 chance. And before we even think about how we would interpret it visually, let's just think about it logically."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Say, OK, it looks like it's about 0.5. And you say, well, I don't know, is it a 0.5 chance? And I would say, no, it is not a 0.5 chance. And before we even think about how we would interpret it visually, let's just think about it logically. What is the probability that tomorrow we have exactly 2 inches of rain? Not 2.01 inches of rain, not 1.99 inches of rain, not 1.9999 inches of rain, not 2.00001 inches of rain, exactly 2 inches of rain. There's not a single extra atom water molecule above the 2 inch mark and not a single water molecule below the 2 inch mark."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And before we even think about how we would interpret it visually, let's just think about it logically. What is the probability that tomorrow we have exactly 2 inches of rain? Not 2.01 inches of rain, not 1.99 inches of rain, not 1.9999 inches of rain, not 2.00001 inches of rain, exactly 2 inches of rain. There's not a single extra atom water molecule above the 2 inch mark and not a single water molecule below the 2 inch mark. It's essentially 0, right? It might not be obvious to you because you probably heard, oh, we had 2 inches of rain last night. But think about the exactly 2 inches, right?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's not a single extra atom water molecule above the 2 inch mark and not a single water molecule below the 2 inch mark. It's essentially 0, right? It might not be obvious to you because you probably heard, oh, we had 2 inches of rain last night. But think about the exactly 2 inches, right? Normally if it's like 2.01, people will say that's 2. But we're saying, no, this does not count. It can't be 2 inches."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But think about the exactly 2 inches, right? Normally if it's like 2.01, people will say that's 2. But we're saying, no, this does not count. It can't be 2 inches. We want exactly 2. 1.99 does not count. Normally, I mean, our measurements, we don't even have tools that can tell us whether it is exactly 2 inches, right?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It can't be 2 inches. We want exactly 2. 1.99 does not count. Normally, I mean, our measurements, we don't even have tools that can tell us whether it is exactly 2 inches, right? No ruler you can even say is exactly 2 inches long. At some point, just the way we manufacture things, there's going to be an extra atom on it here or there. So the odds of actually anything being exactly a measurement to the exact infinite decimal point is actually 0."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Normally, I mean, our measurements, we don't even have tools that can tell us whether it is exactly 2 inches, right? No ruler you can even say is exactly 2 inches long. At some point, just the way we manufacture things, there's going to be an extra atom on it here or there. So the odds of actually anything being exactly a measurement to the exact infinite decimal point is actually 0. The way you would think about a continuous random variable, you could say, what is the probability that y is almost 2? So if we said that the absolute value of y minus 2 is less than some tolerance, is less than, I don't know, 0.1. And if that doesn't make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the odds of actually anything being exactly a measurement to the exact infinite decimal point is actually 0. The way you would think about a continuous random variable, you could say, what is the probability that y is almost 2? So if we said that the absolute value of y minus 2 is less than some tolerance, is less than, I don't know, 0.1. And if that doesn't make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1? These two statements are equivalent. I'll let you think about it a little bit. But now this starts to make a little bit sense."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if that doesn't make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1? These two statements are equivalent. I'll let you think about it a little bit. But now this starts to make a little bit sense. Now we have an interval here. So we want all y's between 1.9 and 2.1. So we are now talking about this whole area."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But now this starts to make a little bit sense. Now we have an interval here. So we want all y's between 1.9 and 2.1. So we are now talking about this whole area. And area is key. So if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point. And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we are now talking about this whole area. And area is key. So if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point. And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point. So from, let me see, I've run out of space down here. So let's say if this graph, let me draw it in a different color, if this line was defined by, I don't know, I'll call it f of x. I could call it p of x or something. The probability of this happening would be equal to the integral, for those of you who have studied calculus, from 1.9 to 2.1 of f of x dx, assuming this is the x-axis."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point. So from, let me see, I've run out of space down here. So let's say if this graph, let me draw it in a different color, if this line was defined by, I don't know, I'll call it f of x. I could call it p of x or something. The probability of this happening would be equal to the integral, for those of you who have studied calculus, from 1.9 to 2.1 of f of x dx, assuming this is the x-axis. So it's a very important thing to realize. Because when a random variable can take on an infinite number of values, or it can take on any value between an interval, to get an exact value, to get exactly 1.999, the probability is actually 0. It's like asking you, what is the area under a curve on just this line?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability of this happening would be equal to the integral, for those of you who have studied calculus, from 1.9 to 2.1 of f of x dx, assuming this is the x-axis. So it's a very important thing to realize. Because when a random variable can take on an infinite number of values, or it can take on any value between an interval, to get an exact value, to get exactly 1.999, the probability is actually 0. It's like asking you, what is the area under a curve on just this line? Or even more specifically, it's like asking you, what's the area of a line? An area of a line, if you were to just draw a line, you'd say, well, area is height times base. Well, the height has some dimension, but the base, what's the width of a line?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's like asking you, what is the area under a curve on just this line? Or even more specifically, it's like asking you, what's the area of a line? An area of a line, if you were to just draw a line, you'd say, well, area is height times base. Well, the height has some dimension, but the base, what's the width of a line? As far as the way we've defined a line, a line has no width, and therefore no area. And it should make intuitive sense that you cannot, the probability of a very super exact thing happening is pretty much 0. That you really have to say, OK, what's the probability that we get close to 2?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the height has some dimension, but the base, what's the width of a line? As far as the way we've defined a line, a line has no width, and therefore no area. And it should make intuitive sense that you cannot, the probability of a very super exact thing happening is pretty much 0. That you really have to say, OK, what's the probability that we get close to 2? And then you can define an area. And if you said, oh, what's the probability that we get someplace between 1 and 3 inches of rain, then of course the probability is much higher. It would be all of this kind of stuff."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "That you really have to say, OK, what's the probability that we get close to 2? And then you can define an area. And if you said, oh, what's the probability that we get someplace between 1 and 3 inches of rain, then of course the probability is much higher. It would be all of this kind of stuff. You could also say, what's the probability we have less than 0.1 inches of rain? Then you would go here, and you would calculate, if this was 0.1, you would calculate this area. And you could say, what's the probability that we have more than 4 inches of rain tomorrow?"}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It would be all of this kind of stuff. You could also say, what's the probability we have less than 0.1 inches of rain? Then you would go here, and you would calculate, if this was 0.1, you would calculate this area. And you could say, what's the probability that we have more than 4 inches of rain tomorrow? Then you would start here, and you would calculate the area on the curve all the way to infinity, if the curve has area all the way to infinity. And hopefully that's not an infinite number, right? Then your probability won't make any sense."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you could say, what's the probability that we have more than 4 inches of rain tomorrow? Then you would start here, and you would calculate the area on the curve all the way to infinity, if the curve has area all the way to infinity. And hopefully that's not an infinite number, right? Then your probability won't make any sense. But hopefully if you take this sum, it comes to some number, and we'll say, oh, there's only a 10% chance that you have more than 4 inches tomorrow. And all of this should kind of immediately lead to one light bulb in your head, is that the probability of all of the events that might occur can't be more than 100%, right? All of the events combined can't, you know, there's a probability of 1 that one of these events will occur."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then your probability won't make any sense. But hopefully if you take this sum, it comes to some number, and we'll say, oh, there's only a 10% chance that you have more than 4 inches tomorrow. And all of this should kind of immediately lead to one light bulb in your head, is that the probability of all of the events that might occur can't be more than 100%, right? All of the events combined can't, you know, there's a probability of 1 that one of these events will occur. So essentially, the whole area under this curve has to be equal to 1. So if we took the integral of f of x from 0 to infinity, this thing, at least as I've drawn it, dx should be equal to 1, for those of you who have studied calculus. For those of you who haven't, an integral is just the area under a curve, and you can watch the calculus videos if you want to learn a little bit more about how to do them."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of the events combined can't, you know, there's a probability of 1 that one of these events will occur. So essentially, the whole area under this curve has to be equal to 1. So if we took the integral of f of x from 0 to infinity, this thing, at least as I've drawn it, dx should be equal to 1, for those of you who have studied calculus. For those of you who haven't, an integral is just the area under a curve, and you can watch the calculus videos if you want to learn a little bit more about how to do them. And this also applies to the discrete probability distributions. Let me draw one. The sum of all of the probabilities have to be equal to 1."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "For those of you who haven't, an integral is just the area under a curve, and you can watch the calculus videos if you want to learn a little bit more about how to do them. And this also applies to the discrete probability distributions. Let me draw one. The sum of all of the probabilities have to be equal to 1. In that example with the dice, or let's say, since it's faster to draw the coin, the two probabilities have to be equal to 1. So if this is 1, 0, where x is equal to 1 if we're heads, or 0 if we're tails, each of these have to be 0.5. Or they don't have to be 0.5, but if 1 was 0.6, the other one would have to be 0.4."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The sum of all of the probabilities have to be equal to 1. In that example with the dice, or let's say, since it's faster to draw the coin, the two probabilities have to be equal to 1. So if this is 1, 0, where x is equal to 1 if we're heads, or 0 if we're tails, each of these have to be 0.5. Or they don't have to be 0.5, but if 1 was 0.6, the other one would have to be 0.4. They have to add to 1. If one of these was, you can't have a 60% probability of getting a heads, and then a 60% probability of getting a tails as well, because then you would have essentially a 120% probability of either of the outcomes happening, which makes no sense at all. So it's important to realize that a probability distribution function, or probability distribution function in this case for a discrete random variable, they all have to add up to 1."}, {"video_title": "Probability density functions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or they don't have to be 0.5, but if 1 was 0.6, the other one would have to be 0.4. They have to add to 1. If one of these was, you can't have a 60% probability of getting a heads, and then a 60% probability of getting a tails as well, because then you would have essentially a 120% probability of either of the outcomes happening, which makes no sense at all. So it's important to realize that a probability distribution function, or probability distribution function in this case for a discrete random variable, they all have to add up to 1. It's 0.5 plus 0.5. And in this case, the area under the probability density function also has to be equal to 1. Anyway, I'm all out of time for now."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "All right, so where we left off, we had simplified our algebraic expression for the squared error to the line from the end data points. And we kind of visualized this expression right here would be a surface in, I guess you could view it as a surface in three dimensions. Where for any m and b is going to be a point on that surface that represents the squared error for that line. And our goal is to find the m and the b, which would define an actual line, to find the m and b that minimize the squared error. And the way that we do that is we find a point where the partial derivative of the squared error with respect to m is 0, and the partial derivative with respect to b is also equal to 0. So it's flat with respect to m. So that means that the slope in this direction is going to be flat. Let me do it in the same color."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And our goal is to find the m and the b, which would define an actual line, to find the m and b that minimize the squared error. And the way that we do that is we find a point where the partial derivative of the squared error with respect to m is 0, and the partial derivative with respect to b is also equal to 0. So it's flat with respect to m. So that means that the slope in this direction is going to be flat. Let me do it in the same color. So the slope in this direction, that's the partial derivative with respect to m, is going to be flat. It's not going to change in that direction. And the partial derivative with respect to b is going to be flat."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Let me do it in the same color. So the slope in this direction, that's the partial derivative with respect to m, is going to be flat. It's not going to change in that direction. And the partial derivative with respect to b is going to be flat. So it will be a flat point right over there. The slope of that point in that direction will also be 0, and that is our minimum point. Let's figure out the m and b's that give us this."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And the partial derivative with respect to b is going to be flat. So it will be a flat point right over there. The slope of that point in that direction will also be 0, and that is our minimum point. Let's figure out the m and b's that give us this. So if I were to take the partial derivative of this expression with respect to m. So the partial derivative of this expression with respect to m. Well, this first term has no m terms in it, so it's a constant from the point of view of m. And just as a reminder, partial derivatives is just like taking a regular derivative, and you're just assuming that everything but the variable that you're taking the partial derivative with respect to, you're assuming everything else is a constant. So in this expression, all the x's, the y's, the b's, the n's, those are all constant. The only variable when we take the partial derivative with respect to m that matters is the m. So this is a constant, there's no m here."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Let's figure out the m and b's that give us this. So if I were to take the partial derivative of this expression with respect to m. So the partial derivative of this expression with respect to m. Well, this first term has no m terms in it, so it's a constant from the point of view of m. And just as a reminder, partial derivatives is just like taking a regular derivative, and you're just assuming that everything but the variable that you're taking the partial derivative with respect to, you're assuming everything else is a constant. So in this expression, all the x's, the y's, the b's, the n's, those are all constant. The only variable when we take the partial derivative with respect to m that matters is the m. So this is a constant, there's no m here. This term right over here, we're taking with respect to m. So the derivative of this with respect to m is negative 2, it's kind of the coefficients on the m. So negative 2 times n times the mean of the xy's, that's the partial of this with respect to m. And then this term right here, this term right here has no m's in it, so it's constant with respect to m, so it's partial derivative with respect to m is 0. Then this term here, you have n times the mean of the x squared times m squared. So this is going to be, we're taking the partial derivative with respect to m, so it's going to be 2 times, so it's going to be plus 2 times, 2 times n times the mean of the x squared times m, right?"}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "The only variable when we take the partial derivative with respect to m that matters is the m. So this is a constant, there's no m here. This term right over here, we're taking with respect to m. So the derivative of this with respect to m is negative 2, it's kind of the coefficients on the m. So negative 2 times n times the mean of the xy's, that's the partial of this with respect to m. And then this term right here, this term right here has no m's in it, so it's constant with respect to m, so it's partial derivative with respect to m is 0. Then this term here, you have n times the mean of the x squared times m squared. So this is going to be, we're taking the partial derivative with respect to m, so it's going to be 2 times, so it's going to be plus 2 times, 2 times n times the mean of the x squared times m, right? The derivative of m squared is 2m, and then you just have this coefficient there as well. Now this term, you also have an m over there, so let's see. Everything else is just kind of a coefficient on this m, so the derivative with respect to m is 2bn times the mean of the x's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So this is going to be, we're taking the partial derivative with respect to m, so it's going to be 2 times, so it's going to be plus 2 times, 2 times n times the mean of the x squared times m, right? The derivative of m squared is 2m, and then you just have this coefficient there as well. Now this term, you also have an m over there, so let's see. Everything else is just kind of a coefficient on this m, so the derivative with respect to m is 2bn times the mean of the x's. If I took the derivative of 3m, the derivative is just 3, it's just the coefficient on it. And then finally, this is a constant with respect to m, so we don't see it. And so this is the partial derivative with respect to m, that's that right over there, and we want to set this equal to 0."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Everything else is just kind of a coefficient on this m, so the derivative with respect to m is 2bn times the mean of the x's. If I took the derivative of 3m, the derivative is just 3, it's just the coefficient on it. And then finally, this is a constant with respect to m, so we don't see it. And so this is the partial derivative with respect to m, that's that right over there, and we want to set this equal to 0. Now let's do the same thing with respect to b. This term, once again, is a constant from the perspective of b. There's no b here, there's no b over here, so the partial derivatives of either of these with respect to b is 0."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And so this is the partial derivative with respect to m, that's that right over there, and we want to set this equal to 0. Now let's do the same thing with respect to b. This term, once again, is a constant from the perspective of b. There's no b here, there's no b over here, so the partial derivatives of either of these with respect to b is 0. Then over here, you have a negative 2n times the mean of y's as a coefficient on a b. So the partial derivative with respect to b is going to be minus 2n, or negative 2n times the mean of the y's. And then there's no b over here, and we do have a b over here, so it's plus 2mn times the mean of the x's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "There's no b here, there's no b over here, so the partial derivatives of either of these with respect to b is 0. Then over here, you have a negative 2n times the mean of y's as a coefficient on a b. So the partial derivative with respect to b is going to be minus 2n, or negative 2n times the mean of the y's. And then there's no b over here, and we do have a b over here, so it's plus 2mn times the mean of the x's. Plus 2mn times the mean of the x's. This is essentially the coefficient on the b over here. It was written in a mixed order, but all of these are constants from the point of view of b."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then there's no b over here, and we do have a b over here, so it's plus 2mn times the mean of the x's. Plus 2mn times the mean of the x's. This is essentially the coefficient on the b over here. It was written in a mixed order, but all of these are constants from the point of view of b. They're the coefficient in front of the b. The partial derivative of that with respect to b is just going to be the coefficient. And then finally, the partial derivative of this with respect to b is going to be 2nb, or 2nb to the first."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "It was written in a mixed order, but all of these are constants from the point of view of b. They're the coefficient in front of the b. The partial derivative of that with respect to b is just going to be the coefficient. And then finally, the partial derivative of this with respect to b is going to be 2nb, or 2nb to the first. You could even say 2bn, or 2nb is the partial derivative of that with respect to b. And we want to set this equal to 0. So it looks very complicated, but remember we're just trying to solve for the m's and the b's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then finally, the partial derivative of this with respect to b is going to be 2nb, or 2nb to the first. You could even say 2bn, or 2nb is the partial derivative of that with respect to b. And we want to set this equal to 0. So it looks very complicated, but remember we're just trying to solve for the m's and the b's. And we have two equations with two unknowns here. We have the m's, we have the m's, and then we have the b's. And to simplify this, both of these equations, actually the top one and the bottom one, both sides are divisible by 2n."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So it looks very complicated, but remember we're just trying to solve for the m's and the b's. And we have two equations with two unknowns here. We have the m's, we have the m's, and then we have the b's. And to simplify this, both of these equations, actually the top one and the bottom one, both sides are divisible by 2n. I mean, 0 is divisible by anything. It'll be just 0. So let's divide the top equation by 2n and see what we get."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And to simplify this, both of these equations, actually the top one and the bottom one, both sides are divisible by 2n. I mean, 0 is divisible by anything. It'll be just 0. So let's divide the top equation by 2n and see what we get. If we divide the top equation by 2n, we can even see it here. This will become just 1. That goes away, and then those go away, and you would just be left with, you are left with negative times the mean, the negative mean of the x, y's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So let's divide the top equation by 2n and see what we get. If we divide the top equation by 2n, we can even see it here. This will become just 1. That goes away, and then those go away, and you would just be left with, you are left with negative times the mean, the negative mean of the x, y's. Plus m times the mean of the x squareds, m times the mean of the x squareds, plus b times the mean of the x's is equal to 0. That's this first expression when you divide both sides by negative 2n. And then the second expression will be, this will go away."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "That goes away, and then those go away, and you would just be left with, you are left with negative times the mean, the negative mean of the x, y's. Plus m times the mean of the x squareds, m times the mean of the x squareds, plus b times the mean of the x's is equal to 0. That's this first expression when you divide both sides by negative 2n. And then the second expression will be, this will go away. This is when you divide it by 2n. I don't want to say negative 2n. That's when you divide it by 2n, you get this."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then the second expression will be, this will go away. This is when you divide it by 2n. I don't want to say negative 2n. That's when you divide it by 2n, you get this. And when you divide this by 2n, that'll go away, that will go away, and then those will go away. And you're just left with negative, the negative mean of the y's, plus m times the mean of the x's, plus b is equal to 0. So if we find the m and the b values that satisfy the system of equations, we have minimized the squared error."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "That's when you divide it by 2n, you get this. And when you divide this by 2n, that'll go away, that will go away, and then those will go away. And you're just left with negative, the negative mean of the y's, plus m times the mean of the x's, plus b is equal to 0. So if we find the m and the b values that satisfy the system of equations, we have minimized the squared error. And we could just solve it in a traditional way, but I want to rewrite this, because I think it's kind of interesting to see what these really represent. So let's add this mean of the x, y's to both sides of this top equation. And then we're going to have, so if we add the mean of the x, y's to both sides of this top equation, what do we get?"}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So if we find the m and the b values that satisfy the system of equations, we have minimized the squared error. And we could just solve it in a traditional way, but I want to rewrite this, because I think it's kind of interesting to see what these really represent. So let's add this mean of the x, y's to both sides of this top equation. And then we're going to have, so if we add the mean of the x, y's to both sides of this top equation, what do we get? We get m times the mean of the x squareds, plus b times the mean of the x's, is equal to, these are going to cancel out, is equal to the mean of the x, y's. That's that top equation. This bottom equation right here, let's add the mean of y to both sides of this equation."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then we're going to have, so if we add the mean of the x, y's to both sides of this top equation, what do we get? We get m times the mean of the x squareds, plus b times the mean of the x's, is equal to, these are going to cancel out, is equal to the mean of the x, y's. That's that top equation. This bottom equation right here, let's add the mean of y to both sides of this equation. And I do that so that that cancels out. And then we're left with m, do that in the blue color, show you the same equation. We have m times the mean of the x's, plus b, is equal to the mean of the y's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This bottom equation right here, let's add the mean of y to both sides of this equation. And I do that so that that cancels out. And then we're left with m, do that in the blue color, show you the same equation. We have m times the mean of the x's, plus b, is equal to the mean of the y's. Now, I actually want to get both of these into mx plus b form. This is actually already there. So you can see that if our line, if our best fitting line is going to be y is equal to mx plus b, we still have to find the m and the b."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We have m times the mean of the x's, plus b, is equal to the mean of the y's. Now, I actually want to get both of these into mx plus b form. This is actually already there. So you can see that if our line, if our best fitting line is going to be y is equal to mx plus b, we still have to find the m and the b. But we see on that best fitting line, because the m and the b that satisfy both of these equations are going to be the m and the b on that best fitting line. So that best fitting line actually contains the point. It actually contains the point."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So you can see that if our line, if our best fitting line is going to be y is equal to mx plus b, we still have to find the m and the b. But we see on that best fitting line, because the m and the b that satisfy both of these equations are going to be the m and the b on that best fitting line. So that best fitting line actually contains the point. It actually contains the point. And we get this from the second equation right here. It contains the point, or the point, I should write it this way, the coordinate mean of x, mean of y, lies on this point. I should say lies on the line."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "It actually contains the point. And we get this from the second equation right here. It contains the point, or the point, I should write it this way, the coordinate mean of x, mean of y, lies on this point. I should say lies on the line. And you could see it right over here. If you put the mean of x in this for the optimal m and b, you're going to get the mean of the y. So that's interesting."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "I should say lies on the line. And you could see it right over here. If you put the mean of x in this for the optimal m and b, you're going to get the mean of the y. So that's interesting. This optimal line, let's never forget what we're even trying to do. This optimal line is going to contain some point on it. Let me do that in a new color."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So that's interesting. This optimal line, let's never forget what we're even trying to do. This optimal line is going to contain some point on it. Let me do that in a new color. It's going to contain some point on it right here that is the mean of all of the x values and the mean of all the y values. That's just interesting. And it kind of makes sense."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Let me do that in a new color. It's going to contain some point on it right here that is the mean of all of the x values and the mean of all the y values. That's just interesting. And it kind of makes sense. It kind of makes intuitive sense. Now, this other thing, just to kind of get it in the same point of view. And then it'll actually become kind of an easier way to solve the system."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And it kind of makes sense. It kind of makes intuitive sense. Now, this other thing, just to kind of get it in the same point of view. And then it'll actually become kind of an easier way to solve the system. You could solve this a million different ways. But just to give us an intuition of what even is going on here. What's another point that's on that line?"}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then it'll actually become kind of an easier way to solve the system. You could solve this a million different ways. But just to give us an intuition of what even is going on here. What's another point that's on that line? Because if you have two points on the line, you know what the equation of the line is going to be. Well, the other point, if we want this to be an mx plus b form. So let's divide both sides of this equation by this term right here, by the mean of the x's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "What's another point that's on that line? Because if you have two points on the line, you know what the equation of the line is going to be. Well, the other point, if we want this to be an mx plus b form. So let's divide both sides of this equation by this term right here, by the mean of the x's. And if we do that, we get m times the mean of the x squareds divided by the mean of the x's plus b is equal to the mean of the xy's divided by the mean of the x's. And so when you write it in this form, this is the exact same equation as that. I just divided both sides by the mean of the x's."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So let's divide both sides of this equation by this term right here, by the mean of the x's. And if we do that, we get m times the mean of the x squareds divided by the mean of the x's plus b is equal to the mean of the xy's divided by the mean of the x's. And so when you write it in this form, this is the exact same equation as that. I just divided both sides by the mean of the x's. You get another interesting point that will lie on this optimal fitting line, at least from the point of view of the squared distances. So another point that will lie on it. So another point that will lie on this optimal line is going to be the point, the x value is going to be this."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "I just divided both sides by the mean of the x's. You get another interesting point that will lie on this optimal fitting line, at least from the point of view of the squared distances. So another point that will lie on it. So another point that will lie on this optimal line is going to be the point, the x value is going to be this. So it's going to be the coordinate, the mean of the x squareds divided by the mean of the x's. And then the y value is going to be the mean of the xy's divided by the mean of the x's. And I'll let you think about that a little bit more."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So another point that will lie on this optimal line is going to be the point, the x value is going to be this. So it's going to be the coordinate, the mean of the x squareds divided by the mean of the x's. And then the y value is going to be the mean of the xy's divided by the mean of the x's. And I'll let you think about that a little bit more. But already, this is actually the two points that lie on the line. So both of these on the line. Both of these on the best fitting line, based on how we're measuring a good fit, which is the square distance."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And I'll let you think about that a little bit more. But already, this is actually the two points that lie on the line. So both of these on the line. Both of these on the best fitting line, based on how we're measuring a good fit, which is the square distance. These are on the line that minimize that square distance. What I'm going to do in the next video, this is turning into like a six or seven video saga on trying to prove the best fitting line, or finding the formula for the best fitting line. But it's interesting."}, {"video_title": "Proof (part 3) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Both of these on the best fitting line, based on how we're measuring a good fit, which is the square distance. These are on the line that minimize that square distance. What I'm going to do in the next video, this is turning into like a six or seven video saga on trying to prove the best fitting line, or finding the formula for the best fitting line. But it's interesting. There's all sorts of kind of neat little mathematical things to ponder over here. But in the next video, we can actually use this information. We could have just solved the system straight up."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that you're curious about studying the dimensions of the cars that happen to sit in the parking lot. And so you measure their lengths. And so let's just make the computation simple. Let's say that there are five cars in the parking lot. The entire size of the population that we care about is five. And you go and measure their lengths. One car is four meters long."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that there are five cars in the parking lot. The entire size of the population that we care about is five. And you go and measure their lengths. One car is four meters long. The other car, another car is 4.2 meters long. Another car is five meters long. The fourth car is 4.3 meters long."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "One car is four meters long. The other car, another car is 4.2 meters long. Another car is five meters long. The fourth car is 4.3 meters long. And then let's say the fifth car is 5.5 meters long. So let's come up with some parameters for this population. So the first one that you might want to figure out is a measure of central tendency."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The fourth car is 4.3 meters long. And then let's say the fifth car is 5.5 meters long. So let's come up with some parameters for this population. So the first one that you might want to figure out is a measure of central tendency. And probably the most popular one is the arithmetic mean. So let's calculate that first. And we're going to do that for the population."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the first one that you might want to figure out is a measure of central tendency. And probably the most popular one is the arithmetic mean. So let's calculate that first. And we're going to do that for the population. So we're going to use mu. So what does the arithmetic mean here? Well, we just have to add all of these data points up and divide by 5."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to do that for the population. So we're going to use mu. So what does the arithmetic mean here? Well, we just have to add all of these data points up and divide by 5. And I'll just get the calculator out just so it's a little bit quicker. And so this is going to be 4 plus 4.2 plus 5 plus 4.3 plus 5.5. And then I'm going to take that sum and then divide by 5."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we just have to add all of these data points up and divide by 5. And I'll just get the calculator out just so it's a little bit quicker. And so this is going to be 4 plus 4.2 plus 5 plus 4.3 plus 5.5. And then I'm going to take that sum and then divide by 5. And I get an arithmetic mean for my population of 4.6. So that's fine. And if we want to put some units there, it's 4.6 meters."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then I'm going to take that sum and then divide by 5. And I get an arithmetic mean for my population of 4.6. So that's fine. And if we want to put some units there, it's 4.6 meters. Now, that's the central tendency or a measure of central tendency. We also might be curious about how dispersed is the data, especially from that central tendency. So what would we use?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we want to put some units there, it's 4.6 meters. Now, that's the central tendency or a measure of central tendency. We also might be curious about how dispersed is the data, especially from that central tendency. So what would we use? Well, we already have a tool at our disposal, the population variance. And the population variance is one of many ways of measuring dispersion. It has some very neat properties, which is why the way we've defined it as the mean of the squared distances from the mean tends to be a useful way of doing it."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what would we use? Well, we already have a tool at our disposal, the population variance. And the population variance is one of many ways of measuring dispersion. It has some very neat properties, which is why the way we've defined it as the mean of the squared distances from the mean tends to be a useful way of doing it. So let's just do that. Let's actually calculate the population variance for this population right over here. Well, all we need to do is find the distance from each of these points to our mean right over here, and then square them and then take the mean of those squared distances."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It has some very neat properties, which is why the way we've defined it as the mean of the squared distances from the mean tends to be a useful way of doing it. So let's just do that. Let's actually calculate the population variance for this population right over here. Well, all we need to do is find the distance from each of these points to our mean right over here, and then square them and then take the mean of those squared distances. So let's do that. So it's going to be 4 minus 4.6 squared plus 4.2 minus 4.6 squared plus 5 minus 4.6 squared plus 4.3 minus 4.6 squared. And then finally, I'm running out of space, plus 5.5 minus 4.6 squared."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, all we need to do is find the distance from each of these points to our mean right over here, and then square them and then take the mean of those squared distances. So let's do that. So it's going to be 4 minus 4.6 squared plus 4.2 minus 4.6 squared plus 5 minus 4.6 squared plus 4.3 minus 4.6 squared. And then finally, I'm running out of space, plus 5.5 minus 4.6 squared. And then we're going to divide all of that by 5 to get our population variance. And so what's that going to give us? Let's get our calculator out."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, I'm running out of space, plus 5.5 minus 4.6 squared. And then we're going to divide all of that by 5 to get our population variance. And so what's that going to give us? Let's get our calculator out. So 4 minus 4.6 squared, that's negative 0.6 squared. Negative 0.6 squared is going to be the exact same thing as 0.6 squared. So let me write that as 0.6 squared plus 4.2 minus 4.6 is negative 0.4."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's get our calculator out. So 4 minus 4.6 squared, that's negative 0.6 squared. Negative 0.6 squared is going to be the exact same thing as 0.6 squared. So let me write that as 0.6 squared plus 4.2 minus 4.6 is negative 0.4. But when we square it, the negative is going to disappear. So it's going to be plus 0.4. I'll just write 0.4 squared."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write that as 0.6 squared plus 4.2 minus 4.6 is negative 0.4. But when we square it, the negative is going to disappear. So it's going to be plus 0.4. I'll just write 0.4 squared. And then we have 5 minus 4.6. That's 0.4, so plus 0.4 squared. 4.3 minus 4.6, that's negative 0.3."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll just write 0.4 squared. And then we have 5 minus 4.6. That's 0.4, so plus 0.4 squared. 4.3 minus 4.6, that's negative 0.3. The negative goes away when you square it. So it's going to be plus 0.3 squared. And then finally, 5.5 minus 4.6 is going to be 0.9."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "4.3 minus 4.6, that's negative 0.3. The negative goes away when you square it. So it's going to be plus 0.3 squared. And then finally, 5.5 minus 4.6 is going to be 0.9. So plus 0.9 squared. And then we will divide by the number of data points we have. And we get 0.316."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, 5.5 minus 4.6 is going to be 0.9. So plus 0.9 squared. And then we will divide by the number of data points we have. And we get 0.316. Or if we want to write it, this is going to be 0.316. Now let me ask you what is a mildly interesting question. What would be the units?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we get 0.316. Or if we want to write it, this is going to be 0.316. Now let me ask you what is a mildly interesting question. What would be the units? What would be the units for this population variance? Since we happen to care about units in this video. Well, up here, this is 4 meters minus 4.6 meters."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What would be the units? What would be the units for this population variance? Since we happen to care about units in this video. Well, up here, this is 4 meters minus 4.6 meters. 4.2 meters minus 4.6 meters. So these are all meters. These are measurements in meters."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, up here, this is 4 meters minus 4.6 meters. 4.2 meters minus 4.6 meters. So these are all meters. These are measurements in meters. We saw it up here. So these are all measurements in meters. When you subtract them, you'll get meters."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are measurements in meters. We saw it up here. So these are all measurements in meters. When you subtract them, you'll get meters. But when you square them, you get meters squared plus meters squared plus meters squared plus meters squared plus meters squared. And then you're just dividing that by a unitless count of the number of data points you have. So the units here are going to be square meters."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "When you subtract them, you'll get meters. But when you square them, you get meters squared plus meters squared plus meters squared plus meters squared plus meters squared. And then you're just dividing that by a unitless count of the number of data points you have. So the units here are going to be square meters. And so you might say, hey, that's kind of a weird unit. If we're trying to figure out, if we're trying to visualize or think about how dispersed we are from the mean, when I visualize it, I visualize dispersion or how varied they are in terms of meters, not meters squared. So what could we do?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the units here are going to be square meters. And so you might say, hey, that's kind of a weird unit. If we're trying to figure out, if we're trying to visualize or think about how dispersed we are from the mean, when I visualize it, I visualize dispersion or how varied they are in terms of meters, not meters squared. So what could we do? And a big hint, this comes out of just even the notation for variance. It's this sigma symbol squared. So why don't we just take the square root of it?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what could we do? And a big hint, this comes out of just even the notation for variance. It's this sigma symbol squared. So why don't we just take the square root of it? So why don't we just take the square root of our variance, which we will denote with just a sigma. Makes a lot of sense. And in this case, what's it going to be?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So why don't we just take the square root of it? So why don't we just take the square root of our variance, which we will denote with just a sigma. Makes a lot of sense. And in this case, what's it going to be? It's going to be the square root of 0.316. And then what are the units going to be? It's going to be just meters."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in this case, what's it going to be? It's going to be the square root of 0.316. And then what are the units going to be? It's going to be just meters. And we end up with, so let me take the square root of 0.316. And I get 0.56. I'll just round to the nearest thousandth."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be just meters. And we end up with, so let me take the square root of 0.316. And I get 0.56. I'll just round to the nearest thousandth. 0.562. So it's approximately 0.562 meters. So you might be saying, Sal, what do we call this thing that we just did, the square root of the variance?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll just round to the nearest thousandth. 0.562. So it's approximately 0.562 meters. So you might be saying, Sal, what do we call this thing that we just did, the square root of the variance? And here we're dealing with the population. We haven't thought about sampling yet. The square root of the population variance, what do we call this thing right over here?"}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you might be saying, Sal, what do we call this thing that we just did, the square root of the variance? And here we're dealing with the population. We haven't thought about sampling yet. The square root of the population variance, what do we call this thing right over here? And this is a very familiar term. Oftentimes when you take an exam, this is calculated for the scores on the exam. This is our population."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The square root of the population variance, what do we call this thing right over here? And this is a very familiar term. Oftentimes when you take an exam, this is calculated for the scores on the exam. This is our population. Let me do this in a new color. I'm using that yellow a little bit too much. This is the population standard deviation."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our population. Let me do this in a new color. I'm using that yellow a little bit too much. This is the population standard deviation. It is a measure of how much the data is varying from the mean. In general, the larger this value, that means that the data is more varied from the population mean, the smaller. It's less varied."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the population standard deviation. It is a measure of how much the data is varying from the mean. In general, the larger this value, that means that the data is more varied from the population mean, the smaller. It's less varied. And these are all somewhat arbitrary definitions of how we've defined variance. We could have taken things to the fourth power. We could have done other things."}, {"video_title": "Population standard deviation   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's less varied. And these are all somewhat arbitrary definitions of how we've defined variance. We could have taken things to the fourth power. We could have done other things. We could have not taken them to a power, but taken the absolute value here. The reason why we do it this way is it has neat statistical properties as we try to build on it. But that's the population standard deviation, which gives us nice units, meters, meters."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And it's a pretty, hopefully you'll find, straightforward idea. If you're doing a trial, something that is probabilistic, a trial or an experiment, a sample space is just all of the, it's a set of the possible outcomes. So a very simple trial might be a coin flip. So if you're talking about a coin flip, well then the sample space is going to be the set of all the possible outcomes. So you could get a heads, or you could get a tails. That right over here is the sample space for the coin flip. So, and it's very useful because, for example, if these are equally likely outcomes, and you say, well what's the probability of the event of a heads?"}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So if you're talking about a coin flip, well then the sample space is going to be the set of all the possible outcomes. So you could get a heads, or you could get a tails. That right over here is the sample space for the coin flip. So, and it's very useful because, for example, if these are equally likely outcomes, and you say, well what's the probability of the event of a heads? You say, okay, that's one out of the two equally likely outcomes. Or you can even construct, once you know all the possible outcomes, even if they aren't equally likely, you could say, well let's create a probability distribution. We at least know what the sample space is."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So, and it's very useful because, for example, if these are equally likely outcomes, and you say, well what's the probability of the event of a heads? You say, okay, that's one out of the two equally likely outcomes. Or you can even construct, once you know all the possible outcomes, even if they aren't equally likely, you could say, well let's create a probability distribution. We at least know what the sample space is. We know what the possible outcomes are, now let's think about the probability of each of those outcomes. But a lot of times when people talk about sample spaces, they're often, they tend to be most useful, I would say, when you have equally likely outcomes, like in the case of a fair coin flip. Because then from the sample space, it's fairly straightforward to think about the probability of various events."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "We at least know what the sample space is. We know what the possible outcomes are, now let's think about the probability of each of those outcomes. But a lot of times when people talk about sample spaces, they're often, they tend to be most useful, I would say, when you have equally likely outcomes, like in the case of a fair coin flip. Because then from the sample space, it's fairly straightforward to think about the probability of various events. But this is a simple sample space right over here, but let's make things a little bit more interesting. Let's imagine a world, so let's just put this aside a little bit. Let's imagine a world where there's a bakery, and at that bakery, there are three types, three flavors of cupcakes, but there's also three different sizes of cupcakes."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Because then from the sample space, it's fairly straightforward to think about the probability of various events. But this is a simple sample space right over here, but let's make things a little bit more interesting. Let's imagine a world, so let's just put this aside a little bit. Let's imagine a world where there's a bakery, and at that bakery, there are three types, three flavors of cupcakes, but there's also three different sizes of cupcakes. So now we're essentially looking at two different ways in which the thing that we're going to be sampling can vary. So what we're doing, let me write this down. So we have our flavors, flavors of cupcakes at this bakery, and let's say that you have chocolate, chocolate."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Let's imagine a world where there's a bakery, and at that bakery, there are three types, three flavors of cupcakes, but there's also three different sizes of cupcakes. So now we're essentially looking at two different ways in which the thing that we're going to be sampling can vary. So what we're doing, let me write this down. So we have our flavors, flavors of cupcakes at this bakery, and let's say that you have chocolate, chocolate. You have, let's say there's strawberry, strawberry. And let's say that there is vanilla. There is vanilla."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So we have our flavors, flavors of cupcakes at this bakery, and let's say that you have chocolate, chocolate. You have, let's say there's strawberry, strawberry. And let's say that there is vanilla. There is vanilla. And it comes in, they come in three different sizes. So sizes. Sizes could be small, I'll just write it out, small, medium, or large."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "There is vanilla. And it comes in, they come in three different sizes. So sizes. Sizes could be small, I'll just write it out, small, medium, or large. So if you were, and let's say, you know, each of these flavors come in each of these sizes, or it could be the other way around. Each of these sizes come in all three flavors. So now how do you construct the sample space?"}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Sizes could be small, I'll just write it out, small, medium, or large. So if you were, and let's say, you know, each of these flavors come in each of these sizes, or it could be the other way around. Each of these sizes come in all three flavors. So now how do you construct the sample space? If you were to say, look, I'm going to go, you know, I'm going to blindfold myself and walk into this bakery, and randomly, you know, somehow pick up a cupcake, and you know, my fingers can't tell the flavor or the size of the cupcake, what are the possible, what are the possible outcomes for the cupcake I'll pick? And the outcome would be both the flavor and the size of the cupcake. Well, there's a bunch of ways to think about this."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So now how do you construct the sample space? If you were to say, look, I'm going to go, you know, I'm going to blindfold myself and walk into this bakery, and randomly, you know, somehow pick up a cupcake, and you know, my fingers can't tell the flavor or the size of the cupcake, what are the possible, what are the possible outcomes for the cupcake I'll pick? And the outcome would be both the flavor and the size of the cupcake. Well, there's a bunch of ways to think about this. One way is you could draw a tree. You could say, okay, well, I'm going to pick three different flavors. I could either pick chocolate, chocolate."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, there's a bunch of ways to think about this. One way is you could draw a tree. You could say, okay, well, I'm going to pick three different flavors. I could either pick chocolate, chocolate. I'm going to pick strawberry, strawberry. Or I'm going to pick vanilla, vanilla. And then for each of those flavors, I'm going to pick a small, medium, or large."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "I could either pick chocolate, chocolate. I'm going to pick strawberry, strawberry. Or I'm going to pick vanilla, vanilla. And then for each of those flavors, I'm going to pick a small, medium, or large. So you could say small, medium, large. Small, and so this is a small chocolate, this is a medium chocolate, this is a large chocolate. This is a small strawberry, medium strawberry, large strawberry."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And then for each of those flavors, I'm going to pick a small, medium, or large. So you could say small, medium, large. Small, and so this is a small chocolate, this is a medium chocolate, this is a large chocolate. This is a small strawberry, medium strawberry, large strawberry. This is a small vanilla, medium vanilla, large vanilla. And so you see there's nine possible outcomes. Once again, this is a medium chocolate."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "This is a small strawberry, medium strawberry, large strawberry. This is a small vanilla, medium vanilla, large vanilla. And so you see there's nine possible outcomes. Once again, this is a medium chocolate. You picked a chocolate and it was a medium one. This is a large vanilla. You picked a vanilla and it is a large one."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Once again, this is a medium chocolate. You picked a chocolate and it was a medium one. This is a large vanilla. You picked a vanilla and it is a large one. And you could have done it the other way around. You could have said, well, okay, I'm going to either pick a small, medium, or large, and then for each of those, I'm going to pick either a chocolate, strawberry, or vanilla. And I'll just use the first letters to, so I'm either going to pick a chocolate, a strawberry, or a vanilla."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You picked a vanilla and it is a large one. And you could have done it the other way around. You could have said, well, okay, I'm going to either pick a small, medium, or large, and then for each of those, I'm going to pick either a chocolate, strawberry, or vanilla. And I'll just use the first letters to, so I'm either going to pick a chocolate, a strawberry, or a vanilla. When I write the S over here in this magenta color, I'm talking about the flavor. And if I write the S in green, I'm talking about small. So here, if you have a medium cupcake, it could be chocolate, it could be strawberry, or it could be vanilla."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And I'll just use the first letters to, so I'm either going to pick a chocolate, a strawberry, or a vanilla. When I write the S over here in this magenta color, I'm talking about the flavor. And if I write the S in green, I'm talking about small. So here, if you have a medium cupcake, it could be chocolate, it could be strawberry, or it could be vanilla. If you have a large cupcake, it could be chocolate, strawberry, or vanilla. So for example, this was a medium chocolate cupcake. Over here, a medium chocolate cupcake is this one."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So here, if you have a medium cupcake, it could be chocolate, it could be strawberry, or it could be vanilla. If you have a large cupcake, it could be chocolate, strawberry, or vanilla. So for example, this was a medium chocolate cupcake. Over here, a medium chocolate cupcake is this one. It's medium chocolate. It would be that one over here. So you could use these, I have a tree diagram like this, to think about the sample space, to think about the nine possible outcomes here."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Over here, a medium chocolate cupcake is this one. It's medium chocolate. It would be that one over here. So you could use these, I have a tree diagram like this, to think about the sample space, to think about the nine possible outcomes here. But you could also do a, I guess you could say a grid, where you could write the flavors. So you could have chocolate, actually let me just write the, hold on, let me write them out. Actually, let me just write the letters, it's gonna take a long time to do."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So you could use these, I have a tree diagram like this, to think about the sample space, to think about the nine possible outcomes here. But you could also do a, I guess you could say a grid, where you could write the flavors. So you could have chocolate, actually let me just write the, hold on, let me write them out. Actually, let me just write the letters, it's gonna take a long time to do. So you could have the flavors, chocolate, strawberry, vanilla. So that's along that axis. And then you have your sizes."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Actually, let me just write the letters, it's gonna take a long time to do. So you could have the flavors, chocolate, strawberry, vanilla. So that's along that axis. And then you have your sizes. You could have a small, a medium, or large. And you can set up a grid here. So this is another way to do it."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And then you have your sizes. You could have a small, a medium, or large. And you can set up a grid here. So this is another way to do it. And notice this grid has nine boxes. So let's look at it. So set up the grid."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So this is another way to do it. And notice this grid has nine boxes. So let's look at it. So set up the grid. Set up the grid. And so what is this one going to be? This is a small chocolate."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So set up the grid. Set up the grid. And so what is this one going to be? This is a small chocolate. Small, small chocolate. Small chocolate. What is this one?"}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "This is a small chocolate. Small, small chocolate. Small chocolate. What is this one? This is going to be a small, a small strawberry. A small strawberry. And you could just keep constructing like this, where everything in this row, this is all about small, whoops."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "What is this one? This is going to be a small, a small strawberry. A small strawberry. And you could just keep constructing like this, where everything in this row, this is all about small, whoops. Let me do the, this is small. I'm having trouble changing colors. All right."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And you could just keep constructing like this, where everything in this row, this is all about small, whoops. Let me do the, this is small. I'm having trouble changing colors. All right. There is a small, and then this is a small, this is a small vanilla. This color changing is really, it's a difficult thing. Small vanilla."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "All right. There is a small, and then this is a small, this is a small vanilla. This color changing is really, it's a difficult thing. Small vanilla. And all of these, these are all, this would be a medium chocolate, medium strawberry, medium vanilla, large chocolate, large strawberry, large vanilla. And once again, you have nine outcomes. This is another way to think about all of the possible outcomes."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Small vanilla. And all of these, these are all, this would be a medium chocolate, medium strawberry, medium vanilla, large chocolate, large strawberry, large vanilla. And once again, you have nine outcomes. This is another way to think about all of the possible outcomes. When you're looking at these two ways in which my cupcakes could vary. Another way, a third way that you could do it is you could literally just construct a table. Well, you could say, okay, I could have a chocolate, actually I'm going to use the letters again."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "This is another way to think about all of the possible outcomes. When you're looking at these two ways in which my cupcakes could vary. Another way, a third way that you could do it is you could literally just construct a table. Well, you could say, okay, I could have a chocolate, actually I'm going to use the letters again. So let's say, let me make, this is the flavor column, and then this is the size column. Size column. And so you could say, I could have a chocolate that is, so let's see, there's three types of chocolate that I could have."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, you could say, okay, I could have a chocolate, actually I'm going to use the letters again. So let's say, let me make, this is the flavor column, and then this is the size column. Size column. And so you could say, I could have a chocolate that is, so let's see, there's three types of chocolate that I could have. They're, and they could be, they could be small, medium, or large. You could say there is three types of, three types of strawberry. It could be small, medium, or large."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And so you could say, I could have a chocolate that is, so let's see, there's three types of chocolate that I could have. They're, and they could be, they could be small, medium, or large. You could say there is three types of, three types of strawberry. It could be small, medium, or large. So let me write that in. Small, medium, large. Or you could say, well, there's three types of vanilla."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "It could be small, medium, or large. So let me write that in. Small, medium, large. Or you could say, well, there's three types of vanilla. There's three types of, color changing again, three types of vanilla. Once again, it could be small, medium, or large. So you have these nine possibilities."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Or you could say, well, there's three types of vanilla. There's three types of, color changing again, three types of vanilla. Once again, it could be small, medium, or large. So you have these nine possibilities. Now, the sample space, the sample space isn't telling you if they're equally likely or not. It's just telling you if you're going to do an experiment, what are all the different possibilities, the possible outcomes for that experiment. Now, in the case where they are equally likely, it can be very, very useful."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So you have these nine possibilities. Now, the sample space, the sample space isn't telling you if they're equally likely or not. It's just telling you if you're going to do an experiment, what are all the different possibilities, the possible outcomes for that experiment. Now, in the case where they are equally likely, it can be very, very useful. Because you could say, you could do something like, if you said that, okay, it's equally likely to pick any one of these nine outcomes, you could say, well, what's the probability of, what's the probability of getting a, something that is either small or chocolate? And so you could see, well, how many of those events out of the total actually meet that constraint. But we'll do more of that in future videos."}, {"video_title": "Compound sample spaces   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now, in the case where they are equally likely, it can be very, very useful. Because you could say, you could do something like, if you said that, okay, it's equally likely to pick any one of these nine outcomes, you could say, well, what's the probability of, what's the probability of getting a, something that is either small or chocolate? And so you could see, well, how many of those events out of the total actually meet that constraint. But we'll do more of that in future videos. That's just a little bit of a clue of why we even care about things like sample spaces, especially sample spaces like this, where we're looking along two ways, or multiple ways, that something can vary. And these types of sample spaces in particular are called compound sample spaces. So these right over here, this is a compound sample space, because we're looking at two different ways that it can vary, not just to heads or tails, it can vary by size or by flavor."}, {"video_title": "Creating a bar chart   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Seven teachers said language, three teachers said history, nine teachers said geometry, one teacher said chemistry, zero teachers said physics. Create a bar chart showing everyone's favorite courses. So we've got the bar chart right over here, and let's see what we need to plot. So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one."}, {"video_title": "Creating a bar chart   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine. One teacher said chemistry, oh, I already read that."}, {"video_title": "Creating a bar chart   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine. One teacher said chemistry, oh, I already read that. History, history, three teachers said history, so let's bring history up to three. And then language, seven teachers said language. So let's move this up to seven."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And we record the number of cars each of the 11 salespeople sold in the past week. So this is, you know, that's one of them, two, three, four, five, six, seven, eight, nine, ten, eleven. So we record how much each of them sold. So maybe one person sold, maybe they sold five cars. Maybe the next person sold seven cars. Maybe the next person sold ten cars. And so that's how we would do it."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So maybe one person sold, maybe they sold five cars. Maybe the next person sold seven cars. Maybe the next person sold ten cars. And so that's how we would do it. So we're going to record all of that. But then we're told that the median car sold is six. So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And so that's how we would do it. So we're going to record all of that. But then we're told that the median car sold is six. So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest. So maybe, and I'm just making up numbers here, maybe this is four, this is four, maybe this is five, five, five, six, seven, seven, eight, nine, ten. The median car sold means that the middle number here is going to be six. Now we don't know whether all of these other numbers are the actual numbers, I just made those up, but we know that the middle number needs to be six."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest. So maybe, and I'm just making up numbers here, maybe this is four, this is four, maybe this is five, five, five, six, seven, seven, eight, nine, ten. The median car sold means that the middle number here is going to be six. Now we don't know whether all of these other numbers are the actual numbers, I just made those up, but we know that the middle number needs to be six. So let me just fill out only the middle number. So the middle number, if you have 11 data points, the middle number is going to have five on either side. So that's that one right over there."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now we don't know whether all of these other numbers are the actual numbers, I just made those up, but we know that the middle number needs to be six. So let me just fill out only the middle number. So the middle number, if you have 11 data points, the middle number is going to have five on either side. So that's that one right over there. One, two, three, four, five. One, two, three, four, five. So we know that the median, we know the median, actually let me do that in purple color, the median is going to be six."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So that's that one right over there. One, two, three, four, five. One, two, three, four, five. So we know that the median, we know the median, actually let me do that in purple color, the median is going to be six. Now the other thing we know is that the range of cars sold is four. And let's remind ourselves, the range is the maximum number of cars sold minus the minimum number of cars sold. And if we have sorted all of these out, this, if we take, if this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we know that the median, we know the median, actually let me do that in purple color, the median is going to be six. Now the other thing we know is that the range of cars sold is four. And let's remind ourselves, the range is the maximum number of cars sold minus the minimum number of cars sold. And if we have sorted all of these out, this, if we take, if this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min. So let me write that down. So you take your min and your max, you have your range, range is equal to the maximum minus the minimum. And they're telling us that that is four, this range is four."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And if we have sorted all of these out, this, if we take, if this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min. So let me write that down. So you take your min and your max, you have your range, range is equal to the maximum minus the minimum. And they're telling us that that is four, this range is four. So the maximum minus the minimum, the difference between the number of cars that the most productive salesperson sold and the number of cars that the least productive salesperson sold, that difference is going to be four. So given all of those assumptions, giving all of that information, I'm now going to give you a statement. And our challenge is, if we assume everything I just said is true, is this statement going to be true?"}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And they're telling us that that is four, this range is four. So the maximum minus the minimum, the difference between the number of cars that the most productive salesperson sold and the number of cars that the least productive salesperson sold, that difference is going to be four. So given all of those assumptions, giving all of that information, I'm now going to give you a statement. And our challenge is, if we assume everything I just said is true, is this statement going to be true? Or is this statement going to be false? Or do we not know? So let me write this down."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And our challenge is, if we assume everything I just said is true, is this statement going to be true? Or is this statement going to be false? Or do we not know? So let me write this down. So is this going to be true? False, false, or do we not know? Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know?"}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let me write this down. So is this going to be true? False, false, or do we not know? Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? And so I encourage you to pause the video now and try it out. All right, so let's work through it together. This is kind of a fun puzzle here where we've been given some clues and then we have a statement and we figure out, can we make this statement?"}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? And so I encourage you to pause the video now and try it out. All right, so let's work through it together. This is kind of a fun puzzle here where we've been given some clues and then we have a statement and we figure out, can we make this statement? Can we say it's true? Or can we say for sure that it's false or do we just not know? So a couple of ways to tackle it."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is kind of a fun puzzle here where we've been given some clues and then we have a statement and we figure out, can we make this statement? Can we say it's true? Or can we say for sure that it's false or do we just not know? So a couple of ways to tackle it. One way to tackle it is, well look, one of the salespeople must have sold six cars. If the median sold is six, and there's 11 salespeople here, the middle number here, if we order it, one of them must have sold six cars. If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars that the middle number of cars sold."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So a couple of ways to tackle it. One way to tackle it is, well look, one of the salespeople must have sold six cars. If the median sold is six, and there's 11 salespeople here, the middle number here, if we order it, one of them must have sold six cars. If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars that the middle number of cars sold. So someone sold six cars. So if we assume, we're trying to find a world where someone sold more than 10 cars. So if someone sold six cars, is there a way with a range of four that someone sold more than 10?"}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars that the middle number of cars sold. So someone sold six cars. So if we assume, we're trying to find a world where someone sold more than 10 cars. So if someone sold six cars, is there a way with a range of four that someone sold more than 10? Well, let's just think about it. Let's just assume that the min is six. We don't know that the min is six, but let's just try it out."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So if someone sold six cars, is there a way with a range of four that someone sold more than 10? Well, let's just think about it. Let's just assume that the min is six. We don't know that the min is six, but let's just try it out. If the minimum is six, what's the maximum going to be? Well, remember, the range is equal to, we know that the range is four, the range is equal to the maximum minus the minimum. Maximum minus the minimum."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We don't know that the min is six, but let's just try it out. If the minimum is six, what's the maximum going to be? Well, remember, the range is equal to, we know that the range is four, the range is equal to the maximum minus the minimum. Maximum minus the minimum. And so the maximum in this case would be 10. Would be 10. Max would be equal to 10."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Maximum minus the minimum. And so the maximum in this case would be 10. Would be 10. Max would be equal to 10. If the min is equal to six, and instead of colons, I'm going to write an equal sign. If the min is equal to six, then the max at most can be equal to 10. And we can't take a higher min because we know that the six is going to be one of the values."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Max would be equal to 10. If the min is equal to six, and instead of colons, I'm going to write an equal sign. If the min is equal to six, then the max at most can be equal to 10. And we can't take a higher min because we know that the six is going to be one of the values. We could try a lower minimum value. We could say the minimum value is five or four or three or two or one, but then the maximum would go down even more because remember, the maximum is no more than four larger than the minimum. So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And we can't take a higher min because we know that the six is going to be one of the values. We could try a lower minimum value. We could say the minimum value is five or four or three or two or one, but then the maximum would go down even more because remember, the maximum is no more than four larger than the minimum. So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10. And so the statement, at least one of the salespeople sold more than 10 cars, that's got to be false. That's got to be false. Now there's another way that you could think about it."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10. And so the statement, at least one of the salespeople sold more than 10 cars, that's got to be false. That's got to be false. Now there's another way that you could think about it. You could assume that someone sold more than 10. You could assume, in the last example or the last way of thinking about it, we assumed that the min was six, but now let's just assume that someone sold, let's just try it out. Let's see if it's possible."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now there's another way that you could think about it. You could assume that someone sold more than 10. You could assume, in the last example or the last way of thinking about it, we assumed that the min was six, but now let's just assume that someone sold, let's just try it out. Let's see if it's possible. Let's just assume that the max is 11, that someone sold more than 10 cars. If the max is equal to 11, what's the min going to need to be? Well, we just have to remind ourselves."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's see if it's possible. Let's just assume that the max is 11, that someone sold more than 10 cars. If the max is equal to 11, what's the min going to need to be? Well, we just have to remind ourselves. Range is equal to max minus min. So four is equal to the maximum, 11 minus the minimum. So 11 minus what is equal to four?"}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, we just have to remind ourselves. Range is equal to max minus min. So four is equal to the maximum, 11 minus the minimum. So 11 minus what is equal to four? Well, 11 minus seven is equal to four. So if we assume that the maximum is 11, then the minimum is going to have to be, the minimum is going to have to be equal to seven. Now can the minimum be seven when your median is six?"}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So 11 minus what is equal to four? Well, 11 minus seven is equal to four. So if we assume that the maximum is 11, then the minimum is going to have to be, the minimum is going to have to be equal to seven. Now can the minimum be seven when your median is six? No. If your median is six, that means you have, that means if you have an odd number of data points, that means one of the data points is six, and if you had an even number of data points, that means that the middle two are going to average out to be six, which means you even have data points that are less than six. So your minimum can't be seven."}, {"video_title": "Median and range puzzle   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now can the minimum be seven when your median is six? No. If your median is six, that means you have, that means if you have an odd number of data points, that means one of the data points is six, and if you had an even number of data points, that means that the middle two are going to average out to be six, which means you even have data points that are less than six. So your minimum can't be seven. You're going to have a value that is at least as low, at least as low as six. So your assumption can't be true, can't be true. So once again, the assumption based on the statement can't be true."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they say the probability, I'll just say P for probability, the probability of picking a yellow marble. And so this is sometimes the event in question right over here, is picking the yellow marble. I'll even write down the word picking. And when you say probability, it's really just a way of measuring the likelihood that something is going to happen. And the way we're going to think about it is how many of the outcomes from this trial, from this picking a marble out of a bag, how many meet our constraints, satisfy this event, and how many possible outcomes are there? So let me write the possible outcomes right over here. So possible outcomes."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And when you say probability, it's really just a way of measuring the likelihood that something is going to happen. And the way we're going to think about it is how many of the outcomes from this trial, from this picking a marble out of a bag, how many meet our constraints, satisfy this event, and how many possible outcomes are there? So let me write the possible outcomes right over here. So possible outcomes. And you'll see it's actually a very straightforward idea, but I'll just make sure that we understand all the words that people might say. So the set of all the possible outcomes, well, there's three yellow marbles, so I could pick that yellow marble, that yellow marble, or that yellow marble. These are clearly all yellow."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So possible outcomes. And you'll see it's actually a very straightforward idea, but I'll just make sure that we understand all the words that people might say. So the set of all the possible outcomes, well, there's three yellow marbles, so I could pick that yellow marble, that yellow marble, or that yellow marble. These are clearly all yellow. There's two red marbles in the bag, so I could pick that red marble or that red marble. There's two green marbles in the bag, so I could pick that green marble or that green marble. And then there's one blue marble in the bag."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are clearly all yellow. There's two red marbles in the bag, so I could pick that red marble or that red marble. There's two green marbles in the bag, so I could pick that green marble or that green marble. And then there's one blue marble in the bag. So this is all the possible outcomes, and sometimes this is referred to as the sample space. Sample, the set of all the possible outcomes. Fancy word for just a simple idea, that the sample space, when I pick something out of the bag and that picking out of the bag is called a trial, there's eight possible things I can do."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then there's one blue marble in the bag. So this is all the possible outcomes, and sometimes this is referred to as the sample space. Sample, the set of all the possible outcomes. Fancy word for just a simple idea, that the sample space, when I pick something out of the bag and that picking out of the bag is called a trial, there's eight possible things I can do. So when I think about the probability of picking a yellow marble, I want to think about, well, what are all of the possibilities? Well, there's eight possibilities. Eight possibilities for my trial."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Fancy word for just a simple idea, that the sample space, when I pick something out of the bag and that picking out of the bag is called a trial, there's eight possible things I can do. So when I think about the probability of picking a yellow marble, I want to think about, well, what are all of the possibilities? Well, there's eight possibilities. Eight possibilities for my trial. So the number of outcomes, number of possible outcomes, you could view it as the size of the sample space, number of possible outcomes. And it's as simple as saying, well, look, I have eight marbles. And then you say, well, how many of those marbles meet my constraint that satisfy this event here?"}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Eight possibilities for my trial. So the number of outcomes, number of possible outcomes, you could view it as the size of the sample space, number of possible outcomes. And it's as simple as saying, well, look, I have eight marbles. And then you say, well, how many of those marbles meet my constraint that satisfy this event here? Well, there's three marbles that satisfy my event. There's three outcomes that will allow this event to occur, I guess is one way to say it. So there's three right over here."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you say, well, how many of those marbles meet my constraint that satisfy this event here? Well, there's three marbles that satisfy my event. There's three outcomes that will allow this event to occur, I guess is one way to say it. So there's three right over here. So number that satisfy the event or the constraint right over here. So it's very simple ideas. Many times, the words make them more complicated than they need to."}, {"video_title": "Finding probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's three right over here. So number that satisfy the event or the constraint right over here. So it's very simple ideas. Many times, the words make them more complicated than they need to. If I say, what's the probability of picking a yellow marble? Well, how many different types of marbles can I pick? Well, there's eight different marbles I could pick."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Our goal is to simplify this expression for the squared error between those end points. Just to remind ourselves what we're doing. We have these end points and we're taking the sum of the squared error between each of those end points and our actual line y equals mx plus b. And we get this expression over here which we've been simplifying over the last couple of videos. We're going to try to simplify the expression as much as possible and then we're going to try to optimize, we're going to try to minimize this expression or find the m and b values that minimize it or that's, I guess you can call it, the best fitting line. Now to do that, it looks like we were just getting, making the algebra even hairier and hairier but this next step is going to simplify things a good bit. So just to show you that, what is, if I want to take the mean of all of the squared values of the y's."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And we get this expression over here which we've been simplifying over the last couple of videos. We're going to try to simplify the expression as much as possible and then we're going to try to optimize, we're going to try to minimize this expression or find the m and b values that minimize it or that's, I guess you can call it, the best fitting line. Now to do that, it looks like we were just getting, making the algebra even hairier and hairier but this next step is going to simplify things a good bit. So just to show you that, what is, if I want to take the mean of all of the squared values of the y's. So that would be this, that would be y1 squared plus y2 squared plus all the way to yn squared. So I've summed n values, n squared values and then I want to divide it by n, since there are n values here, and this is the mean of the y's squared. That's how we can denote it just like that."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So just to show you that, what is, if I want to take the mean of all of the squared values of the y's. So that would be this, that would be y1 squared plus y2 squared plus all the way to yn squared. So I've summed n values, n squared values and then I want to divide it by n, since there are n values here, and this is the mean of the y's squared. That's how we can denote it just like that. That is the mean of the y squared. Or if you multiply both sides of this equation by n, you get y1 squared plus y2 squared plus all the way to, all the way to yn squared, yn squared is equal to n, is equal to n, let me do this in different colors, is equal to, so this is, this is equal to n, this n times the mean of the squared values of y. And notice, this is exactly, this is exactly what we have over here."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "That's how we can denote it just like that. That is the mean of the y squared. Or if you multiply both sides of this equation by n, you get y1 squared plus y2 squared plus all the way to, all the way to yn squared, yn squared is equal to n, is equal to n, let me do this in different colors, is equal to, so this is, this is equal to n, this n times the mean of the squared values of y. And notice, this is exactly, this is exactly what we have over here. That is n times the mean of the squared values of y or the mean of the y squareds. That's exactly what that is and we can do that with each of these terms. What is, what is x1, x1, y1 plus x2, y2 plus all the way to, all the way to xn, yn."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And notice, this is exactly, this is exactly what we have over here. That is n times the mean of the squared values of y or the mean of the y squareds. That's exactly what that is and we can do that with each of these terms. What is, what is x1, x1, y1 plus x2, y2 plus all the way to, all the way to xn, yn. Well if we take this whole sum and we divide it by n terms, this is going to be the mean value for x times y. For each of those points, you multiply x times y and you find the mean of all of those products. That's exactly what this is."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "What is, what is x1, x1, y1 plus x2, y2 plus all the way to, all the way to xn, yn. Well if we take this whole sum and we divide it by n terms, this is going to be the mean value for x times y. For each of those points, you multiply x times y and you find the mean of all of those products. That's exactly what this is. Well once again, you multiply both sides of this equation by n and you get, you get x1, y1 plus x2, y2 plus all the way, all the way to xn, yn is equal to, is equal to n times the mean of xy's, the mean of the xy's. So n times the mean of the xy's and I think you see where this is going. This term right here, so this term right here is going to be equal to n times the mean of the products of xy."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "That's exactly what this is. Well once again, you multiply both sides of this equation by n and you get, you get x1, y1 plus x2, y2 plus all the way, all the way to xn, yn is equal to, is equal to n times the mean of xy's, the mean of the xy's. So n times the mean of the xy's and I think you see where this is going. This term right here, so this term right here is going to be equal to n times the mean of the products of xy. This term right here is n times the mean of the y values. That's what this term right here is. And then this term right here is n times the mean of the x, the mean of the x's squared."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This term right here, so this term right here is going to be equal to n times the mean of the products of xy. This term right here is n times the mean of the y values. That's what this term right here is. And then this term right here is n times the mean of the x, the mean of the x's squared. The mean of the x squared values, I should say. This term right here is the mean of the x's times n. Right, if you divided this by n, you'd get the mean. Since we're not dividing it by n, this is the mean times n. And then this is obviously, we don't have to simplify anything."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then this term right here is n times the mean of the x, the mean of the x's squared. The mean of the x squared values, I should say. This term right here is the mean of the x's times n. Right, if you divided this by n, you'd get the mean. Since we're not dividing it by n, this is the mean times n. And then this is obviously, we don't have to simplify anything. So let's rewrite everything using our new notation, knowing that these are the means of these, you know, of y squared, of xy and all of that. So our squared error of the line, our squared error to the line, from the sum of the squared error to the line from the end points is going to equal to, this term right here is n, let me color code it a little bit. This term right here is n times the mean of the y squared values."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Since we're not dividing it by n, this is the mean times n. And then this is obviously, we don't have to simplify anything. So let's rewrite everything using our new notation, knowing that these are the means of these, you know, of y squared, of xy and all of that. So our squared error of the line, our squared error to the line, from the sum of the squared error to the line from the end points is going to equal to, this term right here is n, let me color code it a little bit. This term right here is n times the mean of the y squared values. This term right here is, I'll do it all in this green color, is equal to negative 2m, that's just that right there, times n times the mean of the xy values, the arithmetic mean. And then we have this term over here. I think you can appreciate this is simplifying the algebraic expression a good bit."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This term right here is n times the mean of the y squared values. This term right here is, I'll do it all in this green color, is equal to negative 2m, that's just that right there, times n times the mean of the xy values, the arithmetic mean. And then we have this term over here. I think you can appreciate this is simplifying the algebraic expression a good bit. This term right over here is going to be minus 2b, minus 2bn, times the mean of the y values. And then we have plus, we have this term right here, plus m squared times n times the mean of the x squared values. And then we have, almost there, home stretch, we have this over here which is plus 2mb times n times the mean of the x values."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "I think you can appreciate this is simplifying the algebraic expression a good bit. This term right over here is going to be minus 2b, minus 2bn, times the mean of the y values. And then we have plus, we have this term right here, plus m squared times n times the mean of the x squared values. And then we have, almost there, home stretch, we have this over here which is plus 2mb times n times the mean of the x values. And then finally we have plus nb squared. So really in the last two, three videos, all we've done is we've simplified the expression for the sum of the squared differences from those end points to this line y equals mx plus b, right over here. So that is kind of, we're finished the hardcore algebra stage."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then we have, almost there, home stretch, we have this over here which is plus 2mb times n times the mean of the x values. And then finally we have plus nb squared. So really in the last two, three videos, all we've done is we've simplified the expression for the sum of the squared differences from those end points to this line y equals mx plus b, right over here. So that is kind of, we're finished the hardcore algebra stage. The next stage we actually want to optimize. We actually want to optimize this. We actually want to optimize, or actually maybe a better way to talk about it, we want to minimize this expression right over here."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So that is kind of, we're finished the hardcore algebra stage. The next stage we actually want to optimize. We actually want to optimize this. We actually want to optimize, or actually maybe a better way to talk about it, we want to minimize this expression right over here. We want to find the m and the b values that minimize it. And to help visualize it, we're going to start breaking into a little bit of three-dimensional calculus here. But hopefully it won't be too daunting."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We actually want to optimize, or actually maybe a better way to talk about it, we want to minimize this expression right over here. We want to find the m and the b values that minimize it. And to help visualize it, we're going to start breaking into a little bit of three-dimensional calculus here. But hopefully it won't be too daunting. If you've done any partial derivatives it won't be difficult. This is a surface. If you view that you have the x and y data points, everything here is a constant except for the m's and the b's."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "But hopefully it won't be too daunting. If you've done any partial derivatives it won't be difficult. This is a surface. If you view that you have the x and y data points, everything here is a constant except for the m's and the b's. We're assuming that we have the x's and y's. So we can figure out the mean of the squared values of y, the mean of the xy product, the mean of the y's, the mean of the x squareds. We assume that those are all actual numbers."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "If you view that you have the x and y data points, everything here is a constant except for the m's and the b's. We're assuming that we have the x's and y's. So we can figure out the mean of the squared values of y, the mean of the xy product, the mean of the y's, the mean of the x squareds. We assume that those are all actual numbers. So this expression right here is going to be, it's actually going to be a surface in three dimensions. So you can imagine this right here, that is the m axis. This right here is the b axis."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We assume that those are all actual numbers. So this expression right here is going to be, it's actually going to be a surface in three dimensions. So you can imagine this right here, that is the m axis. This right here is the b axis. That is the b axis. Let me continue the m axis and let me continue the b axis. And then you can imagine the vertical axis here to be the squared error."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "This right here is the b axis. That is the b axis. Let me continue the m axis and let me continue the b axis. And then you can imagine the vertical axis here to be the squared error. This is the squared error of the line axis. So for any combination of m and b's, if you're in the mb plane, you pick some combination of m and b, you put it into this expression for the squared error of the line, it will give you a point. If you do that for all of the combinations of m's and b's, you're going to get a surface."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And then you can imagine the vertical axis here to be the squared error. This is the squared error of the line axis. So for any combination of m and b's, if you're in the mb plane, you pick some combination of m and b, you put it into this expression for the squared error of the line, it will give you a point. If you do that for all of the combinations of m's and b's, you're going to get a surface. The surface is going to look something like this. I'm going to try my best to draw it. It's going to look something like this."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "If you do that for all of the combinations of m's and b's, you're going to get a surface. The surface is going to look something like this. I'm going to try my best to draw it. It's going to look something like this. You can almost imagine it as somewhat of a kind of a bowl. Or you can even think of it as a three-dimensional parabola, if you want to think of it that way. Instead of a parabola that just goes like this, if you were to kind of rotate it around and distort it a little bit, you would get this thing that looks kind of like a cup or a thimble or whatever."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "It's going to look something like this. You can almost imagine it as somewhat of a kind of a bowl. Or you can even think of it as a three-dimensional parabola, if you want to think of it that way. Instead of a parabola that just goes like this, if you were to kind of rotate it around and distort it a little bit, you would get this thing that looks kind of like a cup or a thimble or whatever. What we want to do is we want to find the m and b values that minimize. Notice this is a three-dimensional surface. I don't know if I'm doing justice to it."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Instead of a parabola that just goes like this, if you were to kind of rotate it around and distort it a little bit, you would get this thing that looks kind of like a cup or a thimble or whatever. What we want to do is we want to find the m and b values that minimize. Notice this is a three-dimensional surface. I don't know if I'm doing justice to it. Let me see if I can imagine a three-dimensional surface that looks something like this. This is the back part that you're not seeing. That's the inside of our three-dimensional surface."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "I don't know if I'm doing justice to it. Let me see if I can imagine a three-dimensional surface that looks something like this. This is the back part that you're not seeing. That's the inside of our three-dimensional surface. We want to find the m and b values that minimize the value on the surface. There's some m and b value right over here that minimizes it. To do that, and I'll actually do the calculation in the next video, we're going to find the partial derivative of this with respect to m, and we're going to find the partial derivative of this with respect to b and set both of them equal to zero."}, {"video_title": "Proof (part 2) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "That's the inside of our three-dimensional surface. We want to find the m and b values that minimize the value on the surface. There's some m and b value right over here that minimizes it. To do that, and I'll actually do the calculation in the next video, we're going to find the partial derivative of this with respect to m, and we're going to find the partial derivative of this with respect to b and set both of them equal to zero. Because at this minimum point, I guess you could say in three dimensions, this minimum point on the surface is going to occur is when the slope with respect to m and the slope with respect to b is zero. At that point, the partial derivative of our squared error with respect to m is going to be equal to zero, and the partial derivative of our squared error with respect to b is going to be equal to zero. All we're going to do in the next video is take the partial derivative of this expression with respect to m, set that equal to zero, and the partial derivative of this with respect to b, set that equal to zero, and then we're ready to solve for the m and the b, or the particular m and b."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And we want to get a sense of how these students feel about the quality of math instruction at this school. So we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do?"}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all."}, {"video_title": "Techniques for random sampling and avoiding bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. So anyway, this is a very high-level overview of how you could think about sampling. You wanna go random because it lowers the probability of their introducing some bias into it. And then these are some techniques."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if we're doing it as a Venn diagram, the universe is usually depicted as some type of a rectangle right over here. And it itself is a set and it usually is denoted with the capital U, U for universe, not to be confused with the union set notation. And you could say that the universe is all possible things that could be in a set, including farm animals and kitchen utensils and emotions and types of Italian food or even types of food. But then that just becomes somewhat crazy because you're literally thinking of all possible things. Normally when people talk about a universal set, they're talking about a universe of things that they care about. So the set of all people or the set of all real numbers or the set of all countries, whatever the discussion is being focused on. But we'll talk about an abstract terms right now."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "But then that just becomes somewhat crazy because you're literally thinking of all possible things. Normally when people talk about a universal set, they're talking about a universe of things that they care about. So the set of all people or the set of all real numbers or the set of all countries, whatever the discussion is being focused on. But we'll talk about an abstract terms right now. And let's say you have a subset of that universal set. So let's say you have a subset of that universal set, set A. And so set A literally contains everything, everything that I have just shaded in."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we'll talk about an abstract terms right now. And let's say you have a subset of that universal set. So let's say you have a subset of that universal set, set A. And so set A literally contains everything, everything that I have just shaded in. What we're gonna talk about now is the idea of a complement or the absolute complement of A. And the way you could think about this is this is the set of all things in the universe, in universe that aren't, that aren't, that aren't in A. And we've already looked at ways of expressing this."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so set A literally contains everything, everything that I have just shaded in. What we're gonna talk about now is the idea of a complement or the absolute complement of A. And the way you could think about this is this is the set of all things in the universe, in universe that aren't, that aren't, that aren't in A. And we've already looked at ways of expressing this. The set of all things in the universe that aren't in A, we could also write as a universal set minus A. Once again, this is a capital U. This is not the union symbol right over here."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we've already looked at ways of expressing this. The set of all things in the universe that aren't in A, we could also write as a universal set minus A. Once again, this is a capital U. This is not the union symbol right over here. Or we could literally write this as U and then we write that little slash looking thing, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U, it would be all the stuff in U that is not in A."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is not the union symbol right over here. Or we could literally write this as U and then we write that little slash looking thing, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U, it would be all the stuff in U that is not in A. One way to think about it, you could think about it as the relative complement of A that is in U, but when you're taking the relative complement of something that is in the universal set, you're really talking about the absolute complement. Or when people just talk about the complement, that's what they're saying. What's the set of all the things in my universe that are not, that are not in A?"}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it would be all the stuff in U, it would be all the stuff in U that is not in A. One way to think about it, you could think about it as the relative complement of A that is in U, but when you're taking the relative complement of something that is in the universal set, you're really talking about the absolute complement. Or when people just talk about the complement, that's what they're saying. What's the set of all the things in my universe that are not, that are not in A? Now let's make things a little bit more concrete by talking about sets of numbers. Once again, our sets, we could have been talking about sets of TV personalities or sets of animals or whatever it might be, but numbers are a nice, simple thing to deal with. And let's say that our universe, our universe that we care about right over here, is the set of integers."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's the set of all the things in my universe that are not, that are not in A? Now let's make things a little bit more concrete by talking about sets of numbers. Once again, our sets, we could have been talking about sets of TV personalities or sets of animals or whatever it might be, but numbers are a nice, simple thing to deal with. And let's say that our universe, our universe that we care about right over here, is the set of integers. So our universe is a set of integers. So I'll write U, capital U, is equal to the set of integers. And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's say that our universe, our universe that we care about right over here, is the set of integers. So our universe is a set of integers. So I'll write U, capital U, is equal to the set of integers. And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. And it's Z for Zoll from German for apparently integer. And the bold is this kind of weird looking, they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. And it's Z for Zoll from German for apparently integer. And the bold is this kind of weird looking, they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers. And in fact, I'll do a little aside here to do that. So for example, they might say, they'll write R like this for the set of real numbers, real numbers. They'll write a Q in that blackboard bold font."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's what mathematicians use for different types of sets of numbers. And in fact, I'll do a little aside here to do that. So for example, they might say, they'll write R like this for the set of real numbers, real numbers. They'll write a Q in that blackboard bold font. So it looks something like this. They'll write the Q, it might look something like this. This would be the set of rational numbers."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "They'll write a Q in that blackboard bold font. So it looks something like this. They'll write the Q, it might look something like this. This would be the set of rational numbers. And you might say, why Q for rational? Well, there's a couple of reasons. One, the R is already taken up."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would be the set of rational numbers. And you might say, why Q for rational? Well, there's a couple of reasons. One, the R is already taken up. And Q for quotient, a rational number can be expressed as a quotient of two integers. And we just saw you have your Z for Zoll, for Zoll or integers, the set of all integers. So our universal set, the universe that we care about right now is integers."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "One, the R is already taken up. And Q for quotient, a rational number can be expressed as a quotient of two integers. And we just saw you have your Z for Zoll, for Zoll or integers, the set of all integers. So our universal set, the universe that we care about right now is integers. And let's define a subset of it. Let's call that subset, I don't know, I've been, let me put a, use a letter that I haven't been using a lot. Let's call it C. The set C, let's say it's equal to negative five, zero, and positive seven."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our universal set, the universe that we care about right now is integers. And let's define a subset of it. Let's call that subset, I don't know, I've been, let me put a, use a letter that I haven't been using a lot. Let's call it C. The set C, let's say it's equal to negative five, zero, and positive seven. And I'm obviously not drawing it to scale. The set of all integers is infinite, whilst the set C is a finite set. But I'll just kind of, just to draw it, that's our set C right over there."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's call it C. The set C, let's say it's equal to negative five, zero, and positive seven. And I'm obviously not drawing it to scale. The set of all integers is infinite, whilst the set C is a finite set. But I'll just kind of, just to draw it, that's our set C right over there. And let's think about what is a member of C and what is not a member of C. So we know that negative five is a member of our set C. This little symbol right here, this denotes membership. Membership, it looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I'll just kind of, just to draw it, that's our set C right over there. And let's think about what is a member of C and what is not a member of C. So we know that negative five is a member of our set C. This little symbol right here, this denotes membership. Membership, it looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set. We know that zero is a member of set of, sorry. We know that zero is a member of our set. We know that seven is a member, is a member of our set."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "This just literally means membership of a set. We know that zero is a member of set of, sorry. We know that zero is a member of our set. We know that seven is a member, is a member of our set. Now we also know some other things. We know that the number negative eight is not, is not a member of our set. We know that the number 53 is not a member, not a member of our set."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that seven is a member, is a member of our set. Now we also know some other things. We know that the number negative eight is not, is not a member of our set. We know that the number 53 is not a member, not a member of our set. 53 is sitting someplace out here. We know the number 42 is not a member of our set. 42 might be sitting someplace out there."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that the number 53 is not a member, not a member of our set. 53 is sitting someplace out here. We know the number 42 is not a member of our set. 42 might be sitting someplace out there. Now let's think about C complement or the complement of C. C complement, which is the same thing as our universe, minus C, which is the same thing as universe, or you could say the relative complement of C in our universe. These are all equivalent notation. What is this, first of all, in our Venn diagram?"}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "42 might be sitting someplace out there. Now let's think about C complement or the complement of C. C complement, which is the same thing as our universe, minus C, which is the same thing as universe, or you could say the relative complement of C in our universe. These are all equivalent notation. What is this, first of all, in our Venn diagram? What's all this stuff outside of our set? Outside of our set C, right over here. And now all of a sudden we know that negative five, negative five is a member of C, so it can't be a member of C complement."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is this, first of all, in our Venn diagram? What's all this stuff outside of our set? Outside of our set C, right over here. And now all of a sudden we know that negative five, negative five is a member of C, so it can't be a member of C complement. So negative five is not a member of C complement. Zero is not a member of C complement. Zero sits in C, not in C complement."}, {"video_title": "Universal set and absolute complement   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now all of a sudden we know that negative five, negative five is a member of C, so it can't be a member of C complement. So negative five is not a member of C complement. Zero is not a member of C complement. Zero sits in C, not in C complement. Negative, or I could say 53, 53 is a member of C complement. It's outside of C, it's in the universe, but outside of C. 42 is a member of C complement. So anyway, hopefully that helps clear things up a little bit."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or from their expected value. So let me just write that down. So if I take, I'll have x first, I'll do this in another color. So it's the expected value of random variable x minus the expected value of x. You could view this as the population mean of x, times, and then this is random variable y, so times the distance from y to its expected value. Or the population mean. The population mean of y."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's the expected value of random variable x minus the expected value of x. You could view this as the population mean of x, times, and then this is random variable y, so times the distance from y to its expected value. Or the population mean. The population mean of y. And if it doesn't make a lot of intuitive sense yet, well one, you can just always kind of think about what it's doing, play around with some numbers here. But the reality is, it's saying how much they vary together. So you always take an x and a y for each of the data points."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "The population mean of y. And if it doesn't make a lot of intuitive sense yet, well one, you can just always kind of think about what it's doing, play around with some numbers here. But the reality is, it's saying how much they vary together. So you always take an x and a y for each of the data points. Let's say you have the whole population, so every x and y that go together with each other, that are a coordinate, you put into this. And what happens is, let's say that x is above its mean when y is below its mean. So let's say in the population, you had the point, so one instantiation of the random variables, you have, you sample once, I guess, from the universe."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you always take an x and a y for each of the data points. Let's say you have the whole population, so every x and y that go together with each other, that are a coordinate, you put into this. And what happens is, let's say that x is above its mean when y is below its mean. So let's say in the population, you had the point, so one instantiation of the random variables, you have, you sample once, I guess, from the universe. And you get x is equal to 1, and that y is equal to, let's say y is equal to 3. And let's say that you knew ahead of time that the expected value of x is 0, and let's say that the expected value of y is equal to 4. So in this situation, what just happened?"}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say in the population, you had the point, so one instantiation of the random variables, you have, you sample once, I guess, from the universe. And you get x is equal to 1, and that y is equal to, let's say y is equal to 3. And let's say that you knew ahead of time that the expected value of x is 0, and let's say that the expected value of y is equal to 4. So in this situation, what just happened? Now we don't know the entire covariance, we only have one sample here of this random variable. But what just happened here? We have 1 minus, so we're not going to calculate the entire expected value, I just want to calculate what happens when we do what's inside the expected value."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in this situation, what just happened? Now we don't know the entire covariance, we only have one sample here of this random variable. But what just happened here? We have 1 minus, so we're not going to calculate the entire expected value, I just want to calculate what happens when we do what's inside the expected value. We'll have 1 minus 0, so you'll have a 1, times a 3 minus 4, times a negative 1. So you're going to have 1 times negative 1, which is negative 1. And what is that telling us?"}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 1 minus, so we're not going to calculate the entire expected value, I just want to calculate what happens when we do what's inside the expected value. We'll have 1 minus 0, so you'll have a 1, times a 3 minus 4, times a negative 1. So you're going to have 1 times negative 1, which is negative 1. And what is that telling us? Well, it's telling us at least for this sample, this one time that we sampled the random variables x and y, x was above its expected value when y was below its expected value. And if we kept doing this, let's say for the entire population this happened, then it would make sense that they have a negative covariance. When one goes up, the other one goes down."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what is that telling us? Well, it's telling us at least for this sample, this one time that we sampled the random variables x and y, x was above its expected value when y was below its expected value. And if we kept doing this, let's say for the entire population this happened, then it would make sense that they have a negative covariance. When one goes up, the other one goes down. When one goes down, the other one goes up. If they both go up together, they would have a positive variance. Or if they both go down together, the degree to which they do it together would tell you the magnitude of the covariance."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "When one goes up, the other one goes down. When one goes down, the other one goes up. If they both go up together, they would have a positive variance. Or if they both go down together, the degree to which they do it together would tell you the magnitude of the covariance. Hopefully that gives you a little bit of intuition about what the covariance is trying to tell us. But the more important thing that I want to do in this video is to connect this formula, is I want to connect this definition of covariance to everything we've been doing with least squared regression. And really it's just kind of a fun math thing to do to show you all of these connections and where really the definition of covariance really becomes useful."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if they both go down together, the degree to which they do it together would tell you the magnitude of the covariance. Hopefully that gives you a little bit of intuition about what the covariance is trying to tell us. But the more important thing that I want to do in this video is to connect this formula, is I want to connect this definition of covariance to everything we've been doing with least squared regression. And really it's just kind of a fun math thing to do to show you all of these connections and where really the definition of covariance really becomes useful. And I really do think it's motivated to a large degree by where it shows up in regressions. And this is all stuff that we've kind of seen before. You're just going to see it in a different way."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And really it's just kind of a fun math thing to do to show you all of these connections and where really the definition of covariance really becomes useful. And I really do think it's motivated to a large degree by where it shows up in regressions. And this is all stuff that we've kind of seen before. You're just going to see it in a different way. So this whole video I'm just going to rewrite this definition of covariance right over here. So this is going to be the same thing as the expected value of, and I'm just going to multiply these two binomials in here. So the expected value of our random variable X times our random variable Y minus, well I'll just do the X first."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're just going to see it in a different way. So this whole video I'm just going to rewrite this definition of covariance right over here. So this is going to be the same thing as the expected value of, and I'm just going to multiply these two binomials in here. So the expected value of our random variable X times our random variable Y minus, well I'll just do the X first. So plus X times the negative expected value of Y. So I'll just say minus X times the expected value of Y. And that negative sign comes from this negative sign right over here."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the expected value of our random variable X times our random variable Y minus, well I'll just do the X first. So plus X times the negative expected value of Y. So I'll just say minus X times the expected value of Y. And that negative sign comes from this negative sign right over here. And then we have minus expected value of X times Y minus the expected value of X times this Y. Just doing the distributed property twice. And then finally you have the negative expected value of X times the negative expected value of Y."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that negative sign comes from this negative sign right over here. And then we have minus expected value of X times Y minus the expected value of X times this Y. Just doing the distributed property twice. And then finally you have the negative expected value of X times the negative expected value of Y. And the negatives cancel out. So you're just going to have plus the expected value of X times the expected value of Y. And of course it's the expected value of this entire thing."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally you have the negative expected value of X times the negative expected value of Y. And the negatives cancel out. So you're just going to have plus the expected value of X times the expected value of Y. And of course it's the expected value of this entire thing. Now let's see if we can rewrite this. Well the expected value of the sum of a bunch of random variables is just the sum or difference of their expected values. So this is going to be the same thing."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And of course it's the expected value of this entire thing. Now let's see if we can rewrite this. Well the expected value of the sum of a bunch of random variables is just the sum or difference of their expected values. So this is going to be the same thing. And remember expected value in a lot of context you can view it as just the arithmetic mean. Or in a continuous distribution you can view it as a probability weighted sum or probability weighted integral. Either way, we've seen it before I think."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be the same thing. And remember expected value in a lot of context you can view it as just the arithmetic mean. Or in a continuous distribution you can view it as a probability weighted sum or probability weighted integral. Either way, we've seen it before I think. So let's rewrite this. So this is equal to the expected value of the random variables X and Y. X times Y. Trying to keep them color coded for you."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Either way, we've seen it before I think. So let's rewrite this. So this is equal to the expected value of the random variables X and Y. X times Y. Trying to keep them color coded for you. Color coded. And then we have minus X times the expected value of Y. So then we're going to have minus the expected value of X times the expected value of Y."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Trying to keep them color coded for you. Color coded. And then we have minus X times the expected value of Y. So then we're going to have minus the expected value of X times the expected value of Y. Of X times the expected value of Y. Times the expected value of Y. Stay with the right colors."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So then we're going to have minus the expected value of X times the expected value of Y. Of X times the expected value of Y. Times the expected value of Y. Stay with the right colors. Then you're going to have minus the expected value of this thing. Minus the expected value of, I'll close the parentheses of this thing right over here. Expected value of X."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Stay with the right colors. Then you're going to have minus the expected value of this thing. Minus the expected value of, I'll close the parentheses of this thing right over here. Expected value of X. Expected value of X times Y. Now this might look really confusing with all the embedded expected values. But one way to think about it is the things that already have the expected values, you can kind of view these as numbers."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Expected value of X. Expected value of X times Y. Now this might look really confusing with all the embedded expected values. But one way to think about it is the things that already have the expected values, you can kind of view these as numbers. You already view them as known. So we're actually going to take them out of the expected value. Because the expected value of an expected value is the same thing as the expected value."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "But one way to think about it is the things that already have the expected values, you can kind of view these as numbers. You already view them as known. So we're actually going to take them out of the expected value. Because the expected value of an expected value is the same thing as the expected value. Actually let me write this over here just to remind ourselves. The expected value of X is just going to be the expected value of X. Think of it this way."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because the expected value of an expected value is the same thing as the expected value. Actually let me write this over here just to remind ourselves. The expected value of X is just going to be the expected value of X. Think of it this way. You could view this as the population mean for the random variable. So that's just going to be a known, it's out there, it's in the universe. So the expected value of that is just going to be itself."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Think of it this way. You could view this as the population mean for the random variable. So that's just going to be a known, it's out there, it's in the universe. So the expected value of that is just going to be itself. If the population mean or the expected value of X is 5, this is like saying the expected value of 5. Well the expected value of 5 is going to be 5. Which is the same thing as the expected value of X. Hopefully that makes sense."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the expected value of that is just going to be itself. If the population mean or the expected value of X is 5, this is like saying the expected value of 5. Well the expected value of 5 is going to be 5. Which is the same thing as the expected value of X. Hopefully that makes sense. We're going to use that in a second. So we're almost done. We did the expected value of this and we have one term left."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which is the same thing as the expected value of X. Hopefully that makes sense. We're going to use that in a second. So we're almost done. We did the expected value of this and we have one term left. And then the final term, the expected value of this guy. And here we can actually use the property right from the get-go. I'll write it down."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We did the expected value of this and we have one term left. And then the final term, the expected value of this guy. And here we can actually use the property right from the get-go. I'll write it down. So the expected value of, put some big brackets up, of this thing right over here. Expected value of X times the expected value of Y. Let's see if we can simplify it right here."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll write it down. So the expected value of, put some big brackets up, of this thing right over here. Expected value of X times the expected value of Y. Let's see if we can simplify it right here. So this is just going to be the expected value of the product of these two random variables. I'll just leave that the way it is. So let me just, the stuff that I'm going to leave the way it is, I'm just going to kind of freeze them."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see if we can simplify it right here. So this is just going to be the expected value of the product of these two random variables. I'll just leave that the way it is. So let me just, the stuff that I'm going to leave the way it is, I'm just going to kind of freeze them. So the expected value of XY. Now what do we have over here? We have the expected value of X times, once again, you can kind of view it if you go back to what we just said, is this is just going to be a number, expected value of Y."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me just, the stuff that I'm going to leave the way it is, I'm just going to kind of freeze them. So the expected value of XY. Now what do we have over here? We have the expected value of X times, once again, you can kind of view it if you go back to what we just said, is this is just going to be a number, expected value of Y. So we can just bring this out. If this was the expected value of 3X, it would be the same thing as 3 times the expected value of X. So we can rewrite this as negative expected value of Y times the expected value of X."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have the expected value of X times, once again, you can kind of view it if you go back to what we just said, is this is just going to be a number, expected value of Y. So we can just bring this out. If this was the expected value of 3X, it would be the same thing as 3 times the expected value of X. So we can rewrite this as negative expected value of Y times the expected value of X. So you can kind of view this as we took it out of the expected value. We factored it out. So just like that."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can rewrite this as negative expected value of Y times the expected value of X. So you can kind of view this as we took it out of the expected value. We factored it out. So just like that. And then you have minus, same thing over here. You can factor out this expected value of X minus the expected value of X times the expected value of Y times the expected value of Y. Let me write it."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just like that. And then you have minus, same thing over here. You can factor out this expected value of X minus the expected value of X times the expected value of Y times the expected value of Y. Let me write it. Times the expected value of Y. This is getting confusing with all the E's laying around. And then finally, you have the expected value of this thing, of two expected values."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write it. Times the expected value of Y. This is getting confusing with all the E's laying around. And then finally, you have the expected value of this thing, of two expected values. Well, that's just going to be the product of those two expected values. So that's just going to be plus, I'll freeze this, expected value of X times the expected value of Y. Now what do we have here?"}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, you have the expected value of this thing, of two expected values. Well, that's just going to be the product of those two expected values. So that's just going to be plus, I'll freeze this, expected value of X times the expected value of Y. Now what do we have here? We have expected value of Y times the expected value of X, and then we are subtracting the expected value of X times the expected value of Y. These two things are the exact same thing. So this is going to be, and actually look at this, we're subtracting it twice and then we have one more."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what do we have here? We have expected value of Y times the expected value of X, and then we are subtracting the expected value of X times the expected value of Y. These two things are the exact same thing. So this is going to be, and actually look at this, we're subtracting it twice and then we have one more. These are all the same thing. This is the expected value of Y times the expected value of X. This is the expected value of Y times the expected value of X, just written in a different order."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be, and actually look at this, we're subtracting it twice and then we have one more. These are all the same thing. This is the expected value of Y times the expected value of X. This is the expected value of Y times the expected value of X, just written in a different order. And this is the expected value of Y times the expected value of X. We're subtracting it twice and then we're adding it. Or, one way to think about it is that this guy and that guy will cancel out."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the expected value of Y times the expected value of X, just written in a different order. And this is the expected value of Y times the expected value of X. We're subtracting it twice and then we're adding it. Or, one way to think about it is that this guy and that guy will cancel out. You could have also picked that guy and that guy. But what do we have left? We have the covariance of these two random variables, X and Y, are equal to the expected value of, I'll switch back to my colors just because this is the final result, the expected value of X times the expected value of the product of XY, the expected value of the product, the expected value of the product, minus, what is this?"}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or, one way to think about it is that this guy and that guy will cancel out. You could have also picked that guy and that guy. But what do we have left? We have the covariance of these two random variables, X and Y, are equal to the expected value of, I'll switch back to my colors just because this is the final result, the expected value of X times the expected value of the product of XY, the expected value of the product, the expected value of the product, minus, what is this? The expected value of Y, the expected value of Y times the expected value of X, times the expected value of X. Now, you can calculate these expected values if you know everything about the probability distribution or density functions for each of these random variables, or if you had the entire population that you're sampling from whenever you take an instantiation of these random variables. But let's say you just had a sample of these random variables."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have the covariance of these two random variables, X and Y, are equal to the expected value of, I'll switch back to my colors just because this is the final result, the expected value of X times the expected value of the product of XY, the expected value of the product, the expected value of the product, minus, what is this? The expected value of Y, the expected value of Y times the expected value of X, times the expected value of X. Now, you can calculate these expected values if you know everything about the probability distribution or density functions for each of these random variables, or if you had the entire population that you're sampling from whenever you take an instantiation of these random variables. But let's say you just had a sample of these random variables. How could you estimate them? Well, if you were estimating it, the expected value, let's say you just have a bunch of data points, a bunch of coordinates, and I think you'll start to see how this relates to what we did with regression. The expected value of X times Y, it can be approximated by the sample mean of the products of X and Y."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "But let's say you just had a sample of these random variables. How could you estimate them? Well, if you were estimating it, the expected value, let's say you just have a bunch of data points, a bunch of coordinates, and I think you'll start to see how this relates to what we did with regression. The expected value of X times Y, it can be approximated by the sample mean of the products of X and Y. This is going to be the sample mean of X and Y. You take each of your XY associations, take their product, and then take the mean of all of them. So that's going to be the product of X and Y, and then this thing right over here, the expected value of Y, that can be approximated by the sample mean of Y, and the expected value of X can be approximated by the sample mean of X."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "The expected value of X times Y, it can be approximated by the sample mean of the products of X and Y. This is going to be the sample mean of X and Y. You take each of your XY associations, take their product, and then take the mean of all of them. So that's going to be the product of X and Y, and then this thing right over here, the expected value of Y, that can be approximated by the sample mean of Y, and the expected value of X can be approximated by the sample mean of X. So what can the covariance of two random variables be approximated by? What can they be approximated by? Well, this right here is the mean of their product from your sample, minus the mean of your sample Ys times the mean of your sample Xs."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's going to be the product of X and Y, and then this thing right over here, the expected value of Y, that can be approximated by the sample mean of Y, and the expected value of X can be approximated by the sample mean of X. So what can the covariance of two random variables be approximated by? What can they be approximated by? Well, this right here is the mean of their product from your sample, minus the mean of your sample Ys times the mean of your sample Xs. And this should start looking familiar. This should look a little bit familiar, because what is this? This was the numerator."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this right here is the mean of their product from your sample, minus the mean of your sample Ys times the mean of your sample Xs. And this should start looking familiar. This should look a little bit familiar, because what is this? This was the numerator. This right here is the numerator when we were trying to figure out the slope of the regression line. So we tried to figure out the slope of the regression line. Let me just rewrite the formula here just to remind you."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This was the numerator. This right here is the numerator when we were trying to figure out the slope of the regression line. So we tried to figure out the slope of the regression line. Let me just rewrite the formula here just to remind you. It was literally the mean of the products of each of our data points, minus, or the XYs, minus the mean of Ys times the mean of the Xs. All of that over the mean of the X squareds, and you could even view it as this, over the mean of the X times the Xs, but I could just write the X squareds over here, minus the mean of X squared. This is how we figured out the slope of our regression line."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me just rewrite the formula here just to remind you. It was literally the mean of the products of each of our data points, minus, or the XYs, minus the mean of Ys times the mean of the Xs. All of that over the mean of the X squareds, and you could even view it as this, over the mean of the X times the Xs, but I could just write the X squareds over here, minus the mean of X squared. This is how we figured out the slope of our regression line. Or maybe a better way to think about it, if we assume in our regression line that the points that we have were a sample from an entire universe of possible points, then you could say that we are approximating the slope of our regression line. You might see this little hat notation in a lot of books. I don't want you to be confused."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is how we figured out the slope of our regression line. Or maybe a better way to think about it, if we assume in our regression line that the points that we have were a sample from an entire universe of possible points, then you could say that we are approximating the slope of our regression line. You might see this little hat notation in a lot of books. I don't want you to be confused. You're saying that you're approximating the population's regression line from a sample of it. Now, this right here, so everything we've learned right now, this right here is the covariance, or this is an estimate of the covariance of X and Y. Now, what is this over here?"}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I don't want you to be confused. You're saying that you're approximating the population's regression line from a sample of it. Now, this right here, so everything we've learned right now, this right here is the covariance, or this is an estimate of the covariance of X and Y. Now, what is this over here? I just said you could rewrite this very easily. This bottom part right here, you could write as the mean of X times X, that's the same thing as X squared, minus the mean of X times the mean of X. That's what the mean of X squared is."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what is this over here? I just said you could rewrite this very easily. This bottom part right here, you could write as the mean of X times X, that's the same thing as X squared, minus the mean of X times the mean of X. That's what the mean of X squared is. What's this? You could view this as the covariance of X with X. We've actually already seen this, and I've actually shown you many, many videos ago when we first learned about it what this is."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what the mean of X squared is. What's this? You could view this as the covariance of X with X. We've actually already seen this, and I've actually shown you many, many videos ago when we first learned about it what this is. The covariance of a random variable with itself is really just the variance of that random variable. You could verify it for yourself. If you change this Y to an X, this becomes X minus the expected value of X times X minus the expected value of X, or that's the expected value of X minus the expected value of X squared."}, {"video_title": "Covariance and the regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We've actually already seen this, and I've actually shown you many, many videos ago when we first learned about it what this is. The covariance of a random variable with itself is really just the variance of that random variable. You could verify it for yourself. If you change this Y to an X, this becomes X minus the expected value of X times X minus the expected value of X, or that's the expected value of X minus the expected value of X squared. That's your definition of variance. Another way of thinking about the slope of our regression line, it can be literally viewed as the covariance of our two random variables over the variance of X. You can view it as the independent random variable."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "City Councilwoman Kelly wants to know how the residents of her district feel about a proposed school redistricting plan. Which of the following survey methods will allow Councilwoman Kelly to make a valid conclusion about how residents of her district feel about the proposed plan? So before we even look at these, we have to realize that if you're trying to make a valid conclusion about how the residents of her entire district feel about the proposed plan, she has to find a representative sample, and not kind of a skewed sample that would just sample parts of her district. So let's look at her choices. Should she just ask her neighbors? Well, her neighbors, she might live in a part of the neighborhood that might unusually benefit from the redistricting plan or might get hurt from the redistricting plan. And so just her neighbors wouldn't be representative of the district as a whole."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look at her choices. Should she just ask her neighbors? Well, her neighbors, she might live in a part of the neighborhood that might unusually benefit from the redistricting plan or might get hurt from the redistricting plan. And so just her neighbors wouldn't be representative of the district as a whole. So just asking her neighbors probably does not make sense. Ask the residents of Whispering Pines Retirement Community. So once again, this is not, the first one's skews by geography."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so just her neighbors wouldn't be representative of the district as a whole. So just asking her neighbors probably does not make sense. Ask the residents of Whispering Pines Retirement Community. So once again, this is not, the first one's skews by geography. She's oversampling her neighbors and not the entire district. Here, she's oversampling a specific age demographic. So here she's oversampling older residents who might have very different opinions than middle-aged or younger residents."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once again, this is not, the first one's skews by geography. She's oversampling her neighbors and not the entire district. Here, she's oversampling a specific age demographic. So here she's oversampling older residents who might have very different opinions than middle-aged or younger residents. So that doesn't make sense either. Ask 200 residents of her district whose names are chosen at random. Well, that seems reasonable."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here she's oversampling older residents who might have very different opinions than middle-aged or younger residents. So that doesn't make sense either. Ask 200 residents of her district whose names are chosen at random. Well, that seems reasonable. It doesn't seem like there's some chance that you somehow oversample one direction or another, but it's most likely to give a reasonably representative sample. And this is a pretty large sample size. So it's important to say what is the random process, where is she getting these names from, but this actually does seem reasonable."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that seems reasonable. It doesn't seem like there's some chance that you somehow oversample one direction or another, but it's most likely to give a reasonably representative sample. And this is a pretty large sample size. So it's important to say what is the random process, where is she getting these names from, but this actually does seem reasonable. Ask a group of parents at the local playground. Well, once again, this is just like asking your neighbors, and it's also sampling a specific demographic. Now, this might be the demographic that cares most about the schools, but she wants to know how her whole district feels about the redistricting plan."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's important to say what is the random process, where is she getting these names from, but this actually does seem reasonable. Ask a group of parents at the local playground. Well, once again, this is just like asking your neighbors, and it's also sampling a specific demographic. Now, this might be the demographic that cares most about the schools, but she wants to know how her whole district feels about the redistricting plan. And once again, this is at a local playground. This isn't at all the playgrounds in the district somehow, so I wouldn't do this one either. Let's do one more of these."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, this might be the demographic that cares most about the schools, but she wants to know how her whole district feels about the redistricting plan. And once again, this is at a local playground. This isn't at all the playgrounds in the district somehow, so I wouldn't do this one either. Let's do one more of these. Mimi wants to conduct a survey of her 300 classmates to determine which candidate for class president Napoleon Dynamite or Blair Waldorf is in the lead in the upcoming election. Mimi will ask the question, if the election were today, which candidate would get your vote? Which of the following methods of surveying her classmates will allow Mimi to make valid conclusions about which candidate is in the lead?"}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do one more of these. Mimi wants to conduct a survey of her 300 classmates to determine which candidate for class president Napoleon Dynamite or Blair Waldorf is in the lead in the upcoming election. Mimi will ask the question, if the election were today, which candidate would get your vote? Which of the following methods of surveying her classmates will allow Mimi to make valid conclusions about which candidate is in the lead? So let's see. Ask all of the students at Blair's lunch table. No, that would skew it in Blair's favor."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which of the following methods of surveying her classmates will allow Mimi to make valid conclusions about which candidate is in the lead? So let's see. Ask all of the students at Blair's lunch table. No, that would skew it in Blair's favor. Probably. That's not a representative sample. Ask all the members of Napoleon's soccer team."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "No, that would skew it in Blair's favor. Probably. That's not a representative sample. Ask all the members of Napoleon's soccer team. Same thing. They're likely to go Napoleon's way, or maybe they don't like Napoleon. Maybe they'll go against Napoleon, but either way, this seems like a skewed sample."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Ask all the members of Napoleon's soccer team. Same thing. They're likely to go Napoleon's way, or maybe they don't like Napoleon. Maybe they'll go against Napoleon, but either way, this seems like a skewed sample. Put the names of all the students in a hat and draw 50 names. Ask those students whose names are drawn. Well, this seems like a nice random sample that could be nicely representative of the entire population."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe they'll go against Napoleon, but either way, this seems like a skewed sample. Put the names of all the students in a hat and draw 50 names. Ask those students whose names are drawn. Well, this seems like a nice random sample that could be nicely representative of the entire population. Ask all students whose names begin with N or B. Well, this could be perceived as kind of random, but notice N is the same starting letter as Napoleon. B is the same starting letter as Blair."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this seems like a nice random sample that could be nicely representative of the entire population. Ask all students whose names begin with N or B. Well, this could be perceived as kind of random, but notice N is the same starting letter as Napoleon. B is the same starting letter as Blair. You might say, well, that's fair. You're doing it for each of their letters, but maybe there's like 10 people whose names start with an N and only two people whose names start with a B. Once again, you're not even getting a large sample."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "B is the same starting letter as Blair. You might say, well, that's fair. You're doing it for each of their letters, but maybe there's like 10 people whose names start with an N and only two people whose names start with a B. Once again, you're not even getting a large sample. And then on top of that, maybe there's some type of people with the same starting letters somehow like each other more. So I would steer clear of this one. Ask every student in the class."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, you're not even getting a large sample. And then on top of that, maybe there's some type of people with the same starting letters somehow like each other more. So I would steer clear of this one. Ask every student in the class. Well, that would work. You know, there's 300 classmates. That might not be that time consuming, and that would be a pretty good."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Ask every student in the class. Well, that would work. You know, there's 300 classmates. That might not be that time consuming, and that would be a pretty good. You can't get a better sample than asking everyone in the population. Which of the following methods of surveying your classmates will allow Mimi to make a valid conclusion about which candidate is in the lead? Well, that's a pretty good conclusion."}, {"video_title": "Reasonable samples   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "That might not be that time consuming, and that would be a pretty good. You can't get a better sample than asking everyone in the population. Which of the following methods of surveying your classmates will allow Mimi to make a valid conclusion about which candidate is in the lead? Well, that's a pretty good conclusion. People might change their minds, so it's not a done deal, but you can't get a better sample size than the entire population. Assign numbers to each student in the class and use a computer program to generate 50 random numbers between 1 and 300. Ask those students whose numbers are selected."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So I have a box and whiskers plot showing us the ages of students at a party. And what I'm hoping to do in this video is get a little bit of practice interpreting this. And what I have here are five different statements. And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct?"}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7. This is going to be 16."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7. This is going to be 16. Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is going to be 16. Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be. It could be 15 as well."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be. It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have an even number right over here. So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that."}, {"video_title": "Interpreting box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that. And that's sometimes OK. But it's very important to think about what types of actual statements you can make and what you can't make. And it's very important when you're looking at statistics to say, well, you know what?"}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And it's called the Monty Hall problem because Monty Hall was the game show host in Let's Make a Deal, where they would set up a situation very similar to the Monty Hall problem that we're about to say. So let's say that on the show, you're presented with three curtains. So you're the contestant, this little chef-looking character right over there. You're presented with three curtains. Curtain number one, curtain number two, and curtain number three. And you're told that behind one of these three curtains, there's a fabulous prize, something that you really want, a car or a vacation or some type of large amount of cash. And then behind the other two, and we don't know which they are, there is something that you do not want, a new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "You're presented with three curtains. Curtain number one, curtain number two, and curtain number three. And you're told that behind one of these three curtains, there's a fabulous prize, something that you really want, a car or a vacation or some type of large amount of cash. And then behind the other two, and we don't know which they are, there is something that you do not want, a new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize. And so your goal is to try to find the cash prize. And they say, guess which one, or which one would you like to select? And so let's say that you select door number one, or curtain number one."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then behind the other two, and we don't know which they are, there is something that you do not want, a new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize. And so your goal is to try to find the cash prize. And they say, guess which one, or which one would you like to select? And so let's say that you select door number one, or curtain number one. Then the Monty Hall and Let's Make a Deal crew will make it a little bit more interesting. They won't just show you whether or not you won. They'll then show you one of the other two doors, or one of the other two curtains."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so let's say that you select door number one, or curtain number one. Then the Monty Hall and Let's Make a Deal crew will make it a little bit more interesting. They won't just show you whether or not you won. They'll then show you one of the other two doors, or one of the other two curtains. And they'll show you one of the other two curtains that does not have the prize. And no matter which one you pick, there will always be at least one other curtain that does not have the prize. There might be two if you picked right, but none of them have the prize."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "They'll then show you one of the other two doors, or one of the other two curtains. And they'll show you one of the other two curtains that does not have the prize. And no matter which one you pick, there will always be at least one other curtain that does not have the prize. There might be two if you picked right, but none of them have the prize. And then they will show it to you. And so let's say that they show you curtain number three, and so curtain number three has the goat. And then they will ask you, do you want to switch to curtain number two?"}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "There might be two if you picked right, but none of them have the prize. And then they will show it to you. And so let's say that they show you curtain number three, and so curtain number three has the goat. And then they will ask you, do you want to switch to curtain number two? And the question here is, does it make a difference? Are you better off holding fast, sticking to your guns, staying with the original guess? Are you better off switching to whatever curtain is left?"}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then they will ask you, do you want to switch to curtain number two? And the question here is, does it make a difference? Are you better off holding fast, sticking to your guns, staying with the original guess? Are you better off switching to whatever curtain is left? Or does it not matter? It's just random probability, and it's not going to make a difference whether you switch or not. So that is the brain teaser."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Are you better off switching to whatever curtain is left? Or does it not matter? It's just random probability, and it's not going to make a difference whether you switch or not. So that is the brain teaser. Pause the video now. I encourage you to think about it. In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So that is the brain teaser. Pause the video now. I encourage you to think about it. In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all. So now I've assumed that you've unpaused it, you've thought deeply about it. Perhaps you have an opinion on it. Now let's work it through a little bit."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all. So now I've assumed that you've unpaused it, you've thought deeply about it. Perhaps you have an opinion on it. Now let's work it through a little bit. And at any point, we can extrapolate beyond what I've already talked about. So let's think about the game show from the show's point of view. So the show knows where there's the goat and where there isn't the goat."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now let's work it through a little bit. And at any point, we can extrapolate beyond what I've already talked about. So let's think about the game show from the show's point of view. So the show knows where there's the goat and where there isn't the goat. So let's door number one, door number two, and door number three. So let's say that our prize is right over here. So our prize is the car."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So the show knows where there's the goat and where there isn't the goat. So let's door number one, door number two, and door number three. So let's say that our prize is right over here. So our prize is the car. And that we have a goat over here, goat over here, and over here we also have two goats in this situation. So what are we going to do as the game show? Remember, the contestants don't know this."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So our prize is the car. And that we have a goat over here, goat over here, and over here we also have two goats in this situation. So what are we going to do as the game show? Remember, the contestants don't know this. We know this. So if the contestant picks door number one right over here, then we can't lift door number two because there's a car back there. We're going to lift door number three and we're going to expose this goat, in which case it probably would be good for the person to switch."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Remember, the contestants don't know this. We know this. So if the contestant picks door number one right over here, then we can't lift door number two because there's a car back there. We're going to lift door number three and we're going to expose this goat, in which case it probably would be good for the person to switch. If the person picks door number two, then we as the game show can show either door number one or door number three, and then it actually does not make sense for the person to switch. If they picked door number three, then we have to show door number one because we can't pick door number two, and in that case it actually makes a lot of sense for the person to switch. Now, with that out of the way, let's think about the probabilities given the two strategies."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We're going to lift door number three and we're going to expose this goat, in which case it probably would be good for the person to switch. If the person picks door number two, then we as the game show can show either door number one or door number three, and then it actually does not make sense for the person to switch. If they picked door number three, then we have to show door number one because we can't pick door number two, and in that case it actually makes a lot of sense for the person to switch. Now, with that out of the way, let's think about the probabilities given the two strategies. If you don't switch, or another way to think about this strategy is you always stick to your guns. You always stick to your first guess. Well, in that situation, what is your probability of winning?"}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now, with that out of the way, let's think about the probabilities given the two strategies. If you don't switch, or another way to think about this strategy is you always stick to your guns. You always stick to your first guess. Well, in that situation, what is your probability of winning? Well, there's three doors. The prize is equally likely to be behind any one of them, so there's three possibilities. One has the outcome that you desire."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, in that situation, what is your probability of winning? Well, there's three doors. The prize is equally likely to be behind any one of them, so there's three possibilities. One has the outcome that you desire. The probability of winning will be 1 3rd if you don't switch. Likewise, your probability of losing, well, there's two ways that you can lose out of three possibilities. It's going to be 2 3rd, and these are the only possibilities, and these add up to one right over here."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "One has the outcome that you desire. The probability of winning will be 1 3rd if you don't switch. Likewise, your probability of losing, well, there's two ways that you can lose out of three possibilities. It's going to be 2 3rd, and these are the only possibilities, and these add up to one right over here. So don't switch 1 3rd chance of winning. Now let's think about the switching situation. So let's say always, when you always switch, let's think about how this might play out."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "It's going to be 2 3rd, and these are the only possibilities, and these add up to one right over here. So don't switch 1 3rd chance of winning. Now let's think about the switching situation. So let's say always, when you always switch, let's think about how this might play out. What is your probability of winning? And before we even think about that, think about how you would win if you always switch. So if you pick wrong the first time, they're going to show you this, and so you should always switch."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let's say always, when you always switch, let's think about how this might play out. What is your probability of winning? And before we even think about that, think about how you would win if you always switch. So if you pick wrong the first time, they're going to show you this, and so you should always switch. So if you pick door number 1, they're going to show you door number 3. You should switch. If you picked wrong door number 3, they're going to show you door number 1."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So if you pick wrong the first time, they're going to show you this, and so you should always switch. So if you pick door number 1, they're going to show you door number 3. You should switch. If you picked wrong door number 3, they're going to show you door number 1. You should switch. So if you picked wrong and switch, you will always win. Let me write this down."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "If you picked wrong door number 3, they're going to show you door number 1. You should switch. So if you picked wrong and switch, you will always win. Let me write this down. And this insight actually came from one of the middle school students in the summer camp that Khan Academy was running. It's actually a fabulous way to think about this. So if you pick wrong, so if it's step 1, so initial pick is wrong."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let me write this down. And this insight actually came from one of the middle school students in the summer camp that Khan Academy was running. It's actually a fabulous way to think about this. So if you pick wrong, so if it's step 1, so initial pick is wrong. So you pick one of the two wrong doors, and then in step 2, you always switch. You will land on the car, because if you picked one of the wrong doors, they're going to have to show the other wrong door, and so if you switch, you're going to end up on the right answer. So what is the probability of winning if you always switch?"}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So if you pick wrong, so if it's step 1, so initial pick is wrong. So you pick one of the two wrong doors, and then in step 2, you always switch. You will land on the car, because if you picked one of the wrong doors, they're going to have to show the other wrong door, and so if you switch, you're going to end up on the right answer. So what is the probability of winning if you always switch? Well, it's going to be the probability that you initially picked wrong. Well, what's the probability that you initially picked wrong? Well, there's two out of the three ways to initially pick wrong."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So what is the probability of winning if you always switch? Well, it's going to be the probability that you initially picked wrong. Well, what's the probability that you initially picked wrong? Well, there's two out of the three ways to initially pick wrong. So you actually have a 2 3rds chance of winning. There's a 2 3rds chance you're going to pick wrong and then switch into the right one. Likewise, what's your probability of losing?"}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, there's two out of the three ways to initially pick wrong. So you actually have a 2 3rds chance of winning. There's a 2 3rds chance you're going to pick wrong and then switch into the right one. Likewise, what's your probability of losing? Given that you're always going to switch? Well, the way that you would lose is you pick right, you pick correctly, and step 2, they're going to show one of the two empty or non-prized doors, and then step 3, you're going to switch into the other empty. But either way, you're definitely going to switch, so the only way to lose if you're always going to switch is to pick right the first time."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Likewise, what's your probability of losing? Given that you're always going to switch? Well, the way that you would lose is you pick right, you pick correctly, and step 2, they're going to show one of the two empty or non-prized doors, and then step 3, you're going to switch into the other empty. But either way, you're definitely going to switch, so the only way to lose if you're always going to switch is to pick right the first time. Well, what's the probability of you picking right the first time? Well, that is only 1 3rd. So you see it here, and it's sometimes counterintuitive, but hopefully this makes sense as to why it isn't."}, {"video_title": "Probability and the Monty Hall problem   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "But either way, you're definitely going to switch, so the only way to lose if you're always going to switch is to pick right the first time. Well, what's the probability of you picking right the first time? Well, that is only 1 3rd. So you see it here, and it's sometimes counterintuitive, but hopefully this makes sense as to why it isn't. You have a 1 3rd chance of winning if you stick to your guns and a 2 3rds chance of winning if you always switch. Another way to think about it is when you first make your initial pick, there's a 1 3rd chance that it's there, and there's a 2 3rds chance that it's in one of the other two doors, and they're going to empty out one of them, so when you switch, you essentially are capturing that 2 3rd probability, and we see that right there. I hope you enjoyed that."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "Now, let's say you have a hunch that, well, maybe it is skewed towards one letter or another. How could you test this? Well, you could start with a null and alternative hypothesis, and then we can actually do a hypothesis test. So let's say that our null hypothesis is equal distribution, equal distribution of correct choices, correct choices. Or another way of thinking about it is A would be correct 25% of the time, B would be correct 25% of the time, C would be correct 25% of the time, and D would be correct 25% of the time. Now, what would be our alternative hypothesis? Well, our alternative hypothesis would be not equal distribution, not equal distribution."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say that our null hypothesis is equal distribution, equal distribution of correct choices, correct choices. Or another way of thinking about it is A would be correct 25% of the time, B would be correct 25% of the time, C would be correct 25% of the time, and D would be correct 25% of the time. Now, what would be our alternative hypothesis? Well, our alternative hypothesis would be not equal distribution, not equal distribution. Now, how are we going to actually test this? Well, we've seen this show before, at least the beginnings of the show. You have the population of all of your potential items here, and you could take a sample."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "Well, our alternative hypothesis would be not equal distribution, not equal distribution. Now, how are we going to actually test this? Well, we've seen this show before, at least the beginnings of the show. You have the population of all of your potential items here, and you could take a sample. And so let's say we take a sample of 100 items. So n is equal to 100. And let's write down the data that we get when we look at that sample."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "You have the population of all of your potential items here, and you could take a sample. And so let's say we take a sample of 100 items. So n is equal to 100. And let's write down the data that we get when we look at that sample. So this is the correct choice, correct choice. And then this would be the expected number that you would expect. And then this is the actual number."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And let's write down the data that we get when we look at that sample. So this is the correct choice, correct choice. And then this would be the expected number that you would expect. And then this is the actual number. And if this doesn't make sense yet, we'll see it in a second. So there's four different choices. A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And then this is the actual number. And if this doesn't make sense yet, we'll see it in a second. So there's four different choices. A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. So the expected number where A is the correct choice would be 25% of this 100. So you'd expect 25 times the A to be the correct choice, 25 times B to be the correct choice, 25 times C to be the correct choice, and 25 times D to be the correct choice. But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. So the expected number where A is the correct choice would be 25% of this 100. So you'd expect 25 times the A to be the correct choice, 25 times B to be the correct choice, 25 times C to be the correct choice, and 25 times D to be the correct choice. But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. So if you just look at this, you say, hey, maybe there's a higher frequency of D. But maybe you say, well, this is just a sample, and just a random chance, it might have just gotten more Ds than not. There's some probability of getting this result, even assuming that the null hypothesis is true. And that's the goal of these hypothesis tests."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. So if you just look at this, you say, hey, maybe there's a higher frequency of D. But maybe you say, well, this is just a sample, and just a random chance, it might have just gotten more Ds than not. There's some probability of getting this result, even assuming that the null hypothesis is true. And that's the goal of these hypothesis tests. They say, what's the probability of getting a result at least this extreme? And if that probability is below some threshold, then we tend to reject the null hypothesis and accept an alternative. And those thresholds you have seen before, we've seen these significance levels."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And that's the goal of these hypothesis tests. They say, what's the probability of getting a result at least this extreme? And if that probability is below some threshold, then we tend to reject the null hypothesis and accept an alternative. And those thresholds you have seen before, we've seen these significance levels. Let's say we set a significance level of 5%, 0.05. So if the probability of getting this result, or something even more different than what's expected, is less than the significance level, then we'd reject the null hypothesis. But this all leads to one really interesting question."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And those thresholds you have seen before, we've seen these significance levels. Let's say we set a significance level of 5%, 0.05. So if the probability of getting this result, or something even more different than what's expected, is less than the significance level, then we'd reject the null hypothesis. But this all leads to one really interesting question. How do we calculate a probability of getting a result this extreme or more extreme? How do we even measure that? And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "But this all leads to one really interesting question. How do we calculate a probability of getting a result this extreme or more extreme? How do we even measure that? And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. And that is the capital Greek letter chi, which might look like an X to you, but it's a little bit curvier, and you could look up more on that. You kind of curve that part of the X. But it's a chi, not an X."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. And that is the capital Greek letter chi, which might look like an X to you, but it's a little bit curvier, and you could look up more on that. You kind of curve that part of the X. But it's a chi, not an X. And the statistic is called chi squared. And it's a way of taking the difference between the actual and the expected, and translating that into a number. And the chi squared distribution is well, I really should say distributions, are well studied."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "But it's a chi, not an X. And the statistic is called chi squared. And it's a way of taking the difference between the actual and the expected, and translating that into a number. And the chi squared distribution is well, I really should say distributions, are well studied. And we can use that to figure out what is the probability of getting a result this extreme or more extreme? And if that's lower than our significance level, we reject the null hypothesis, and it suggests the alternative. But how do we calculate the chi squared statistic here?"}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And the chi squared distribution is well, I really should say distributions, are well studied. And we can use that to figure out what is the probability of getting a result this extreme or more extreme? And if that's lower than our significance level, we reject the null hypothesis, and it suggests the alternative. But how do we calculate the chi squared statistic here? Well, it's reasonably intuitive. What we do is, for each of these categories, in this case, it's for each of these choices, we look at the difference between the actual and the expected. So for choice A, we'd say 20 is the actual minus the expected."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "But how do we calculate the chi squared statistic here? Well, it's reasonably intuitive. What we do is, for each of these categories, in this case, it's for each of these choices, we look at the difference between the actual and the expected. So for choice A, we'd say 20 is the actual minus the expected. And then we're going to square that. And then we're going to divide by what was expected. And then we're gonna do that for choice B."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "So for choice A, we'd say 20 is the actual minus the expected. And then we're going to square that. And then we're going to divide by what was expected. And then we're gonna do that for choice B. So we're going to say the actual was 20, expected is 25, so 20 minus 25 squared over the expected over 25. Plus, then we do that for choice C, 25 minus 25, we know where that one will end up, squared over the expected over 25. And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And then we're gonna do that for choice B. So we're going to say the actual was 20, expected is 25, so 20 minus 25 squared over the expected over 25. Plus, then we do that for choice C, 25 minus 25, we know where that one will end up, squared over the expected over 25. And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. And we are now, let's see, if we calculate this, this is going to be negative five squared, so that's going to be 25. This is going to be 25. This is going to be zero."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. And we are now, let's see, if we calculate this, this is going to be negative five squared, so that's going to be 25. This is going to be 25. This is going to be zero. 35 minus 25 is 10 squared, that is 100. So this is one plus one plus zero plus four. So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "This is going to be zero. 35 minus 25 is 10 squared, that is 100. So this is one plus one plus zero plus four. So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. So what do we make of this? Well, what we can do is then look at a chi squared distribution for the appropriate degrees of freedom, and we'll talk about that in a second, and say what is the probability of getting a chi squared statistic six or larger? And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. So what do we make of this? Well, what we can do is then look at a chi squared distribution for the appropriate degrees of freedom, and we'll talk about that in a second, and say what is the probability of getting a chi squared statistic six or larger? And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. And to calculate the degrees of freedom, you look at the number of categories. In this case, we have four categories, and you subtract one. Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. And to calculate the degrees of freedom, you look at the number of categories. In this case, we have four categories, and you subtract one. Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. That's why it is four minus one degrees of freedom. So in this case, our degrees of freedom are going to be equal to three. Over here, sometimes you'll see it described as k. So k is equal to three."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. That's why it is four minus one degrees of freedom. So in this case, our degrees of freedom are going to be equal to three. Over here, sometimes you'll see it described as k. So k is equal to three. So if we look at, that's that little light blue, so we're looking at this chi squared distribution where the degree of freedom is three, and we wanna figure out what is the probability of getting a chi squared statistic that is six or greater? So we would be looking at this area right over here. And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "Over here, sometimes you'll see it described as k. So k is equal to three. So if we look at, that's that little light blue, so we're looking at this chi squared distribution where the degree of freedom is three, and we wanna figure out what is the probability of getting a chi squared statistic that is six or greater? So we would be looking at this area right over here. And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. And so a table like this could be quite useful. Remember, we're dealing with a situation where we have three degrees of freedom. We have four categories, so four minus one is three."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. And so a table like this could be quite useful. Remember, we're dealing with a situation where we have three degrees of freedom. We have four categories, so four minus one is three. And we got a chi squared value. Our chi squared statistic was six. So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "We have four categories, so four minus one is three. And we got a chi squared value. Our chi squared statistic was six. So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. If we go back to this chart, we just learned that this probability from 6.25 and up, when we have three degrees of freedom, that this right over here is 10%. Well, if that's 10%, then the probability, the probability of getting a chi squared value greater than or equal to six is going to be greater than 10%, greater than 10%. And we could also view this as our p-value."}, {"video_title": "Chi-square statistic for hypothesis testing   AP Statistics   Khan Academy.mp3", "Sentence": "So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. If we go back to this chart, we just learned that this probability from 6.25 and up, when we have three degrees of freedom, that this right over here is 10%. Well, if that's 10%, then the probability, the probability of getting a chi squared value greater than or equal to six is going to be greater than 10%, greater than 10%. And we could also view this as our p-value. And so if our probability, assuming the null hypothesis, is greater than 10%, well, it's definitely going to be greater than our significance level. And because of that, we will fail to reject, fail to reject. And so this is an example of, even though in your sample you just happened to get more Ds, the probability of getting a result at least as extreme as what you saw is going to be a little bit over 10%."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is a histogram here. And in each bucket it tells us the number of guests that are in that age bucket. So we don't have any guests that are under the age of 20. We have a reasonable number between 20 and 30. We have a lot of guests in that bucket between 30 and 40, reasonable number between 40 and 50, and then as we get older we have fewer and fewer guests. So just when you look at something like this, a distribution like this, something might pop out at you. It kind of looks like, if you were to imagine this were an armadillo, this would be the body of the armadillo, and then what we see to the right kind of looks like the tail of the armadillo."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have a reasonable number between 20 and 30. We have a lot of guests in that bucket between 30 and 40, reasonable number between 40 and 50, and then as we get older we have fewer and fewer guests. So just when you look at something like this, a distribution like this, something might pop out at you. It kind of looks like, if you were to imagine this were an armadillo, this would be the body of the armadillo, and then what we see to the right kind of looks like the tail of the armadillo. And we actually use those types of words to describe distributions. So this distribution right over here, it looks like it has a tail to the right. It doesn't have a tail to the left."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It kind of looks like, if you were to imagine this were an armadillo, this would be the body of the armadillo, and then what we see to the right kind of looks like the tail of the armadillo. And we actually use those types of words to describe distributions. So this distribution right over here, it looks like it has a tail to the right. It doesn't have a tail to the left. In fact, we have no one under the age of 20. But here when we have a few people between 60 and 70, even fewer between 70 and 80, even fewer between 80 and 90, and if it just kind of keeps going like this, this is a tail and it's on the right side. It's a right-tailed distribution."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It doesn't have a tail to the left. In fact, we have no one under the age of 20. But here when we have a few people between 60 and 70, even fewer between 70 and 80, even fewer between 80 and 90, and if it just kind of keeps going like this, this is a tail and it's on the right side. It's a right-tailed distribution. So I'd call this distribution right-tailed. And I'm using Khan Academy exercises because it's a good way to see a lot of examples, and frankly, you should too because it'll help you test your knowledge. But it's not left-tailed."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's a right-tailed distribution. So I'd call this distribution right-tailed. And I'm using Khan Academy exercises because it's a good way to see a lot of examples, and frankly, you should too because it'll help you test your knowledge. But it's not left-tailed. Left-tailed we would see a tail going like that. And frankly, if you're left-tailed and right-tailed, you're likely to be approximately symmetrical. Remember, symmetry, you define a line of symmetry, and one type of symmetry is one where if you were to, where both sides of that line of symmetry are kind of mirror images of each other."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But it's not left-tailed. Left-tailed we would see a tail going like that. And frankly, if you're left-tailed and right-tailed, you're likely to be approximately symmetrical. Remember, symmetry, you define a line of symmetry, and one type of symmetry is one where if you were to, where both sides of that line of symmetry are kind of mirror images of each other. You could fold over the line of symmetry and they'll roughly meet. And this one does not meet that because if you were to say, hey, maybe there's a line of symmetry here and you were trying to fold this over, it wouldn't match up. The two sides would not match up."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Remember, symmetry, you define a line of symmetry, and one type of symmetry is one where if you were to, where both sides of that line of symmetry are kind of mirror images of each other. You could fold over the line of symmetry and they'll roughly meet. And this one does not meet that because if you were to say, hey, maybe there's a line of symmetry here and you were trying to fold this over, it wouldn't match up. The two sides would not match up. So I feel good saying that it is right-tailed. So let's see. Retirement of age of each guest."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The two sides would not match up. So I feel good saying that it is right-tailed. So let's see. Retirement of age of each guest. Well, yeah, these names aren't that great, but let's actually see what they're saying. They're saying by age, they're telling us the number of guests. So this is the number of guests at a Logan-assisted living."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Retirement of age of each guest. Well, yeah, these names aren't that great, but let's actually see what they're saying. They're saying by age, they're telling us the number of guests. So this is the number of guests at a Logan-assisted living. So we have a lot of guests that are between 60 and 70 years old, or reasonable that are between 50 and 60, or 70 or 80. And this distribution actually looks pretty symmetrical. If I were to draw a line of symmetry right down here, right at around an age of, you know, I guess the line would be right at an age of 65."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is the number of guests at a Logan-assisted living. So we have a lot of guests that are between 60 and 70 years old, or reasonable that are between 50 and 60, or 70 or 80. And this distribution actually looks pretty symmetrical. If I were to draw a line of symmetry right down here, right at around an age of, you know, I guess the line would be right at an age of 65. I guess you could say, oh, this is a bucket for ages 60 to 70. Then you could flip it over and it looks pretty symmetrical. Not exactly."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "If I were to draw a line of symmetry right down here, right at around an age of, you know, I guess the line would be right at an age of 65. I guess you could say, oh, this is a bucket for ages 60 to 70. Then you could flip it over and it looks pretty symmetrical. Not exactly. This bucket doesn't quite match up to this one, but it's pretty close. These roughly match each other. These roughly match each other."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Not exactly. This bucket doesn't quite match up to this one, but it's pretty close. These roughly match each other. These roughly match each other. So I feel good about saying it is approximately symmetrical. Now, just to know what these other words mean, skewed to the left, or skewed to the right. These actually have fairly technical definitions when you get further in statistics, but a, I guess, easier to process version of them are when you have a left tail, you tend to be, when you are left-tailed, you also tend to be skewed to the left."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "These roughly match each other. So I feel good about saying it is approximately symmetrical. Now, just to know what these other words mean, skewed to the left, or skewed to the right. These actually have fairly technical definitions when you get further in statistics, but a, I guess, easier to process version of them are when you have a left tail, you tend to be, when you are left-tailed, you also tend to be skewed to the left. And when you are right-tailed, you tend to be skewed to the right. Another way to think about skewed to the left is that your mean is to the left of your median in mode. That might not make any sense to you."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "These actually have fairly technical definitions when you get further in statistics, but a, I guess, easier to process version of them are when you have a left tail, you tend to be, when you are left-tailed, you also tend to be skewed to the left. And when you are right-tailed, you tend to be skewed to the right. Another way to think about skewed to the left is that your mean is to the left of your median in mode. That might not make any sense to you. You might just want to go after the tail. If you're left-tailed, you're probably left-skewed. If you're right-tailed, you're probably right-skewed."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "That might not make any sense to you. You might just want to go after the tail. If you're left-tailed, you're probably left-skewed. If you're right-tailed, you're probably right-skewed. So let's keep going. Let's see if we can see, let's actually see another example. So this is interesting."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "If you're right-tailed, you're probably right-skewed. So let's keep going. Let's see if we can see, let's actually see another example. So this is interesting. This is not, we're not given a histogram here. We're not given a bar graph. We're given a box and whiskers plot, which is really just telling us the different quartiles."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is interesting. This is not, we're not given a histogram here. We're not given a bar graph. We're given a box and whiskers plot, which is really just telling us the different quartiles. So just to remind ourselves, this tells us the minimum of our data set, the bottom of our range. So the minimum value in our data set, we have at least 111. And then the maximum value of our data set, we have at least 125."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We're given a box and whiskers plot, which is really just telling us the different quartiles. So just to remind ourselves, this tells us the minimum of our data set, the bottom of our range. So the minimum value in our data set, we have at least 111. And then the maximum value of our data set, we have at least 125. Now this line right over here is the median. The middle number is 21. And then the box defines the middle 50% of our numbers."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then the maximum value of our data set, we have at least 125. Now this line right over here is the median. The middle number is 21. And then the box defines the middle 50% of our numbers. So it's kind of the meat of our distribution. So if we were to try to visualize what this would look like as maybe a histogram, and we don't know for sure, because we might have a whole bunch of 11s, not so much that it skews this, but we could have more than one. But a distribution that this could match up with is something that looks like having a tail down here, and then you kind of bump up here."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then the box defines the middle 50% of our numbers. So it's kind of the meat of our distribution. So if we were to try to visualize what this would look like as maybe a histogram, and we don't know for sure, because we might have a whole bunch of 11s, not so much that it skews this, but we could have more than one. But a distribution that this could match up with is something that looks like having a tail down here, and then you kind of bump up here. This is the meat of the distribution. It kind of looks something like that. And I can't draw, because I'm doing this on the exercises right now."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But a distribution that this could match up with is something that looks like having a tail down here, and then you kind of bump up here. This is the meat of the distribution. It kind of looks something like that. And I can't draw, because I'm doing this on the exercises right now. But for something like that, well, something like that would have a tail to the left. Would have a tail to the left. It has, its range goes, you know, fairly low to the left, but it might not have a lot of values there."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I can't draw, because I'm doing this on the exercises right now. But for something like that, well, something like that would have a tail to the left. Would have a tail to the left. It has, its range goes, you know, fairly low to the left, but it might not have a lot of values there. If it had more values on the left side, this box would have been shifted over, because a larger percentage would have fit, would have been on the left, so to speak. And so this one, I feel pretty good about saying this is skewed to the left. It's definitely not symmetrical."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It has, its range goes, you know, fairly low to the left, but it might not have a lot of values there. If it had more values on the left side, this box would have been shifted over, because a larger percentage would have fit, would have been on the left, so to speak. And so this one, I feel pretty good about saying this is skewed to the left. It's definitely not symmetrical. If it was symmetrical, the median would be pretty close to the center. The box would be pretty centered. And it's not skewed to the right."}, {"video_title": "Thinking about shapes of distributions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's definitely not symmetrical. If it was symmetrical, the median would be pretty close to the center. The box would be pretty centered. And it's not skewed to the right. If it was skewed to the right, you would have a tail to the right. You would have, this whisker would likely be much, much, much longer. And we're done."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "There's a parameter here. Let's say it's the population mean. We do not know what this is, so we take a sample. Here we're gonna take a sample of 15. So n is equal to 15, and from that sample, we can calculate a sample mean. But we also wanna construct a 98% confidence interval about that sample mean. So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Here we're gonna take a sample of 15. So n is equal to 15, and from that sample, we can calculate a sample mean. But we also wanna construct a 98% confidence interval about that sample mean. So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error. Now, in other videos, we have talked about that we wanna use the t distribution here because we don't want to underestimate the margin of error. So it's going to be t star times the sample standard deviation divided by the square root of our sample size, which in this case was going to be 15, so the square root of n. But what they're asking us is, well, what is the appropriate critical value? What is the t star that we should use in this situation?"}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error. Now, in other videos, we have talked about that we wanna use the t distribution here because we don't want to underestimate the margin of error. So it's going to be t star times the sample standard deviation divided by the square root of our sample size, which in this case was going to be 15, so the square root of n. But what they're asking us is, well, what is the appropriate critical value? What is the t star that we should use in this situation? And so we're about to look at a, I guess we call it a t table instead of a z table, but the key thing to realize is there's one extra variable to take into consideration when we're looking up the appropriate critical value on a t table, and that's this notion of degree of freedom. Sometimes it's abbreviated DF. And I'm not gonna go in-depth on degrees of freedom."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "What is the t star that we should use in this situation? And so we're about to look at a, I guess we call it a t table instead of a z table, but the key thing to realize is there's one extra variable to take into consideration when we're looking up the appropriate critical value on a t table, and that's this notion of degree of freedom. Sometimes it's abbreviated DF. And I'm not gonna go in-depth on degrees of freedom. It's actually a pretty deep concept, but it's this idea that you actually have a different t distribution depending on the different sample sizes, depending on the degrees of freedom. And your degree of freedom is going to be your sample size minus one. So in this situation, our degree of freedom is going to be 15 minus one."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And I'm not gonna go in-depth on degrees of freedom. It's actually a pretty deep concept, but it's this idea that you actually have a different t distribution depending on the different sample sizes, depending on the degrees of freedom. And your degree of freedom is going to be your sample size minus one. So in this situation, our degree of freedom is going to be 15 minus one. So in this situation, our degree of freedom is going to be equal to 14. And this isn't the first time that we have seen this. We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So in this situation, our degree of freedom is going to be 15 minus one. So in this situation, our degree of freedom is going to be equal to 14. And this isn't the first time that we have seen this. We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation. And in future videos, we'll go into more advanced conversations about degrees of freedom. But for the purposes of this example, you need to know that when you're looking at the t distribution for a given degree of freedom, your degree of freedom is based on the sample size, and it's going to be your sample size minus one. When we're thinking about a confidence interval for your mean."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation. And in future videos, we'll go into more advanced conversations about degrees of freedom. But for the purposes of this example, you need to know that when you're looking at the t distribution for a given degree of freedom, your degree of freedom is based on the sample size, and it's going to be your sample size minus one. When we're thinking about a confidence interval for your mean. So now let's look at the t table. So we want a 98% confidence interval. And we want a degree of freedom of 14."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "When we're thinking about a confidence interval for your mean. So now let's look at the t table. So we want a 98% confidence interval. And we want a degree of freedom of 14. So let's get our t table out. And I actually copy and pasted this bottom part, moved it up so that you could see the whole thing here. And what's useful about this t table is they actually give our confidence levels right over here."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And we want a degree of freedom of 14. So let's get our t table out. And I actually copy and pasted this bottom part, moved it up so that you could see the whole thing here. And what's useful about this t table is they actually give our confidence levels right over here. So if you want a confidence level of 98%, you're going to look at this column. You're going to look at this column right over here. Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And what's useful about this t table is they actually give our confidence levels right over here. So if you want a confidence level of 98%, you're going to look at this column. You're going to look at this column right over here. Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail. And so if you're looking at your t distribution, everything up to and including that top 1%, you would look for a tail probability of 0.01, which is, you can't see it right over there. Let me do it in a slightly brighter color, which would be that tail probability to the right. But either way, we're in this column right over here."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail. And so if you're looking at your t distribution, everything up to and including that top 1%, you would look for a tail probability of 0.01, which is, you can't see it right over there. Let me do it in a slightly brighter color, which would be that tail probability to the right. But either way, we're in this column right over here. We have a confidence level of 98%. And remember, our degrees of freedom, our degree of freedom here is, we have 14 degrees of freedom. And so we'll look at this row right over here."}, {"video_title": "Example finding critical t value   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "But either way, we're in this column right over here. We have a confidence level of 98%. And remember, our degrees of freedom, our degree of freedom here is, we have 14 degrees of freedom. And so we'll look at this row right over here. And so there you have it. This is our critical t value, 2.624. And so let's just go back here."}, {"video_title": "Conditional Probability.mp3", "Sentence": "Consider the following story. Bob is in a room and he has two coins. One fair coin and one double-sided coin. He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree."}, {"video_title": "Conditional Probability.mp3", "Sentence": "He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin."}, {"video_title": "Conditional Probability.mp3", "Sentence": "To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads."}, {"video_title": "Conditional Probability.mp3", "Sentence": "The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, Heads!"}, {"video_title": "Conditional Probability.mp3", "Sentence": "While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, Heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it."}, {"video_title": "Conditional Probability.mp3", "Sentence": "He says, Heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, Heads! Remember, after each event, our tree grows."}, {"video_title": "Conditional Probability.mp3", "Sentence": "And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, Heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads, we cut any branches leading to tails."}, {"video_title": "Conditional Probability.mp3", "Sentence": "Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads, we cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row, is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair."}, {"video_title": "Conditional Probability.mp3", "Sentence": "After we hear the second, heads, we cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row, is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins."}, {"video_title": "Conditional Probability.mp3", "Sentence": "No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it."}, {"video_title": "Conditional Probability.mp3", "Sentence": "Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree."}, {"video_title": "Conditional Probability.mp3", "Sentence": "He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes. Fair coin, fair coin, and unfair coin. The next event, the coin is flipped."}, {"video_title": "Conditional Probability.mp3", "Sentence": "Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes. Fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves. Two representing heads and one representing tails."}, {"video_title": "Conditional Probability.mp3", "Sentence": "The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves. Two representing heads and one representing tails. Now the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six."}, {"video_title": "Conditional Probability.mp3", "Sentence": "Two representing heads and one representing tails. Now the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it."}, {"video_title": "Conditional Probability.mp3", "Sentence": "For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, Heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So the probability that he chose the biased coin, given heads occur?"}, {"video_title": "Conditional Probability.mp3", "Sentence": "For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, Heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem."}, {"video_title": "Conditional Probability.mp3", "Sentence": "So the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem. It tells us the probability of event A, given some new evidence B. Though if you forgot it, no worries. You need only know how to grow stories with trimmed trees."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "What I wanna talk about in this video, it's really about building even more intuition, is get a gut feeling for why this independence is important for making this claim. And to get that intuition, let's look at two random variables that are definitely random variables, but that are definitely not independent. So let's say, let's let X is equal to the number of hours that the next person you meet, so I'll say random person, random person slept yesterday. And let's say that Y is equal to the number of hours that same person, person was awake yesterday. And appreciate why these are not independent random variables. One of them is going to completely determine the other. If I slept eight hours yesterday, then I would have been awake for 16 hours."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "And let's say that Y is equal to the number of hours that same person, person was awake yesterday. And appreciate why these are not independent random variables. One of them is going to completely determine the other. If I slept eight hours yesterday, then I would have been awake for 16 hours. If I slept for 16 hours, then I would have been awake for eight hours. We know that X plus Y, even though they're random variables, and there could be variation in X, and there could be variation in Y, but for any given person, remember, these are still based on that same person, X plus Y is always going to be equal to 24 hours. So these are not independent, not independent."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "If I slept eight hours yesterday, then I would have been awake for 16 hours. If I slept for 16 hours, then I would have been awake for eight hours. We know that X plus Y, even though they're random variables, and there could be variation in X, and there could be variation in Y, but for any given person, remember, these are still based on that same person, X plus Y is always going to be equal to 24 hours. So these are not independent, not independent. If you're given one of the variables, it would completely determine what the other variable is. The probability of getting a certain value for one variable is going to be very different given what value you got for the other variable. So they're not independent at all."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "So these are not independent, not independent. If you're given one of the variables, it would completely determine what the other variable is. The probability of getting a certain value for one variable is going to be very different given what value you got for the other variable. So they're not independent at all. So in this situation, if someone said, let's just say, for the sake of argument, that the variance of X, the variance of X, is equal to, I don't know, let's say it's equal to four, and the units for variance would be squared hours, so four hours squared. We could say that the standard deviation for X in this case would be two hours. And let's say that the variance, let's say the standard deviation of Y is also equal to two hours."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "So they're not independent at all. So in this situation, if someone said, let's just say, for the sake of argument, that the variance of X, the variance of X, is equal to, I don't know, let's say it's equal to four, and the units for variance would be squared hours, so four hours squared. We could say that the standard deviation for X in this case would be two hours. And let's say that the variance, let's say the standard deviation of Y is also equal to two hours. And let's say that the variance of Y, variance of Y, well, it would be the square of the standard deviation, so it would be four hours, four hours squared would be our units. So if we just tried to blindly say, oh, I'm just gonna apply this little expression, this claim we had, without thinking about the independence, we would try to say, well, then the variance of X plus Y, the variance of X plus Y, must be equal to the sum of their variances. So it would be four plus four."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "And let's say that the variance, let's say the standard deviation of Y is also equal to two hours. And let's say that the variance of Y, variance of Y, well, it would be the square of the standard deviation, so it would be four hours, four hours squared would be our units. So if we just tried to blindly say, oh, I'm just gonna apply this little expression, this claim we had, without thinking about the independence, we would try to say, well, then the variance of X plus Y, the variance of X plus Y, must be equal to the sum of their variances. So it would be four plus four. So is it equal to eight hours squared? Well, that doesn't make any sense, because we know that a random variable that is equal to X plus Y, that this is always going to be 24 hours. In fact, it's not going to have any variation."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "So it would be four plus four. So is it equal to eight hours squared? Well, that doesn't make any sense, because we know that a random variable that is equal to X plus Y, that this is always going to be 24 hours. In fact, it's not going to have any variation. X plus Y is always gonna be 24 hours. So for these two random variables, because they are so connected, they are not independent at all, this is actually going to be zero. There is zero variance here."}, {"video_title": "Intuition for why independence matters for variance of sum   AP Statistics   Khan Academy.mp3", "Sentence": "In fact, it's not going to have any variation. X plus Y is always gonna be 24 hours. So for these two random variables, because they are so connected, they are not independent at all, this is actually going to be zero. There is zero variance here. X plus Y is always going to be 24, at least on Earth, where we have a 24-hour day. I guess if someone lived on another planet or something, then it could be slightly different. And we're assuming that we have an exactly 24-hour day on Earth."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "The two-way frequency table below shows data on type of vehicle driven, so this is type of vehicle driven, and whether there was an accident in the last year. So whether there was an accident in the last year for customers of all American auto insurance. Complete the following two-way table of column relative frequencies, so that's what they're talking here, this is a two-way table of column relative frequencies, if necessary, round your answers to the nearest hundred. So let's see what they're saying. They're saying, let's see, of the accidents within the last year, 28 were the people were driving an SUV, a sport utility vehicle, and 35 were in a sports car. Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So let's see what they're saying. They're saying, let's see, of the accidents within the last year, 28 were the people were driving an SUV, a sport utility vehicle, and 35 were in a sports car. Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year. And so what they want us to do is put those relative frequencies in here. So the way we could think about it, one right over here, this represents all of the sport utility vehicles."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year. And so what they want us to do is put those relative frequencies in here. So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97. And so in each of these, we want to put the relative frequency."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97. And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first. This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first. This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here. So one way to think about this is 22% of the sport utility vehicles had an accident within the last year, or you could say 0.22 of them."}, {"video_title": "Two-way relative frequency tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here. So one way to think about this is 22% of the sport utility vehicles had an accident within the last year, or you could say 0.22 of them. And you could say 78%, or 0.78, of the sport utility vehicles had no accidents. Likewise, you could say 25% of the sports cars had an accident within the last year, and 75% did not have an accident in the last year. So it allows you to think more in terms of the relative frequencies, the whole, the percentages, however you want to think about it, while this gives you the actual numbers."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "So he took a random sample of 24 games and recorded their outcomes. Here are his results. So out of the 24 games, he won four, lost 13, and tied seven times. He wants to use these results to carry out a chi-squared goodness-of-fit test to determine if the distribution of his outcomes disagrees with an even distribution. What are the values of the test statistic, the chi-squared test statistic, and p-value for Kenny's test? So pause this video and see if you can figure that out. Okay, so he's essentially just doing a hypothesis test using the chi-squared statistic because it's a hypothesis that's thinking about multiple categories."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "He wants to use these results to carry out a chi-squared goodness-of-fit test to determine if the distribution of his outcomes disagrees with an even distribution. What are the values of the test statistic, the chi-squared test statistic, and p-value for Kenny's test? So pause this video and see if you can figure that out. Okay, so he's essentially just doing a hypothesis test using the chi-squared statistic because it's a hypothesis that's thinking about multiple categories. So what would his null hypothesis be? Well, his null hypothesis would be that all of the outcomes are equal probability. Outcomes equal, equal probability."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "Okay, so he's essentially just doing a hypothesis test using the chi-squared statistic because it's a hypothesis that's thinking about multiple categories. So what would his null hypothesis be? Well, his null hypothesis would be that all of the outcomes are equal probability. Outcomes equal, equal probability. And then his alternative hypothesis would be that his outcomes have not equal, not equal probability. Remember, we assume that the null hypothesis is true. And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then we would reject our null hypothesis."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "Outcomes equal, equal probability. And then his alternative hypothesis would be that his outcomes have not equal, not equal probability. Remember, we assume that the null hypothesis is true. And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then we would reject our null hypothesis. Another way to think about it is if our p-value is below a threshold, we would reject our null hypothesis. And so what he did is he took a sample of 24 games, so n is equal to 24, and then this was the data that he got. Now, before we even calculate our chi-squared statistic and figure out what's the probability of getting a chi-squared statistic that large or greater, let's make sure we meet the conditions for inference for a chi-squared goodness-of-fit test."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then we would reject our null hypothesis. Another way to think about it is if our p-value is below a threshold, we would reject our null hypothesis. And so what he did is he took a sample of 24 games, so n is equal to 24, and then this was the data that he got. Now, before we even calculate our chi-squared statistic and figure out what's the probability of getting a chi-squared statistic that large or greater, let's make sure we meet the conditions for inference for a chi-squared goodness-of-fit test. So you've seen some of them, but some of them are a little bit different. One is the random condition. I'll write them up here."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "Now, before we even calculate our chi-squared statistic and figure out what's the probability of getting a chi-squared statistic that large or greater, let's make sure we meet the conditions for inference for a chi-squared goodness-of-fit test. So you've seen some of them, but some of them are a little bit different. One is the random condition. I'll write them up here. The random condition. And that would be that this is truly a random sample of games. And it tells us right here, he took a random sample of his 24 games, so we meet that condition."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "I'll write them up here. The random condition. And that would be that this is truly a random sample of games. And it tells us right here, he took a random sample of his 24 games, so we meet that condition. The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts condition. And this is an important one to appreciate. This is that the expected number of each category of outcomes is at least equal to five."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "And it tells us right here, he took a random sample of his 24 games, so we meet that condition. The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts condition. And this is an important one to appreciate. This is that the expected number of each category of outcomes is at least equal to five. Now, you might say, hey, wait, wait, I only got four wins, or Kenny only got four wins out of his sample of 24. But that does not violate the large counts condition. Remember, what is the expected number of wins, losses, and ties?"}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "This is that the expected number of each category of outcomes is at least equal to five. Now, you might say, hey, wait, wait, I only got four wins, or Kenny only got four wins out of his sample of 24. But that does not violate the large counts condition. Remember, what is the expected number of wins, losses, and ties? Well, if you were assuming the null hypothesis, where the outcomes have equal probability, so the expected, the expected, I could write right over here, it would be that it's 1 3rd, 1 3rd, 1 3rd. And so 1 3rd of 24 is eight, eight and eight. That's what Kenny would expect."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "Remember, what is the expected number of wins, losses, and ties? Well, if you were assuming the null hypothesis, where the outcomes have equal probability, so the expected, the expected, I could write right over here, it would be that it's 1 3rd, 1 3rd, 1 3rd. And so 1 3rd of 24 is eight, eight and eight. That's what Kenny would expect. And since, because all of these are at least equal to five, we meet the large counts condition. And then the last condition is the independence condition. If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "That's what Kenny would expect. And since, because all of these are at least equal to five, we meet the large counts condition. And then the last condition is the independence condition. If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population. And he can definitely play more than 240 games in his life, so we would assume that we meet that condition as well. And so with that out of the way, we can actually calculate our chi-squared statistic and try to make some inference based on it. And so, let's see, our chi-squared statistic is going to be equal to, so for each category, it's going to be the difference between the expected and what he got in that sample, squared divided by the expected."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population. And he can definitely play more than 240 games in his life, so we would assume that we meet that condition as well. And so with that out of the way, we can actually calculate our chi-squared statistic and try to make some inference based on it. And so, let's see, our chi-squared statistic is going to be equal to, so for each category, it's going to be the difference between the expected and what he got in that sample, squared divided by the expected. So the first category is wins. So that's going to be four minus eight, four minus eight squared over an expected number of wins of eight, plus losses, so that's 13 minus eight. 13 is how many he got, how many he lost, minus eight expected, squared over the number expected, plus he got seven ties, he would have expected eight squared, all of that over eight."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "And so, let's see, our chi-squared statistic is going to be equal to, so for each category, it's going to be the difference between the expected and what he got in that sample, squared divided by the expected. So the first category is wins. So that's going to be four minus eight, four minus eight squared over an expected number of wins of eight, plus losses, so that's 13 minus eight. 13 is how many he got, how many he lost, minus eight expected, squared over the number expected, plus he got seven ties, he would have expected eight squared, all of that over eight. And so let's see, what is this? Four minus eight is negative four. You square that, you get 16."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "13 is how many he got, how many he lost, minus eight expected, squared over the number expected, plus he got seven ties, he would have expected eight squared, all of that over eight. And so let's see, what is this? Four minus eight is negative four. You square that, you get 16. 13 minus eight is five. You square that, you get 25. Seven minus eight is negative one."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "You square that, you get 16. 13 minus eight is five. You square that, you get 25. Seven minus eight is negative one. Square that, you get one. And 16 divided by eight is going to be two. 25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "Seven minus eight is negative one. Square that, you get one. And 16 divided by eight is going to be two. 25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. And then 1 8 is 0.125, 0.125. You add these together, so let's see, it's gonna be two plus 3.125, 5.125, plus another.125, so that's going to be 5.25. So our chi-squared statistic is 5.25."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. And then 1 8 is 0.125, 0.125. You add these together, so let's see, it's gonna be two plus 3.125, 5.125, plus another.125, so that's going to be 5.25. So our chi-squared statistic is 5.25. And now to figure out our p-value, our p-value is going to be equal to the probability of getting a chi-squared statistic greater than or equal to 5.25. And you could use a chi-squared table for that. And we always have to think about our degrees of freedom."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "So our chi-squared statistic is 5.25. And now to figure out our p-value, our p-value is going to be equal to the probability of getting a chi-squared statistic greater than or equal to 5.25. And you could use a chi-squared table for that. And we always have to think about our degrees of freedom. We have one, two, three categories. So our degrees of freedom is going to be one less than that, or three minus one, which is two. So our degrees of freedom is going to be equal to two."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "And we always have to think about our degrees of freedom. We have one, two, three categories. So our degrees of freedom is going to be one less than that, or three minus one, which is two. So our degrees of freedom is going to be equal to two. And that makes sense, because you know for a certain number of games, if you know the number of wins, and you know the certain number of losses, you can figure out the number of ties. Or if you know any two of these categories, you can always figure out the third. So that's why you have two degrees of freedom."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "So our degrees of freedom is going to be equal to two. And that makes sense, because you know for a certain number of games, if you know the number of wins, and you know the certain number of losses, you can figure out the number of ties. Or if you know any two of these categories, you can always figure out the third. So that's why you have two degrees of freedom. And so let's get out our chi-squared table. So we have two degrees of freedom. So we are in this row."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "So that's why you have two degrees of freedom. And so let's get out our chi-squared table. So we have two degrees of freedom. So we are in this row. And where is 5.25? So 5.25 is right over there. And so our probability is going to be between 0.10 and 0.05."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "So we are in this row. And where is 5.25? So 5.25 is right over there. And so our probability is going to be between 0.10 and 0.05. So our p-value is going to be greater than 0.05 and less than 0.10. And so for example, if ahead of time, and he should have done this ahead of time, he set a significance level of 5%, and our p-value here is greater than 5%, which we just saw, he would fail to reject in this situation the null hypothesis. But they're not asking us that here."}, {"video_title": "Chi-square goodness-of-fit example   AP Statistics   Khan Academy.mp3", "Sentence": "And so our probability is going to be between 0.10 and 0.05. So our p-value is going to be greater than 0.05 and less than 0.10. And so for example, if ahead of time, and he should have done this ahead of time, he set a significance level of 5%, and our p-value here is greater than 5%, which we just saw, he would fail to reject in this situation the null hypothesis. But they're not asking us that here. All they're asking us is what is our chi-squared value and what range is our p-value in? Well, let's see, 5.25 are both of these values. And we saw we got a p-value between 5% and 10%."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "For the antibiotic to be sufficiently effective, it has to kill at least 90% of bacteria when applied to a harmful bacteria culture. She applied her antibiotic to a Petri dish full of harmful bacteria, waited for it to take effect, and then tried to estimate the percentage of dead bacteria in it. She took a random sample of 300 bacteria and found that 94% of them were dead. Then she calculated the margin of error and found that the true percentage of dead bacteria is most likely to be above 90%. So what's happening over here, she's trying to figure out what percentage of the total population of bacteria died. And maybe there's something about this bacteria, maybe just when you look at it from the naked eye, you can't tell whether the bacteria died or not. And so she decides to estimate the true percentage by sampling 300 individual bacterium, or I always forget the singular case, by sampling 300 bacteria."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then she calculated the margin of error and found that the true percentage of dead bacteria is most likely to be above 90%. So what's happening over here, she's trying to figure out what percentage of the total population of bacteria died. And maybe there's something about this bacteria, maybe just when you look at it from the naked eye, you can't tell whether the bacteria died or not. And so she decides to estimate the true percentage by sampling 300 individual bacterium, or I always forget the singular case, by sampling 300 bacteria. And then in her sample, she found that 94% of them were dead and then the margin of error tells us, because the margin of error says that it's unlikely, or that it's very likely that the true percentage is above 90%, that means that given that you sampled 300 bacteria, it's very unlikely that the true percentage is below 90%. So she could feel reasonably confident that in her Petri dish, more than 90% of the population did indeed die. Now let's answer these questions."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so she decides to estimate the true percentage by sampling 300 individual bacterium, or I always forget the singular case, by sampling 300 bacteria. And then in her sample, she found that 94% of them were dead and then the margin of error tells us, because the margin of error says that it's unlikely, or that it's very likely that the true percentage is above 90%, that means that given that you sampled 300 bacteria, it's very unlikely that the true percentage is below 90%. So she could feel reasonably confident that in her Petri dish, more than 90% of the population did indeed die. Now let's answer these questions. What type of statistical study did Alma use? Well, she used a, she's trying to estimate a parameter for a population, in this case, the parameter was the percentage of all of the bacteria that died, she couldn't observe that directly, so instead, she took a random sample of the bacteria in the Petri dish, and she used, she calculated a statistic for them, 94% of them were dead, and that's her estimate for the population parameter, the percentage of the population that died. So this is, when you're using a random sample to generate a statistic which estimates a parameter for a population, that's a sample study."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's answer these questions. What type of statistical study did Alma use? Well, she used a, she's trying to estimate a parameter for a population, in this case, the parameter was the percentage of all of the bacteria that died, she couldn't observe that directly, so instead, she took a random sample of the bacteria in the Petri dish, and she used, she calculated a statistic for them, 94% of them were dead, and that's her estimate for the population parameter, the percentage of the population that died. So this is, when you're using a random sample to generate a statistic which estimates a parameter for a population, that's a sample study. So she ran a sample study. Now the next question is, is the study appropriate for the statistical questions it's supposed to answer? So what was the question that she's trying to answer?"}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is, when you're using a random sample to generate a statistic which estimates a parameter for a population, that's a sample study. So she ran a sample study. Now the next question is, is the study appropriate for the statistical questions it's supposed to answer? So what was the question that she's trying to answer? Well, at least the way it's written, it seems like she's trying to answer whether or not her antibiotic works, whether it's an effective antibiotic, whether it's capable of killing bacteria. And you might be tempted to say, okay, well look, it looks like it killed, it killed more than 90% of the bacteria, or it very likely killed more than 90% of the bacteria, given the sample size and the margin error and all of that, but even if it is indeed the case that 95% of all of the bacteria died, it doesn't necessarily mean that it was caused by the antibiotic. Maybe it was caused by the plastic in the Petri dish."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what was the question that she's trying to answer? Well, at least the way it's written, it seems like she's trying to answer whether or not her antibiotic works, whether it's an effective antibiotic, whether it's capable of killing bacteria. And you might be tempted to say, okay, well look, it looks like it killed, it killed more than 90% of the bacteria, or it very likely killed more than 90% of the bacteria, given the sample size and the margin error and all of that, but even if it is indeed the case that 95% of all of the bacteria died, it doesn't necessarily mean that it was caused by the antibiotic. Maybe it was caused by the plastic in the Petri dish. Maybe the air in the Petri dish was too cold or went bad, or maybe it was handled in a weird way, or maybe that bacteria was just a bad culture and it somehow just spontaneously died on its own. She can't say with confidence that it was definitely the antibiotic. In order for her to make that statement, she would have to run a proper experiment."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe it was caused by the plastic in the Petri dish. Maybe the air in the Petri dish was too cold or went bad, or maybe it was handled in a weird way, or maybe that bacteria was just a bad culture and it somehow just spontaneously died on its own. She can't say with confidence that it was definitely the antibiotic. In order for her to make that statement, she would have to run a proper experiment. She would have to have a control and a treatment group where everything is equal except for the treatment group has the treatment. So if she had two Petri dishes that were kept in the same conditions with the same lighting, the same air, the same material that the bacteria is growing on, everything the same, except for the treatment group has the antibiotic applied to it, and then she saw that in the treatment group that most of the bacteria died while in the control group most of the bacteria didn't die, then she could say, okay, it looks like the antibiotic caused the bacteria to die, that there was actual causality here. So she would have had to run an experiment."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "In order for her to make that statement, she would have to run a proper experiment. She would have to have a control and a treatment group where everything is equal except for the treatment group has the treatment. So if she had two Petri dishes that were kept in the same conditions with the same lighting, the same air, the same material that the bacteria is growing on, everything the same, except for the treatment group has the antibiotic applied to it, and then she saw that in the treatment group that most of the bacteria died while in the control group most of the bacteria didn't die, then she could say, okay, it looks like the antibiotic caused the bacteria to die, that there was actual causality here. So she would have had to run an experiment. The most appropriate statistical study, or the most appropriate study would have been a proper controlled experiment where you have a control group where they don't have the antibiotic and a treatment group where they do have the antibiotic. So let's see what are the choices here where they say is the study appropriate. So yes, because she's in a proper study, no, I don't like that answer."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So she would have had to run an experiment. The most appropriate statistical study, or the most appropriate study would have been a proper controlled experiment where you have a control group where they don't have the antibiotic and a treatment group where they do have the antibiotic. So let's see what are the choices here where they say is the study appropriate. So yes, because she's in a proper study, no, I don't like that answer. No, because she can't know for certain that the true percentage of dead bacteria is above 90%. Well, I'm not gonna click on that because even if she knew for certain that the true percentage of dead bacteria were 95%, she can't feel confident that it was due to the antibiotic. Once again, it could be caused by the air conditioner, could have been caused by the Petri dish, could have been caused by the lighting in the room."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So yes, because she's in a proper study, no, I don't like that answer. No, because she can't know for certain that the true percentage of dead bacteria is above 90%. Well, I'm not gonna click on that because even if she knew for certain that the true percentage of dead bacteria were 95%, she can't feel confident that it was due to the antibiotic. Once again, it could be caused by the air conditioner, could have been caused by the Petri dish, could have been caused by the lighting in the room. So no, because the study didn't have a treatment and a control group. Yeah, I would go with that one right over there. Yes, because she found the antibiotic kills more than 90% of the harmful bacteria."}, {"video_title": "Appropriate statistical study example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, it could be caused by the air conditioner, could have been caused by the Petri dish, could have been caused by the lighting in the room. So no, because the study didn't have a treatment and a control group. Yeah, I would go with that one right over there. Yes, because she found the antibiotic kills more than 90% of the harmful bacteria. Once again, even if she knew for sure that more than 90% of the population had been killed, she doesn't know that it was caused by the antibiotic. It could have been caused by a whole bunch of things. If she had a control group that had the exact same conditions and the bacteria didn't die, then she could feel better that the bacteria death was due to the antibiotic."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "You're impatient. You want your frozen yogurt immediately. And so you decide to conduct a study. You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of the line."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of the line. So let me make two columns here. Line size is the left column. And on the right column, let's say this is the number of times observed."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "You observe the size of the line. So let me make two columns here. Line size is the left column. And on the right column, let's say this is the number of times observed. So times, times observed. Observed. Alright, times observed."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "And on the right column, let's say this is the number of times observed. So times, times observed. Observed. Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times. And you see two people in line, exactly eight times. And in your 50 visits, you don't see more than, you never see more than two people in line. I guess this is a very efficient cashier at this frozen yogurt store."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "You see one person in line, exactly 18 times. And you see two people in line, exactly eight times. And in your 50 visits, you don't see more than, you never see more than two people in line. I guess this is a very efficient cashier at this frozen yogurt store. So based on this, based on what you have observed, what would be your estimate of the probability of finding no people in line, one people in line, or two people in line, at 4 p.m. on the days after school that you visit the frozen yogurt store? And you'll say you only visited on weekdays where there are school days. So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "I guess this is a very efficient cashier at this frozen yogurt store. So based on this, based on what you have observed, what would be your estimate of the probability of finding no people in line, one people in line, or two people in line, at 4 p.m. on the days after school that you visit the frozen yogurt store? And you'll say you only visited on weekdays where there are school days. So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day? Well, all you can do is estimate the true probability, the true theoretical probability. We don't know what that is, but you've done 50 observations here, right? This is, and notice this adds up to 50."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day? Well, all you can do is estimate the true probability, the true theoretical probability. We don't know what that is, but you've done 50 observations here, right? This is, and notice this adds up to 50. 18 plus eight is 26, 26 plus 24 is 50. So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? What is the relative frequency of one person, or the relative frequency of two people in line?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "This is, and notice this adds up to 50. 18 plus eight is 26, 26 plus 24 is 50. So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? What is the relative frequency of one person, or the relative frequency of two people in line? And then we can use that as the estimates for the probability. So let's do that. So probability estimate."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "What is the relative frequency of one person, or the relative frequency of two people in line? And then we can use that as the estimates for the probability. So let's do that. So probability estimate. I'll do it in the next column. So probability, probability estimate. And once again, we can do that by looking at the relative frequency."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "So probability estimate. I'll do it in the next column. So probability, probability estimate. And once again, we can do that by looking at the relative frequency. The relative frequency of zero, well, we observe that 24 times out of 50, and so 24 out of 50 is the same thing as 0.48, or you could even say that this is 48%. Now, what's the relative frequency of seeing one person in line? Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "And once again, we can do that by looking at the relative frequency. The relative frequency of zero, well, we observe that 24 times out of 50, and so 24 out of 50 is the same thing as 0.48, or you could even say that this is 48%. Now, what's the relative frequency of seeing one person in line? Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits. And then finally, the relative frequency of seeing a two-person line, that was eight out of the 50 visits, and so that is 0.16, and that is equal to 16% of the visits. And so there's interesting things here. Remember, these are estimates of the probability."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits. And then finally, the relative frequency of seeing a two-person line, that was eight out of the 50 visits, and so that is 0.16, and that is equal to 16% of the visits. And so there's interesting things here. Remember, these are estimates of the probability. You're doing this by essentially sampling what the line on 50 different days. You don't know, it's not gonna always be exactly this, but it's a good estimate. You did it 50 times."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "Remember, these are estimates of the probability. You're doing this by essentially sampling what the line on 50 different days. You don't know, it's not gonna always be exactly this, but it's a good estimate. You did it 50 times. And so based on this, you'd say, well, I'd estimate the probability of having a zero-person line is 48%. I'd estimate that the probability of having a one-person line is 36%. I'd estimate that the probability of having a two-person line is 16%, or 0.16."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "You did it 50 times. And so based on this, you'd say, well, I'd estimate the probability of having a zero-person line is 48%. I'd estimate that the probability of having a one-person line is 36%. I'd estimate that the probability of having a two-person line is 16%, or 0.16. And it's important to realize that these are legitimate probabilities. Remember, to be a probability, it has to be between zero and one. It has to be zero and one."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "I'd estimate that the probability of having a two-person line is 16%, or 0.16. And it's important to realize that these are legitimate probabilities. Remember, to be a probability, it has to be between zero and one. It has to be zero and one. And if you look at all of the possible events, it should add up to one, because at least based on your observations, these are the possibilities. Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. Let's see, 36 plus 16 is 52."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "It has to be zero and one. And if you look at all of the possible events, it should add up to one, because at least based on your observations, these are the possibilities. Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. Let's see, 36 plus 16 is 52. 52 plus 48, they add up to one. Now, once you do this, you might do something interesting. You might say, okay, you know what?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "Let's see, 36 plus 16 is 52. 52 plus 48, they add up to one. Now, once you do this, you might do something interesting. You might say, okay, you know what? Over the next two years, you plan on visiting 500 times. So visiting 500 times. So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "You might say, okay, you know what? Over the next two years, you plan on visiting 500 times. So visiting 500 times. So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far? Well, it's reasonable to say, well, a good estimate of the number of times you'll see a two-person line when you visit 500 times. Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. And so what is this going to be?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far? Well, it's reasonable to say, well, a good estimate of the number of times you'll see a two-person line when you visit 500 times. Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. And so what is this going to be? Let's see, 500 divided by 50 is just 10. So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. Now, to be clear, I would be shocked if it's exactly 80 ends up being the case, but this is actually a very good expectation, based on your observations."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy.mp3", "Sentence": "And so what is this going to be? Let's see, 500 divided by 50 is just 10. So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. Now, to be clear, I would be shocked if it's exactly 80 ends up being the case, but this is actually a very good expectation, based on your observations. It is completely possible, first of all, that your observations were off, that this is just a random chance that you happened to observe this many, or this few times that there were two people in line. So that could be off, but even if these are very good estimates, it's possible that something, that you see a two-person line 85 out of the 500 times, or 65 out of the 500 times. All of those things are possible."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "Nutritionists measured the sugar content in grams for 32 drinks at Starbucks. A cumulative relative frequency graph, let me underline that, a cumulative relative frequency graph for the data is shown below. So they have different, on the horizontal axis, different amounts of sugar in grams, and then we have the cumulative relative frequency. So let's just make sure we understand how to read this. This is saying that zero, or 0%, of the drinks have a sugar content, have no sugar content. This right over here, this data point, this looks like it's at the.5 grams, and then this looks like it's at 0.1. This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just make sure we understand how to read this. This is saying that zero, or 0%, of the drinks have a sugar content, have no sugar content. This right over here, this data point, this looks like it's at the.5 grams, and then this looks like it's at 0.1. This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less. This data point tells us that 100% of the drinks at Starbucks has 50 grams of sugar or less. The cumulative relative frequency, that's why we, for each of these points, we say this is the frequency that has that much sugar or less, and that's why it just keeps on increasing and increasing. As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less. This data point tells us that 100% of the drinks at Starbucks has 50 grams of sugar or less. The cumulative relative frequency, that's why we, for each of these points, we say this is the frequency that has that much sugar or less, and that's why it just keeps on increasing and increasing. As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less. So let's read the first question. An iced coffee has 15 grams of sugar. Estimate the percentile of this drink to the nearest whole percent."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less. So let's read the first question. An iced coffee has 15 grams of sugar. Estimate the percentile of this drink to the nearest whole percent. So iced coffee has 15 grams of sugar, which would be right over here. And so let's estimate the percentile. So we can see they actually have a data point right over here."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "Estimate the percentile of this drink to the nearest whole percent. So iced coffee has 15 grams of sugar, which would be right over here. And so let's estimate the percentile. So we can see they actually have a data point right over here. And we can see that 20%, or 0.2, 20% of the drinks that Starbucks offers has 15 grams of sugar or less. So the percentile of this drink, if I were to estimate it, it looks like it's the relative frequency, 0.2, has that much sugar or less, and so this percentile would be 20%. Once again, another way to think about it, to read this, you could convert these to percentages."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So we can see they actually have a data point right over here. And we can see that 20%, or 0.2, 20% of the drinks that Starbucks offers has 15 grams of sugar or less. So the percentile of this drink, if I were to estimate it, it looks like it's the relative frequency, 0.2, has that much sugar or less, and so this percentile would be 20%. Once again, another way to think about it, to read this, you could convert these to percentages. You could say that 20% has this much sugar or less, 15 grams of sugar or less, so an iced coffee is in the 20th percentile. Let's do another question. So here we are asked to estimate the median of the distribution of drinks."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, another way to think about it, to read this, you could convert these to percentages. You could say that 20% has this much sugar or less, 15 grams of sugar or less, so an iced coffee is in the 20th percentile. Let's do another question. So here we are asked to estimate the median of the distribution of drinks. Hint, think about the 50th percentile. So the median, if you were to line up all of the drinks, you would take the middle drink. And so you could view that as, well, what drink is exactly at the 50th percentile?"}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So here we are asked to estimate the median of the distribution of drinks. Hint, think about the 50th percentile. So the median, if you were to line up all of the drinks, you would take the middle drink. And so you could view that as, well, what drink is exactly at the 50th percentile? So now let's look at the 50th percentile would be a cumulative relative frequency of 0.5, which would be right over here on our vertical axis. Another way to think about it is 0.5, or 50% of the drinks are going, if we go to this point right over here, what has a cumulative relative frequency of 0.5? We see that we are right at, looks like this is 25 grams."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "And so you could view that as, well, what drink is exactly at the 50th percentile? So now let's look at the 50th percentile would be a cumulative relative frequency of 0.5, which would be right over here on our vertical axis. Another way to think about it is 0.5, or 50% of the drinks are going, if we go to this point right over here, what has a cumulative relative frequency of 0.5? We see that we are right at, looks like this is 25 grams. So one way to interpret this is 50% of the drinks have less than, or have 25 grams of sugar or less. So this looks like a pretty good estimate for the median, for the middle data point. So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "We see that we are right at, looks like this is 25 grams. So one way to interpret this is 50% of the drinks have less than, or have 25 grams of sugar or less. So this looks like a pretty good estimate for the median, for the middle data point. So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar. Let's do one more based on the same data set. So here we're asked, what is the best estimate for the interquartile range of the distribution of drinks? So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile?"}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar. Let's do one more based on the same data set. So here we're asked, what is the best estimate for the interquartile range of the distribution of drinks? So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile? And we wanna think about what's at the 75th percentile. And then we want to take the difference. That's what the interquartile range is."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile? And we wanna think about what's at the 75th percentile. And then we want to take the difference. That's what the interquartile range is. So let's do that. So first, the 25th percentile, we'd wanna look at the cumulative relative frequency. So 25th, this would be 30th, so 25th would be right around here."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "That's what the interquartile range is. So let's do that. So first, the 25th percentile, we'd wanna look at the cumulative relative frequency. So 25th, this would be 30th, so 25th would be right around here. And so it looks like the 25th percentile is, that looks like about, I don't know. And we're estimating here, so that looks like it's about, this would be 15, looks like I would say maybe 18 grams. So approximately 18 grams."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So 25th, this would be 30th, so 25th would be right around here. And so it looks like the 25th percentile is, that looks like about, I don't know. And we're estimating here, so that looks like it's about, this would be 15, looks like I would say maybe 18 grams. So approximately 18 grams. Once again, one way to think about it is, 25% of the drinks have 18 grams of sugar or less. And let's look at the 75th percentile. So this is 70th, 75th would be right over there."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So approximately 18 grams. Once again, one way to think about it is, 25% of the drinks have 18 grams of sugar or less. And let's look at the 75th percentile. So this is 70th, 75th would be right over there. Actually, I can draw a straighter line than that. I have a line tool here. So 75th percentile would put me right over there."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So this is 70th, 75th would be right over there. Actually, I can draw a straighter line than that. I have a line tool here. So 75th percentile would put me right over there. I don't know, that looks like, well, I'll go with 39 grams, roughly 39 grams. And so what's the difference between these two? Well, the difference between these two, it looks like it's about 21 grams."}, {"video_title": "Analyzing a cumulative relative frequency graph   AP Statistics   Khan Academy.mp3", "Sentence": "So 75th percentile would put me right over there. I don't know, that looks like, well, I'll go with 39 grams, roughly 39 grams. And so what's the difference between these two? Well, the difference between these two, it looks like it's about 21 grams. So our interquartile range, our estimate of our interquartile range, looking at this cumulative relative frequency distribution, because we're saying, hey, look, it looks like the 25th percentile, it looks like 25% of the drinks have 18 grams or less. 75% of the drinks have 39 grams or less. If we take the difference between these two quartiles, this is the first quartile, this is our third quartile, we're gonna get 21 grams."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "So this first one, Roy's Toys received a shipment of 100,000 rubber duckies from the factory. The factory couldn't promise that all rubber duckies are in perfect form, but they promised that the percentage of defective toys won't exceed 5%. Let me underline that. They promised that the percentage of defective toys won't exceed 5%. Roy wanted to get an estimation of the percentage of defective toys, and since he couldn't go over the entire 100,000 duckies, he took a random sample of 10 duckies. He found that 10% of them were defective. So what's going on here?"}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "They promised that the percentage of defective toys won't exceed 5%. Roy wanted to get an estimation of the percentage of defective toys, and since he couldn't go over the entire 100,000 duckies, he took a random sample of 10 duckies. He found that 10% of them were defective. So what's going on here? Roy gets a shipment. There's 100,000 ducks in the shipment. He wants to figure out what percentage of them are defective."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "So what's going on here? Roy gets a shipment. There's 100,000 ducks in the shipment. He wants to figure out what percentage of them are defective. He can't look at all 100,000 ducks. It's not practical. So he samples 10 of them."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "He wants to figure out what percentage of them are defective. He can't look at all 100,000 ducks. It's not practical. So he samples 10 of them. One, two, three, four, five, six, seven, eight, nine, 10, and he finds that one out of those 10 are defective, 10% of the 10. So first of all, this is clearly a sample study. This is a sample study."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "So he samples 10 of them. One, two, three, four, five, six, seven, eight, nine, 10, and he finds that one out of those 10 are defective, 10% of the 10. So first of all, this is clearly a sample study. This is a sample study. How do we know that? Well, he is taking a sample from a broader population in order to estimate a parameter. The parameter is the percentage of those 100,000 duckies that are actually defective."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "This is a sample study. How do we know that? Well, he is taking a sample from a broader population in order to estimate a parameter. The parameter is the percentage of those 100,000 duckies that are actually defective. Now, the next question is, is what kind of a conclusion can you make? Roy, since he got the shipment and he took a sample and he found that 10% of the sample was defective, he might be all up in arms and say, oh, this toy shipment from the factory, they violated this promise that the percentage of defective toys won't exceed 5% because I sampled 10 toys and 10% of those 10 toys were defective. Well, that isn't a reasonable conclusion because this is a small sample."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "The parameter is the percentage of those 100,000 duckies that are actually defective. Now, the next question is, is what kind of a conclusion can you make? Roy, since he got the shipment and he took a sample and he found that 10% of the sample was defective, he might be all up in arms and say, oh, this toy shipment from the factory, they violated this promise that the percentage of defective toys won't exceed 5% because I sampled 10 toys and 10% of those 10 toys were defective. Well, that isn't a reasonable conclusion because this is a small sample. This is a small sample. Think about it. He could have sampled five duckies and if he just happened to get one of the defective ones, he would have said, oh, maybe 20% are defective."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "Well, that isn't a reasonable conclusion because this is a small sample. This is a small sample. Think about it. He could have sampled five duckies and if he just happened to get one of the defective ones, he would have said, oh, maybe 20% are defective. What he's really gotta do is sample, take a larger sample. And once again, whenever you're sampling, there's always a probability that your estimate is going to be not close or definitely not the same as the parameter for the population. But the larger your sample, the higher probability that your estimate is close to the actual parameter for the population."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "He could have sampled five duckies and if he just happened to get one of the defective ones, he would have said, oh, maybe 20% are defective. What he's really gotta do is sample, take a larger sample. And once again, whenever you're sampling, there's always a probability that your estimate is going to be not close or definitely not the same as the parameter for the population. But the larger your sample, the higher probability that your estimate is close to the actual parameter for the population. And 10 in this is just too low. In future videos, we'll talk about how you can estimate the probability or how you can figure out whether your sample seems sufficient. But for this one, for what Roy did, I don't think 10 duckies is enough."}, {"video_title": "Worked example identifying sample study.mp3", "Sentence": "But the larger your sample, the higher probability that your estimate is close to the actual parameter for the population. And 10 in this is just too low. In future videos, we'll talk about how you can estimate the probability or how you can figure out whether your sample seems sufficient. But for this one, for what Roy did, I don't think 10 duckies is enough. If he sampled maybe 100 duckies or more than that and he found that 10% of them were defective, well, that seems less likely to happen just purely due to chance. Let's do a few more of these. Actually, I'll do those in the next videos."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "They're closed on Sunday. So this is 100% of their customers for a week. If you add that up, you get 100%. I obviously am a little bit suspicious, so I decide to see how good this distribution that he's describing actually fits observed data. So I actually observe the number of customers when they come in during the week, and this is what I get for my observed data. So to figure out whether I want to accept or reject his hypothesis right here, I'm going to do a little bit of a hypothesis test. So I'll make the null hypothesis that the owner's distribution, so that's this thing right here, is correct."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "I obviously am a little bit suspicious, so I decide to see how good this distribution that he's describing actually fits observed data. So I actually observe the number of customers when they come in during the week, and this is what I get for my observed data. So to figure out whether I want to accept or reject his hypothesis right here, I'm going to do a little bit of a hypothesis test. So I'll make the null hypothesis that the owner's distribution, so that's this thing right here, is correct. And then the alternative hypothesis is going to be that it is not correct, that it is not a correct distribution, that I should not feel reasonably okay relying on this. It's not correct. I should reject the owner's distribution."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll make the null hypothesis that the owner's distribution, so that's this thing right here, is correct. And then the alternative hypothesis is going to be that it is not correct, that it is not a correct distribution, that I should not feel reasonably okay relying on this. It's not correct. I should reject the owner's distribution. And I want to do this with a significance level of 5%. Or another way of thinking about it, I'm going to calculate a statistic based on this data right here. And it's going to be a chi-square statistic, or another way to view it is that that statistic that I'm going to calculate has approximately a chi-square distribution."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "I should reject the owner's distribution. And I want to do this with a significance level of 5%. Or another way of thinking about it, I'm going to calculate a statistic based on this data right here. And it's going to be a chi-square statistic, or another way to view it is that that statistic that I'm going to calculate has approximately a chi-square distribution. And given that it does have a chi-square distribution with a certain number of degrees of freedom, and we're going to calculate that, what I want to see is the probability of getting this result, or getting a result like this or a result more extreme, less than 5%. If the probability of getting a result like this or something less likely than this is less than 5%, then I'm going to reject the null hypothesis, which is essentially just rejecting the owner's distribution. If I don't get that, if I say, hey, the probability of getting a chi-square statistic that is this extreme or more is greater than my alpha, than my significance level, then I'm not going to reject it."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's going to be a chi-square statistic, or another way to view it is that that statistic that I'm going to calculate has approximately a chi-square distribution. And given that it does have a chi-square distribution with a certain number of degrees of freedom, and we're going to calculate that, what I want to see is the probability of getting this result, or getting a result like this or a result more extreme, less than 5%. If the probability of getting a result like this or something less likely than this is less than 5%, then I'm going to reject the null hypothesis, which is essentially just rejecting the owner's distribution. If I don't get that, if I say, hey, the probability of getting a chi-square statistic that is this extreme or more is greater than my alpha, than my significance level, then I'm not going to reject it. I'm going to say, well, I have no reason to really assume that he's lying. So let's do that. So to calculate the chi-square statistic, what I'm going to do is, so here we're assuming the owner's distribution is correct."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I don't get that, if I say, hey, the probability of getting a chi-square statistic that is this extreme or more is greater than my alpha, than my significance level, then I'm not going to reject it. I'm going to say, well, I have no reason to really assume that he's lying. So let's do that. So to calculate the chi-square statistic, what I'm going to do is, so here we're assuming the owner's distribution is correct. So assuming the owner's distribution was correct, what would have been the expected observed? So we have the expected percentage here, but what would have been the expected observed? So let me write this right here, expected."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So to calculate the chi-square statistic, what I'm going to do is, so here we're assuming the owner's distribution is correct. So assuming the owner's distribution was correct, what would have been the expected observed? So we have the expected percentage here, but what would have been the expected observed? So let me write this right here, expected. I'll add another row. Expected. So we would have expected 10% of the total customers in that week to come in on Monday, 10% of the total customers of that week to come in on Tuesday, 15% to come in on Wednesday."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write this right here, expected. I'll add another row. Expected. So we would have expected 10% of the total customers in that week to come in on Monday, 10% of the total customers of that week to come in on Tuesday, 15% to come in on Wednesday. Now to figure out what that actual number is, we need to figure out the total number of customers. So let's add up these numbers right here. So we have, let me get the calculator out."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we would have expected 10% of the total customers in that week to come in on Monday, 10% of the total customers of that week to come in on Tuesday, 15% to come in on Wednesday. Now to figure out what that actual number is, we need to figure out the total number of customers. So let's add up these numbers right here. So we have, let me get the calculator out. So we have 30 plus 14 plus 34 plus 45 plus 57 plus 20. So there's a total of 200 customers who came into the restaurant that week. So let me write this down."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have, let me get the calculator out. So we have 30 plus 14 plus 34 plus 45 plus 57 plus 20. So there's a total of 200 customers who came into the restaurant that week. So let me write this down. So this is equal to, so I wrote the total over here, total. Ignore this right here. I had 200 customers come in for the week."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write this down. So this is equal to, so I wrote the total over here, total. Ignore this right here. I had 200 customers come in for the week. So what was the expected number on Monday? Well, on Monday we would have expected 10% of the 200 to come in. So this would have been 20 customers, 10% times 200."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "I had 200 customers come in for the week. So what was the expected number on Monday? Well, on Monday we would have expected 10% of the 200 to come in. So this would have been 20 customers, 10% times 200. On Tuesday, another 10%. So we would have expected 20 customers. Wednesday, 15% of 200."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this would have been 20 customers, 10% times 200. On Tuesday, another 10%. So we would have expected 20 customers. Wednesday, 15% of 200. That's 30 customers. On Thursday, we would have expected 20% of 200 customers. So that would have been 40 customers."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Wednesday, 15% of 200. That's 30 customers. On Thursday, we would have expected 20% of 200 customers. So that would have been 40 customers. Then on Friday, 30%. That would have been 60 customers. And then on Friday, 15% again."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that would have been 40 customers. Then on Friday, 30%. That would have been 60 customers. And then on Friday, 15% again. So 15% of 200 would have been 30 customers. So if this distribution is correct, this is the actual number that I would have expected. Now, to calculate our chi-squared statistic, we essentially just take, let me just show it to you."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then on Friday, 15% again. So 15% of 200 would have been 30 customers. So if this distribution is correct, this is the actual number that I would have expected. Now, to calculate our chi-squared statistic, we essentially just take, let me just show it to you. And I'll write it, instead of writing chi, I'm going to write a capital X squared. Sometimes someone will write the actual Greek letter chi here. But I'll write the X squared here so that it will be, and let me write it this way."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, to calculate our chi-squared statistic, we essentially just take, let me just show it to you. And I'll write it, instead of writing chi, I'm going to write a capital X squared. Sometimes someone will write the actual Greek letter chi here. But I'll write the X squared here so that it will be, and let me write it this way. This is our chi-squared statistic. But I'm going to write it with a capital X instead of a chi because this is going to have approximately a chi-squared distribution. I can't assume that it's exactly."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I'll write the X squared here so that it will be, and let me write it this way. This is our chi-squared statistic. But I'm going to write it with a capital X instead of a chi because this is going to have approximately a chi-squared distribution. I can't assume that it's exactly. So we're dealing with approximations right here. But it's fairly straightforward to calculate. We take, for each of the days, we take the difference between the observed and the expected."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "I can't assume that it's exactly. So we're dealing with approximations right here. But it's fairly straightforward to calculate. We take, for each of the days, we take the difference between the observed and the expected. So it's going to be 30 minus 20. I'll do the first one color-coded. Squared divided by the expected."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We take, for each of the days, we take the difference between the observed and the expected. So it's going to be 30 minus 20. I'll do the first one color-coded. Squared divided by the expected. So we're essentially taking the square of, almost you could kind of view it, the error between what we observed and expected, or the difference between what we observed and expected. And we're kind of normalizing it by the expected right over here. But we want to take the sum of all of these."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Squared divided by the expected. So we're essentially taking the square of, almost you could kind of view it, the error between what we observed and expected, or the difference between what we observed and expected. And we're kind of normalizing it by the expected right over here. But we want to take the sum of all of these. I'll just do all of those in yellow. Plus 14 minus 20 squared over 20. Plus 34 minus 30 squared over 30."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we want to take the sum of all of these. I'll just do all of those in yellow. Plus 14 minus 20 squared over 20. Plus 34 minus 30 squared over 30. Plus, I'll continue over here, 45 minus 40 squared over 40. Plus 57 minus 60 squared over 60. And then finally, 20 minus 30 squared over 30."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 34 minus 30 squared over 30. Plus, I'll continue over here, 45 minus 40 squared over 40. Plus 57 minus 60 squared over 60. And then finally, 20 minus 30 squared over 30. I just took the observed minus the expected squared over the expected. I took the sum of it. And this is what gives us our chi-squared statistic."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, 20 minus 30 squared over 30. I just took the observed minus the expected squared over the expected. I took the sum of it. And this is what gives us our chi-squared statistic. Now let's just calculate what this number is going to be. So this is going to be equal to, I'll do it over here so we don't run out of space. Let me do this in a new color."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is what gives us our chi-squared statistic. Now let's just calculate what this number is going to be. So this is going to be equal to, I'll do it over here so we don't run out of space. Let me do this in a new color. I'll do it in the orange. This is going to be equal to, this is what? 30 minus 20 is 10 squared, which is 100 divided by 20, which is 5."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do this in a new color. I'll do it in the orange. This is going to be equal to, this is what? 30 minus 20 is 10 squared, which is 100 divided by 20, which is 5. I might not be able to do all of them in my head like this. Plus, actually let me just write it this way, just so you see what I'm doing. This is going to be 100, this right here is 100 over 20."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "30 minus 20 is 10 squared, which is 100 divided by 20, which is 5. I might not be able to do all of them in my head like this. Plus, actually let me just write it this way, just so you see what I'm doing. This is going to be 100, this right here is 100 over 20. Plus 14 minus 20 is negative 6 squared is positive 36. So plus 36 over 20. Plus 34 minus 30 is 4 squared is 16."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be 100, this right here is 100 over 20. Plus 14 minus 20 is negative 6 squared is positive 36. So plus 36 over 20. Plus 34 minus 30 is 4 squared is 16. So plus 16 over 30. Plus 45 minus 40 is 5 squared is 25. So plus 25 over 40."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 34 minus 30 is 4 squared is 16. So plus 16 over 30. Plus 45 minus 40 is 5 squared is 25. So plus 25 over 40. Plus, the difference here is 3 squared is 9. So it's 9 over 60. Plus, we have a difference of 10 squared is 100 over 30."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So plus 25 over 40. Plus, the difference here is 3 squared is 9. So it's 9 over 60. Plus, we have a difference of 10 squared is 100 over 30. Plus 100 over 30. And this is equal to, and I'll just get the calculator out for this. This is equal to, this is equal to, we have 100 divided by 20."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus, we have a difference of 10 squared is 100 over 30. Plus 100 over 30. And this is equal to, and I'll just get the calculator out for this. This is equal to, this is equal to, we have 100 divided by 20. Plus 36 divided by 20. Plus 16 divided by 30. Plus 25 divided by 40."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is equal to, this is equal to, we have 100 divided by 20. Plus 36 divided by 20. Plus 16 divided by 30. Plus 25 divided by 40. Plus 9 divided by 60. Plus 100 divided by 30. Gives us 11.44."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 25 divided by 40. Plus 9 divided by 60. Plus 100 divided by 30. Gives us 11.44. So let me write that down. So this right here is going to be 11.44. This is my chi-square statistic, or we could call it a big capital X squared."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Gives us 11.44. So let me write that down. So this right here is going to be 11.44. This is my chi-square statistic, or we could call it a big capital X squared. Sometimes you'll have it written as a chi-square. But this is approximately, this statistic is going to have approximately a chi-square distribution. Anyway, with that said, let's figure out, if we assume that it has roughly a chi-square distribution, what is the probability of getting a result this extreme, or at least this extreme, I guess is another way of thinking about it."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is my chi-square statistic, or we could call it a big capital X squared. Sometimes you'll have it written as a chi-square. But this is approximately, this statistic is going to have approximately a chi-square distribution. Anyway, with that said, let's figure out, if we assume that it has roughly a chi-square distribution, what is the probability of getting a result this extreme, or at least this extreme, I guess is another way of thinking about it. Or, another way of saying, is this a more extreme result than the critical chi-square value that there's a 5% chance of getting a result that extreme. So let's do it that way. Let's figure out the critical chi-square value, and if this is more extreme than that, then we will reject our null hypothesis."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Anyway, with that said, let's figure out, if we assume that it has roughly a chi-square distribution, what is the probability of getting a result this extreme, or at least this extreme, I guess is another way of thinking about it. Or, another way of saying, is this a more extreme result than the critical chi-square value that there's a 5% chance of getting a result that extreme. So let's do it that way. Let's figure out the critical chi-square value, and if this is more extreme than that, then we will reject our null hypothesis. So let's figure out our critical chi-square value. So we have an alpha 5%. And actually, the other thing we have to figure out is the degrees of freedom."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's figure out the critical chi-square value, and if this is more extreme than that, then we will reject our null hypothesis. So let's figure out our critical chi-square value. So we have an alpha 5%. And actually, the other thing we have to figure out is the degrees of freedom. The degrees of freedom here, we're taking 1, 2, 3, 4, 5, 6 sums. So you might be tempted to say the degrees of freedom are 6. But one thing to realize is that if you had all of this information over here, you could actually figure out this last piece of information."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, the other thing we have to figure out is the degrees of freedom. The degrees of freedom here, we're taking 1, 2, 3, 4, 5, 6 sums. So you might be tempted to say the degrees of freedom are 6. But one thing to realize is that if you had all of this information over here, you could actually figure out this last piece of information. So you actually have 5 degrees of freedom. When you have just kind of n data points like this, and you're measuring kind of the observed versus the expected, your degrees of freedom are going to be n minus 1, because you could figure out that nth data point just based on everything else that you have, all of the other information. So our degrees of freedom here are going to be 5."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But one thing to realize is that if you had all of this information over here, you could actually figure out this last piece of information. So you actually have 5 degrees of freedom. When you have just kind of n data points like this, and you're measuring kind of the observed versus the expected, your degrees of freedom are going to be n minus 1, because you could figure out that nth data point just based on everything else that you have, all of the other information. So our degrees of freedom here are going to be 5. It's n minus 1. Our significance level is 5%. And our degrees of freedom is also going to be equal to 5."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our degrees of freedom here are going to be 5. It's n minus 1. Our significance level is 5%. And our degrees of freedom is also going to be equal to 5. So let's look at our chi-square distribution. We have a degree of freedom of 5. We have a significance level of 5%."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And our degrees of freedom is also going to be equal to 5. So let's look at our chi-square distribution. We have a degree of freedom of 5. We have a significance level of 5%. We have a significance level of 5%. And so the critical chi-square value is 11.07. So let's go to this chart."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have a significance level of 5%. We have a significance level of 5%. And so the critical chi-square value is 11.07. So let's go to this chart. So we have a chi-square distribution with a degree of freedom of 5. So that's this distribution over here in magenta. And we care about a critical value of 11.07."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's go to this chart. So we have a chi-square distribution with a degree of freedom of 5. So that's this distribution over here in magenta. And we care about a critical value of 11.07. So this is right here. You actually even can't see it on this. So if I were to keep drawing this magenta thing all the way over here, if the magenta line just kept the distribution, just kept going over here, you'd have 8."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we care about a critical value of 11.07. So this is right here. You actually even can't see it on this. So if I were to keep drawing this magenta thing all the way over here, if the magenta line just kept the distribution, just kept going over here, you'd have 8. Over here you'd have 10. Over here you'd have 12. 11.07 is maybe someplace right over there."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I were to keep drawing this magenta thing all the way over here, if the magenta line just kept the distribution, just kept going over here, you'd have 8. Over here you'd have 10. Over here you'd have 12. 11.07 is maybe someplace right over there. So what it's saying is the probability of getting a result at least as extreme as 11.07 is 5%. Our result, so our critical chi-square value, so we could write even here, our critical chi-square value is equal to, we just saw, 11.07. Let me look at the chart again."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "11.07 is maybe someplace right over there. So what it's saying is the probability of getting a result at least as extreme as 11.07 is 5%. Our result, so our critical chi-square value, so we could write even here, our critical chi-square value is equal to, we just saw, 11.07. Let me look at the chart again. 11.07 is equal to 11.07. The result we got for our statistic is even less likely than that. It's even less likely than that."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me look at the chart again. 11.07 is equal to 11.07. The result we got for our statistic is even less likely than that. It's even less likely than that. The probability is less than our significance level. So then we are going to reject. So the probability of getting that is, let me put it this way, 11.44 is more extreme than our critical chi-square level."}, {"video_title": "Pearson's chi square test (goodness of fit)   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's even less likely than that. The probability is less than our significance level. So then we are going to reject. So the probability of getting that is, let me put it this way, 11.44 is more extreme than our critical chi-square level. So it's very unlikely that this distribution is true. So we will reject what he's telling us. We will reject this distribution."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "She wants to test whether this holds true for teachers in her state. So she is going to take a random sample of these teachers and see what percent of them are members of a union. Let P represent the proportion of teachers in her state that are members of a union. Write an appropriate set of hypotheses for her significance test. So pause this video and see if you can do that. All right, now let's do it together. So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Write an appropriate set of hypotheses for her significance test. So pause this video and see if you can do that. All right, now let's do it together. So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that, hey, there's no news here. It's what you would expect it to be. And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that, hey, there's no news here. It's what you would expect it to be. And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union. So that percentage, that proportion is P. So this would be the null hypothesis, that the proportion in her state is also 49%. And now what would the alternative be? Well, the alternative is that the proportion in her state is not 49%."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union. So that percentage, that proportion is P. So this would be the null hypothesis, that the proportion in her state is also 49%. And now what would the alternative be? Well, the alternative is that the proportion in her state is not 49%. This is the thing that, hey, there would be news here. There'd be something interesting to report. There's something different about her state."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the alternative is that the proportion in her state is not 49%. This is the thing that, hey, there would be news here. There'd be something interesting to report. There's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample proportion, figure out the probability of getting that sample proportion if we were to assume that the null hypothesis is true. If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "There's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample proportion, figure out the probability of getting that sample proportion if we were to assume that the null hypothesis is true. If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative. Let's do another example here. According to a very large poll in 2015, about 90% of homes in California had access to the internet. Market researchers want to test if that proportion is now higher."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative. Let's do another example here. According to a very large poll in 2015, about 90% of homes in California had access to the internet. Market researchers want to test if that proportion is now higher. So they take a random sample of 1,000 homes in California and find that 920, or 92% of homes sampled, have access to the internet. Let P represent the proportion of homes in California that have access to the internet. Write an appropriate set of hypotheses for their significance test."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Market researchers want to test if that proportion is now higher. So they take a random sample of 1,000 homes in California and find that 920, or 92% of homes sampled, have access to the internet. Let P represent the proportion of homes in California that have access to the internet. Write an appropriate set of hypotheses for their significance test. So once again, pause this video and see if you can figure it out. So once again, we want to have a null hypothesis and we want to have an alternative hypothesis. The null hypothesis is the, hey, there's no news here."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Write an appropriate set of hypotheses for their significance test. So once again, pause this video and see if you can figure it out. So once again, we want to have a null hypothesis and we want to have an alternative hypothesis. The null hypothesis is the, hey, there's no news here. And so that would say that, you know, it's kind of the status quo, that the proportion of people who have internet is still the same as the last study, is still the same at 90%. Or I could write 90%, or I could write 0.9 right over here. Now some of you might have been tempted to put 92% there."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "The null hypothesis is the, hey, there's no news here. And so that would say that, you know, it's kind of the status quo, that the proportion of people who have internet is still the same as the last study, is still the same at 90%. Or I could write 90%, or I could write 0.9 right over here. Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in California that now have the internet?"}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access to the internet."}, {"video_title": "Constructing hypotheses for a significance test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access to the internet. So this right over here is my sample proportion. And then they're gonna figure out, well, what's the probability of getting this sample proportion for this sample size if we were to assume that the null hypothesis is true? If this probability of getting this is below a threshold, it's below alpha, below our significance level, then we'll reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, one driver drives one hour a day. Two drivers drive two hours a day. One driver drives three hours a day. It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours?"}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile."}, {"video_title": "Calculating percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile. Now if you look at these choices here, lucky for us, there's only one choice that's reasonably close to either one of those and that's the 55th percentile. And it looks like the people who wrote this question went with the calculation of percentile where they include the data point in question. So everything at six hours or less, what percentage of the total data is that?"}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look at the frequency table below. So let's see, that's the frequency table, and let's see, there's three categories of computer time, just like they told us, minimal, moderate, and extreme. This is before they go to bed or at night. And then they have the three categories of how much they're sleeping, five or few hours per night, five to seven hours per night, or seven or more hours. Okay, so that's fair enough, so let's see what they want us to do. So they tell us, suppose there were 17 people in the study who were both in moderate computer users and got five to seven hours of sleep. So moderate, five to seven."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then they have the three categories of how much they're sleeping, five or few hours per night, five to seven hours per night, or seven or more hours. Okay, so that's fair enough, so let's see what they want us to do. So they tell us, suppose there were 17 people in the study who were both in moderate computer users and got five to seven hours of sleep. So moderate, five to seven. So this category right over here, there were 17 people in this category over here. And just to mark that, let me, I copied and pasted this chart onto my scratch pad so I can write on it. So this group, they're telling us this group, and my pen is really acting up, I don't understand what's going on."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So moderate, five to seven. So this category right over here, there were 17 people in this category over here. And just to mark that, let me, I copied and pasted this chart onto my scratch pad so I can write on it. So this group, they're telling us this group, and my pen is really acting up, I don't understand what's going on. This group right over here, there are 17 people. So that group right over there is 17 people. Now what are they asking us?"}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this group, they're telling us this group, and my pen is really acting up, I don't understand what's going on. This group right over here, there are 17 people. So that group right over there is 17 people. Now what are they asking us? They're saying, so they're saying, how many people in the study were both extreme computer users and got five to seven hours of sleep round to the nearest whole number? So extreme and got five to seven. Get my scratch pad out."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what are they asking us? They're saying, so they're saying, how many people in the study were both extreme computer users and got five to seven hours of sleep round to the nearest whole number? So extreme and got five to seven. Get my scratch pad out. So they're saying, how many people are in, how many people are in this bucket, in this bucket right over here? I think I have to replace my pen tablet or something, I don't know why it's getting all splotchy like this. So how would we think about this?"}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "Get my scratch pad out. So they're saying, how many people are in, how many people are in this bucket, in this bucket right over here? I think I have to replace my pen tablet or something, I don't know why it's getting all splotchy like this. So how would we think about this? There's 17 people in this group, how many people are in this group? Well, they tell us that 17 is 34% of the row, of the row total, so I guess you could say 17 is 34.3% of the moderate, of the moderate computer users. Or you could say that 17 is 30% of the people who slept five to seven hours each night."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how would we think about this? There's 17 people in this group, how many people are in this group? Well, they tell us that 17 is 34% of the row, of the row total, so I guess you could say 17 is 34.3% of the moderate, of the moderate computer users. Or you could say that 17 is 30% of the people who slept five to seven hours each night. Or you could say 17 is 10%, is 10% of the total, of the total number of people. So let's just go with that. We could figure out the total number of people."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or you could say that 17 is 30% of the people who slept five to seven hours each night. Or you could say 17 is 10%, is 10% of the total, of the total number of people. So let's just go with that. We could figure out the total number of people. So 10%, actually let me write it this way. So 10%, 10% of the total, 10% of total is going to be equal to 17, or that the total, just divide both sides by 10%, is equal to 17 divided by 10%, which is the same thing as 17 over 0.1, which of course is equal to 170. So the total is 170, and they tell us that extreme, extreme computer users represent, extreme computer users who sleep five to seven hours per night represent 11.7%, 11.7% of the total."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could figure out the total number of people. So 10%, actually let me write it this way. So 10%, 10% of the total, 10% of total is going to be equal to 17, or that the total, just divide both sides by 10%, is equal to 17 divided by 10%, which is the same thing as 17 over 0.1, which of course is equal to 170. So the total is 170, and they tell us that extreme, extreme computer users represent, extreme computer users who sleep five to seven hours per night represent 11.7%, 11.7% of the total. So to answer their question of how many people are extreme computer users who sleep five to seven hours per night, that's 11.7% of 170. So let's go back over here. So we actually have a little calculator tool here."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the total is 170, and they tell us that extreme, extreme computer users represent, extreme computer users who sleep five to seven hours per night represent 11.7%, 11.7% of the total. So to answer their question of how many people are extreme computer users who sleep five to seven hours per night, that's 11.7% of 170. So let's go back over here. So we actually have a little calculator tool here. So it's 11.7%, which is 0.117, times 170, times 170 is, and let me make sure that you can see what I'm doing by scrolling over a little bit, times 170 is equal to 19.89. So if we're rounding to the nearest whole, that's going to be 20 people, 20 people. And then they are going to ask us some questions."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we actually have a little calculator tool here. So it's 11.7%, which is 0.117, times 170, times 170 is, and let me make sure that you can see what I'm doing by scrolling over a little bit, times 170 is equal to 19.89. So if we're rounding to the nearest whole, that's going to be 20 people, 20 people. And then they are going to ask us some questions. They say, does the table show evidence of an association between being a minimal computer user and getting seven hours of sleep or more? So let's just look at the chart. So an association between being a minimal computer user and getting seven hours of sleep or more."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then they are going to ask us some questions. They say, does the table show evidence of an association between being a minimal computer user and getting seven hours of sleep or more? So let's just look at the chart. So an association between being a minimal computer user and getting seven hours of sleep or more. So it looks like minimal computer users, so these are the minimal computer users who get seven or more hours of sleep. And there's a couple of ways to read this. So you could say that 51% of minimal computer users get seven or more hours of sleep."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So an association between being a minimal computer user and getting seven hours of sleep or more. So it looks like minimal computer users, so these are the minimal computer users who get seven or more hours of sleep. And there's a couple of ways to read this. So you could say that 51% of minimal computer users get seven or more hours of sleep. You could say that of the people who get seven or more hours of sleep, 55% are minimal computer users. And of course, this one just says that minimal computer users who get seven or more hours of sleep represent 18.3% of all of the people who were surveyed. So let's look at the choices."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could say that 51% of minimal computer users get seven or more hours of sleep. You could say that of the people who get seven or more hours of sleep, 55% are minimal computer users. And of course, this one just says that minimal computer users who get seven or more hours of sleep represent 18.3% of all of the people who were surveyed. So let's look at the choices. And when I just, actually, before I even look at the choices, let's see if there's an association. It does look like, if you look at minimal computer users, it looks like a small percentage, only 16% get five or few hours. A higher percentage gets five to seven hours."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look at the choices. And when I just, actually, before I even look at the choices, let's see if there's an association. It does look like, if you look at minimal computer users, it looks like a small percentage, only 16% get five or few hours. A higher percentage gets five to seven hours. And 51%, the highest percentage, gets seven or more. So it looks like for minimal computer users, it looks like the distribution is definitely weighted towards getting more sleep. And for example, if we look at the extreme computer users, it's the opposite trend."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "A higher percentage gets five to seven hours. And 51%, the highest percentage, gets seven or more. So it looks like for minimal computer users, it looks like the distribution is definitely weighted towards getting more sleep. And for example, if we look at the extreme computer users, it's the opposite trend. 47% have five or few hours per night, 33% five to seven hours, and then only 19% get seven or more. And it looks like the moderate is someplace in between. So just looking at this, just looking at each of these rows, it looks like there's a trend where if you use less, if you use a computer for less time, you're more likely to have more sleep."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "And for example, if we look at the extreme computer users, it's the opposite trend. 47% have five or few hours per night, 33% five to seven hours, and then only 19% get seven or more. And it looks like the moderate is someplace in between. So just looking at this, just looking at each of these rows, it looks like there's a trend where if you use less, if you use a computer for less time, you're more likely to have more sleep. And likewise, if you use a computer more, you're more likely to have less sleep. Another way to think about it, when you look at the people who are getting seven or more hours of sleep, a majority of them, a majority of them, are minimal computer users. And there's three categories."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just looking at this, just looking at each of these rows, it looks like there's a trend where if you use less, if you use a computer for less time, you're more likely to have more sleep. And likewise, if you use a computer more, you're more likely to have less sleep. Another way to think about it, when you look at the people who are getting seven or more hours of sleep, a majority of them, a majority of them, are minimal computer users. And there's three categories. So for 55% to be minimal computer users, it really does feel like the minimal computer users are more, they're definitely more, disproportionately representing the people who are getting seven, or disproportionately represented in this category of seven or more hours of sleep. And you see that the extreme computer users in this category they represent only 20% of this category. So it does look like there is an association between minimal computer use and getting seven or more hours of sleep."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "And there's three categories. So for 55% to be minimal computer users, it really does feel like the minimal computer users are more, they're definitely more, disproportionately representing the people who are getting seven, or disproportionately represented in this category of seven or more hours of sleep. And you see that the extreme computer users in this category they represent only 20% of this category. So it does look like there is an association between minimal computer use and getting seven or more hours of sleep. But let's look at the actual choices they give us. Does the table show evidence of an association between a minimal computer user and getting seven hours of sleep or more? So yes, because 35.1% are extreme computer users and 29.1% of people are moderate."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it does look like there is an association between minimal computer use and getting seven or more hours of sleep. But let's look at the actual choices they give us. Does the table show evidence of an association between a minimal computer user and getting seven hours of sleep or more? So yes, because 35.1% are extreme computer users and 29.1% of people are moderate. Well, I go with the yes, but this doesn't seem to really back up the claim. This is just giving us some kind of random data about the percentage that are extreme computer users or moderate computer users. No, well, I already explained why I go with yes, that there does seem to be a trend."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So yes, because 35.1% are extreme computer users and 29.1% of people are moderate. Well, I go with the yes, but this doesn't seem to really back up the claim. This is just giving us some kind of random data about the percentage that are extreme computer users or moderate computer users. No, well, I already explained why I go with yes, that there does seem to be a trend. And I don't even believe what the statement is because the total column percentages are essentially equal. We see that the total column, that the column percentages, the column percentages are not equal for the various, for people who are getting seven hours or more of sleep. So we see that right over here, 55, 25, 20."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "No, well, I already explained why I go with yes, that there does seem to be a trend. And I don't even believe what the statement is because the total column percentages are essentially equal. We see that the total column, that the column percentages, the column percentages are not equal for the various, for people who are getting seven hours or more of sleep. So we see that right over here, 55, 25, 20. So I won't go with that one either. Yes, because 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours. Yeah, I mean, that seems pretty good."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we see that right over here, 55, 25, 20. So I won't go with that one either. Yes, because 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours. Yeah, I mean, that seems pretty good. 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours of sleep. So that looks like a pretty good explanation. So I'll check that, but let's just review all of them."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yeah, I mean, that seems pretty good. 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours of sleep. So that looks like a pretty good explanation. So I'll check that, but let's just review all of them. No, well, I already said, I think it's yes, but because the total percentage of extreme users who get five to seven hours of sleep is the same as the total percentage of moderate computer users who get five to seven hours of sleep. So that doesn't really mean, it's not touching on the point that we're looking for. Yes, because 55% of people who get 75, who get seven or more hours of minimal computer users and only 35% of all people are minimal computer users."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll check that, but let's just review all of them. No, well, I already said, I think it's yes, but because the total percentage of extreme users who get five to seven hours of sleep is the same as the total percentage of moderate computer users who get five to seven hours of sleep. So that doesn't really mean, it's not touching on the point that we're looking for. Yes, because 55% of people who get 75, who get seven or more hours of minimal computer users and only 35% of all people are minimal computer users. Actually, I'll go with this as well. Oh yeah, this is a multi-select here, so I could select that one as well. So this one, we're looking at, so here we looked at the percentage of minimal computer users who get seven hours of sleep and we saw that percentage is higher than for the whole population."}, {"video_title": "Analyzing trends in categorical data   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yes, because 55% of people who get 75, who get seven or more hours of minimal computer users and only 35% of all people are minimal computer users. Actually, I'll go with this as well. Oh yeah, this is a multi-select here, so I could select that one as well. So this one, we're looking at, so here we looked at the percentage of minimal computer users who get seven hours of sleep and we saw that percentage is higher than for the whole population. Here we're looking at the people who get seven or more hours of sleep and we're saying, wow, 55% of them, 55% of them are minimal computer users even though only 35% of all the people are minimal computer users. So I would go with both of these. And so let us check our answer and we got it right."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do it in that same shade of green. I've already defined set A here. And in both cases, I've defined these sets with numbers. Instead of having numbers as being the objects in the set, I could have had farm animals there or famous presidents, but numbers hopefully keep things fairly simple. So I'm going to start with set A. And from set A, I'm going to subtract set B. So this is one way of thinking about the difference between set A and set B."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Instead of having numbers as being the objects in the set, I could have had farm animals there or famous presidents, but numbers hopefully keep things fairly simple. So I'm going to start with set A. And from set A, I'm going to subtract set B. So this is one way of thinking about the difference between set A and set B. And when I've written it this way, this essentially says, give me the set of all of the objects that are in A with the things that are in B taken out of that set. So let's think about what that means. So what's in set A with the things that are in B taken out?"}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is one way of thinking about the difference between set A and set B. And when I've written it this way, this essentially says, give me the set of all of the objects that are in A with the things that are in B taken out of that set. So let's think about what that means. So what's in set A with the things that are in B taken out? Well, that means let's take set A and take out a 17, a 19, or take out the 17s, the 19s, and the 6s. So we're going to be left with, we're going to have the 5, we're going to have the 3. We're not going to have the 17 because we subtracted out set B."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's in set A with the things that are in B taken out? Well, that means let's take set A and take out a 17, a 19, or take out the 17s, the 19s, and the 6s. So we're going to be left with, we're going to have the 5, we're going to have the 3. We're not going to have the 17 because we subtracted out set B. 17 is in set B, so take out anything that is in set B. So you get the 5, the 3. See, the 12 is not in set B, so we can keep that in there."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not going to have the 17 because we subtracted out set B. 17 is in set B, so take out anything that is in set B. So you get the 5, the 3. See, the 12 is not in set B, so we can keep that in there. And then the 19 is in set B, so we're going to take out the 19 as well. And so that is, this right over here is, you could view it as set B subtracted from set A. So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "See, the 12 is not in set B, so we can keep that in there. And then the 19 is in set B, so we're going to take out the 19 as well. And so that is, this right over here is, you could view it as set B subtracted from set A. So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. Another way you could think about it is, these are all of the elements that are not in set B but also in set A. So let me make it clear. You could view this as B subtracted from A, or you could view this as the relative complement of set B in A."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. Another way you could think about it is, these are all of the elements that are not in set B but also in set A. So let me make it clear. You could view this as B subtracted from A, or you could view this as the relative complement of set B in A. And we're going to talk a lot more about complements in the future, but the complement is the things that are not in B. And so this is saying, look, what are all of the things that are not in B? So you could say, what are all of the things not in B but are in A?"}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could view this as B subtracted from A, or you could view this as the relative complement of set B in A. And we're going to talk a lot more about complements in the future, but the complement is the things that are not in B. And so this is saying, look, what are all of the things that are not in B? So you could say, what are all of the things not in B but are in A? So once again, if you said all of the things that aren't in B, then you're thinking about all of the numbers in the whole universe that aren't 17, 19, or 6. And actually, you could even think broader. You're not even just thinking about numbers."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could say, what are all of the things not in B but are in A? So once again, if you said all of the things that aren't in B, then you're thinking about all of the numbers in the whole universe that aren't 17, 19, or 6. And actually, you could even think broader. You're not even just thinking about numbers. You could even be the color orange is not in set B, so that would be in the absolute complement of B. I don't see a zebra here in set B, so that would be its complement. But we're saying, what are the things that are not in B but are in A? And that would be the numbers 5, 3, and 12."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're not even just thinking about numbers. You could even be the color orange is not in set B, so that would be in the absolute complement of B. I don't see a zebra here in set B, so that would be its complement. But we're saying, what are the things that are not in B but are in A? And that would be the numbers 5, 3, and 12. Now, when we visualized this as B subtracted from A, you might be saying, hey, wait, look, look. OK, I could imagine you took the 17 out, you took the 19 out, but what about taking the 6 out? Maybe you've taken a 6 out, or in traditional subtraction, maybe we would end up with a negative number or something."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that would be the numbers 5, 3, and 12. Now, when we visualized this as B subtracted from A, you might be saying, hey, wait, look, look. OK, I could imagine you took the 17 out, you took the 19 out, but what about taking the 6 out? Maybe you've taken a 6 out, or in traditional subtraction, maybe we would end up with a negative number or something. And when you subtract a set, if the set you're subtracting from does not have that element, then taking that element out of it doesn't change it. If I start with set A and take a 6, if I take all the 6s out of set A, it doesn't change it. There was no 6 to begin with."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe you've taken a 6 out, or in traditional subtraction, maybe we would end up with a negative number or something. And when you subtract a set, if the set you're subtracting from does not have that element, then taking that element out of it doesn't change it. If I start with set A and take a 6, if I take all the 6s out of set A, it doesn't change it. There was no 6 to begin with. I could take all the zebras out of set A. It will not change it. Now, another way to denote the relative complement of set B in A or B subtracted from A is the notation that I'm about to write."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "There was no 6 to begin with. I could take all the zebras out of set A. It will not change it. Now, another way to denote the relative complement of set B in A or B subtracted from A is the notation that I'm about to write. We could have written it this way, A, and then we would have had this little figure like this that looks eerily like a division sign, but this also means the difference between set A and B where we're talking about, when we write it this way, we're talking about all of the things in set A that are not in set B or the things in set B taken out of set A or the relative complement of B in A. Now, with that out of the way, let's think about things the other way around. What would B slash, I'll just call it a slash right over here, what would B minus A be?"}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, another way to denote the relative complement of set B in A or B subtracted from A is the notation that I'm about to write. We could have written it this way, A, and then we would have had this little figure like this that looks eerily like a division sign, but this also means the difference between set A and B where we're talking about, when we write it this way, we're talking about all of the things in set A that are not in set B or the things in set B taken out of set A or the relative complement of B in A. Now, with that out of the way, let's think about things the other way around. What would B slash, I'll just call it a slash right over here, what would B minus A be? What would B minus A be, which we could also write as B minus A. What would this be equal to? Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it or all of the things, the complement of A that happens to be in B."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "What would B slash, I'll just call it a slash right over here, what would B minus A be? What would B minus A be, which we could also write as B minus A. What would this be equal to? Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it or all of the things, the complement of A that happens to be in B. Let's think of it as the set B with all the things in A taken out of it. We start with set B, we have a 17, but a 17 is in set A so we have to take the 17 out. Then we have a 19, but there's a 19 in set A so we have to take the 19 out."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it or all of the things, the complement of A that happens to be in B. Let's think of it as the set B with all the things in A taken out of it. We start with set B, we have a 17, but a 17 is in set A so we have to take the 17 out. Then we have a 19, but there's a 19 in set A so we have to take the 19 out. Then we have a 6. We don't have to take a 6 out of B because the 6 is not in set A so we're left with just the 6. This would be just the set with a single element in it, set 6."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we have a 19, but there's a 19 in set A so we have to take the 19 out. Then we have a 6. We don't have to take a 6 out of B because the 6 is not in set A so we're left with just the 6. This would be just the set with a single element in it, set 6. Now, let me ask another question. What would the relative complement of A in A be? This is the same thing as A minus A."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would be just the set with a single element in it, set 6. Now, let me ask another question. What would the relative complement of A in A be? This is the same thing as A minus A. This is literally saying let's take set A and then take all of the things that are in set A out of it. I start with a 5, but there's a 5 in set A so I have to take the 5 out. There's a 3, but there's a 3 in set A so I have to take a 3 out."}, {"video_title": "Relative complement or difference between sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the same thing as A minus A. This is literally saying let's take set A and then take all of the things that are in set A out of it. I start with a 5, but there's a 5 in set A so I have to take the 5 out. There's a 3, but there's a 3 in set A so I have to take a 3 out. I'm going to take all of these things out. I'm just going to be left with the empty set, often called the null set. Sometimes the notation for that will look like this."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that you know your probability of making a free throw. You know that the probability, the probability, and let's say the probability of scoring a free throw, just because if I say make and miss, they both start with M, it'll get confusing. So let's say the probability of scoring, score, you know, free throw, is equal to, is going to be, let's say 70%. If we want to write it as a percent, or we could write it as 0.7 if we write it as a decimal. And let's say the probability of missing a free throw then, and this is just gonna come straight out of what we just wrote down. So the probability of missing, of missing a free throw is just going to be 100% minus this. You're either gonna make or miss, you're either gonna score or miss."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we want to write it as a percent, or we could write it as 0.7 if we write it as a decimal. And let's say the probability of missing a free throw then, and this is just gonna come straight out of what we just wrote down. So the probability of missing, of missing a free throw is just going to be 100% minus this. You're either gonna make or miss, you're either gonna score or miss. I don't wanna use make and miss because they both start with M. So this is going to be a 30% probability, or if we write it as a decimal, 0.3. One minus 0.7. These are the only two possibilities, so they have to add up to 100%, or they have to add up to one."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're either gonna make or miss, you're either gonna score or miss. I don't wanna use make and miss because they both start with M. So this is going to be a 30% probability, or if we write it as a decimal, 0.3. One minus 0.7. These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. Now let's say that you are going to take six attempts. And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts. In six, in six attempts."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. Now let's say that you are going to take six attempts. And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts. In six, in six attempts. So let's think about what that is. And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now. So this is what we wanna figure out."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "In six, in six attempts. So let's think about what that is. And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now. So this is what we wanna figure out. The probability of exactly two scores in six attempts. So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts? So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is what we wanna figure out. The probability of exactly two scores in six attempts. So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts? So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. So score, score, and then it's miss, miss, miss, and miss. So what's the probability of this exact thing happening, this exact thing? Well you have a 0.7 chance of making, of scoring on the first one."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. So score, score, and then it's miss, miss, miss, and miss. So what's the probability of this exact thing happening, this exact thing? Well you have a 0.7 chance of making, of scoring on the first one. Then you have a 0.7 chance of scoring on the second one. And then you have a 0.3 chance of missing the next four. So the probability of this exact circumstance is going to be what I'm writing down."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well you have a 0.7 chance of making, of scoring on the first one. Then you have a 0.7 chance of scoring on the second one. And then you have a 0.3 chance of missing the next four. So the probability of this exact circumstance is going to be what I'm writing down. And hopefully you don't get the multiplication symbols confused with the decimals. I'm trying to write them a little bit higher. So times 0.3."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of this exact circumstance is going to be what I'm writing down. And hopefully you don't get the multiplication symbols confused with the decimals. I'm trying to write them a little bit higher. So times 0.3. And what is this going to be equal to? Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts?"}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So times 0.3. And what is this going to be equal to? Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two. So then you miss and you miss."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two. So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again going to be 0.7, if you just rearrange the order in which you're multiplying, this is going to be 0.7 squared times 0.3 to the fourth power. So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again going to be 0.7, if you just rearrange the order in which you're multiplying, this is going to be 0.7 squared times 0.3 to the fourth power. So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. Well, if you have, out of six attempts, you're choosing two of them to have scores, how many ways are there? Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. Well, if you have, out of six attempts, you're choosing two of them to have scores, how many ways are there? Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. This is gonna tell us the number of different ways you can make two scores in six attempts. And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. This is gonna tell us the number of different ways you can make two scores in six attempts. And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the reasoning behind this formula. It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the reasoning behind this formula. It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to?"}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to be four factorial. So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to be four factorial. So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16?"}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 different ways to make two out of six free throws. Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 different ways to make two out of six free throws. Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then it's 0.3, 0.3 to the, 0.3 to the fourth power. Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then it's 0.3, 0.3 to the, 0.3 to the fourth power. Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81. Three to the fourth power would be 81, but I'm multiplying four decimals, each of them have one space to the right of the decimal point, so I'm gonna have, this is gonna, I'm gonna have four spaces to the right of the decimal, so 0.0081. So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's see, three to the fourth power would be 81. Three to the fourth power would be 81, but I'm multiplying four decimals, each of them have one space to the right of the decimal point, so I'm gonna have, this is gonna, I'm gonna have four spaces to the right of the decimal, so 0.0081. So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I wish I had a little bit more real estate right over here, but I'll write it in a very bold color. This is going to be, oh, actually, I'm kind of out of bold colors. I'll write it in a slightly less bold color."}, {"video_title": "Probability of making 2 shots in 6 attempts   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I wish I had a little bit more real estate right over here, but I'll write it in a very bold color. This is going to be, oh, actually, I'm kind of out of bold colors. I'll write it in a slightly less bold color. This is going to be equal to 0.05935, if we wanted the exact number, or we could say this is approximately, if we round to the nearest percentage, this is approximately a 6% chance, 6% probability of getting exactly two scores in the six attempts. I didn't say two or more, I just said exactly two scores in the six attempts, and actually, it's a fairly low probability because I have a pretty high free throw percentage. If someone has this high of a free throw percentage, it's actually reasonably unlikely that they're only going to make two scores in the six attempts."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so like all random variables, this is taking particular outcomes and converting them into numbers. And this random variable, it could take on the value X equals zero, one, two, three, four, or five. And what I wanna do is figure out, well, what's the probability that this random variable takes on zero, can be one, can be two, can be three, can be four, can be five. And so to do that, first let's think about how many possible outcomes are there from flipping a fair coin five times. So let's think about this. So let's write possible outcomes, possible outcomes from five flips. From five flips."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so to do that, first let's think about how many possible outcomes are there from flipping a fair coin five times. So let's think about this. So let's write possible outcomes, possible outcomes from five flips. From five flips. These aren't the possible outcomes for the random variable, this is literally the number of possible outcomes for flipping a coin five times. For example, one possible outcome could be tails, heads, tails, heads, tails. Another possible outcome could be heads, heads, heads, tails, tails."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "From five flips. These aren't the possible outcomes for the random variable, this is literally the number of possible outcomes for flipping a coin five times. For example, one possible outcome could be tails, heads, tails, heads, tails. Another possible outcome could be heads, heads, heads, tails, tails. That is one of the equally likely outcomes, that's another one of the equally likely outcomes. How many of these are there? Well, for each flip, you have two possibilities."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Another possible outcome could be heads, heads, heads, tails, tails. That is one of the equally likely outcomes, that's another one of the equally likely outcomes. How many of these are there? Well, for each flip, you have two possibilities. So let's write this down. So let me, let me, so the first flip, the first flip, there's two possibilities, times two for the second flip, times two for the third flip, actually, I'm not gonna use the time notation, you might get confused with the random variable. Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, for each flip, you have two possibilities. So let's write this down. So let me, let me, so the first flip, the first flip, there's two possibilities, times two for the second flip, times two for the third flip, actually, I'm not gonna use the time notation, you might get confused with the random variable. Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip. Or two to the fifth equally likely possibilities from flipping a coin five times, which of course is equal to 32. And so this is going to be helpful because for each of the values that the random variable can take on, we just have to think about, well, how many of these equally likely possibilities would result in the random variable taking on that value? And let's just delve into it to see what we're actually talking about."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip. Or two to the fifth equally likely possibilities from flipping a coin five times, which of course is equal to 32. And so this is going to be helpful because for each of the values that the random variable can take on, we just have to think about, well, how many of these equally likely possibilities would result in the random variable taking on that value? And let's just delve into it to see what we're actually talking about. All right, and I'll do it in this light. Let me do it in, I'll start in blue. All right, so let's think about the probability that our random variable X is equal to one."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's just delve into it to see what we're actually talking about. All right, and I'll do it in this light. Let me do it in, I'll start in blue. All right, so let's think about the probability that our random variable X is equal to one. Well, actually, let me start with zero. The probability that our random variable X is equal to zero. So that would mean that you got no heads out of the five flips."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, so let's think about the probability that our random variable X is equal to one. Well, actually, let me start with zero. The probability that our random variable X is equal to zero. So that would mean that you got no heads out of the five flips. Well, there's only one way, one out of the 32 equally likely possibilities that you get no heads. That's the one where you just get five, where you get five tails. So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that would mean that you got no heads out of the five flips. Well, there's only one way, one out of the 32 equally likely possibilities that you get no heads. That's the one where you just get five, where you get five tails. So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities. Now, for this case, to kind of think in terms of kind of the binomial coefficients and combinatorics and all of that, it's much easier to just reason through it. But just so we can think in those terms, it'll be more useful as we go into higher values for our random variable. And it's also, this is all a buildup for the binomial distribution, so you get a sense of where the name comes from."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities. Now, for this case, to kind of think in terms of kind of the binomial coefficients and combinatorics and all of that, it's much easier to just reason through it. But just so we can think in those terms, it'll be more useful as we go into higher values for our random variable. And it's also, this is all a buildup for the binomial distribution, so you get a sense of where the name comes from. Let's write it in those terms. So this one, this one, this one right over here, the one way to think about that in combinatorics is that you had five flips and you're choosing zero of them to be heads. Five flips and you're choosing zero of them to be heads."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's also, this is all a buildup for the binomial distribution, so you get a sense of where the name comes from. Let's write it in those terms. So this one, this one, this one right over here, the one way to think about that in combinatorics is that you had five flips and you're choosing zero of them to be heads. Five flips and you're choosing zero of them to be heads. And let's verify that five choose zero is indeed one. So five choose zero, let me write it right over here, five choose zero is equal to five factorial over, over, over five minus zero factorial, over, actually over zero factorial times five minus zero factorial. Five minus zero factorial."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Five flips and you're choosing zero of them to be heads. And let's verify that five choose zero is indeed one. So five choose zero, let me write it right over here, five choose zero is equal to five factorial over, over, over five minus zero factorial, over, actually over zero factorial times five minus zero factorial. Five minus zero factorial. Well this is zero factorial is one by definition and so this is going to be five factorial over five factorial which is going to be equal to one. Once again, I like reasoning through it instead of blindly applying a formula, but I just wanted to show you that these, these two ideas are consistent. So let's keep going."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Five minus zero factorial. Well this is zero factorial is one by definition and so this is going to be five factorial over five factorial which is going to be equal to one. Once again, I like reasoning through it instead of blindly applying a formula, but I just wanted to show you that these, these two ideas are consistent. So let's keep going. And I'm going to do x equals one all the way up to x equals five and if you are inspired, and I encourage you to be inspired, try to fill out the whole thing. What's the probability that x equals one, two, three, four, or five? So let's go to the probability that x equals two, oh sorry, x equals one."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's keep going. And I'm going to do x equals one all the way up to x equals five and if you are inspired, and I encourage you to be inspired, try to fill out the whole thing. What's the probability that x equals one, two, three, four, or five? So let's go to the probability that x equals two, oh sorry, x equals one. So the probability that x equals one is going to be equal to, well how do you get one head? Well it could be the first one could be head and then the rest of them are going to be tails. The second one could be head and then the rest of them are going to be tails."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's go to the probability that x equals two, oh sorry, x equals one. So the probability that x equals one is going to be equal to, well how do you get one head? Well it could be the first one could be head and then the rest of them are going to be tails. The second one could be head and then the rest of them are going to be tails. I could write them all out, but you could see that there's just five different places to have that one head. So five out of the 32 equally likely outcomes involve one head. So let me write that down."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The second one could be head and then the rest of them are going to be tails. I could write them all out, but you could see that there's just five different places to have that one head. So five out of the 32 equally likely outcomes involve one head. So let me write that down. So this is going to be equal to, this is going to be equal to five out of the 32 equally likely outcomes which of course is the same thing. This is going to be the same thing as saying, look I got five flips and I'm choosing one of them. I'm choosing one of them to be heads."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write that down. So this is going to be equal to, this is going to be equal to five out of the 32 equally likely outcomes which of course is the same thing. This is going to be the same thing as saying, look I got five flips and I'm choosing one of them. I'm choosing one of them to be heads. So that over 32. And you could verify that five factorial over one factorial times five minus, actually let me just do it just so that you don't have to take my word for it. So five choose one is equal to five factorial over one factorial, which is just one, times five minus four, sorry five minus one factorial, which is equal to five factorial over four factorial, which is just going to be equal to five."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm choosing one of them to be heads. So that over 32. And you could verify that five factorial over one factorial times five minus, actually let me just do it just so that you don't have to take my word for it. So five choose one is equal to five factorial over one factorial, which is just one, times five minus four, sorry five minus one factorial, which is equal to five factorial over four factorial, which is just going to be equal to five. All right, we're making good progress. So now let in purple, let's think about the probability that our random variable X is equal to two. Well this is going to be equal to, and now I'll actually resort to the combinatorics."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So five choose one is equal to five factorial over one factorial, which is just one, times five minus four, sorry five minus one factorial, which is equal to five factorial over four factorial, which is just going to be equal to five. All right, we're making good progress. So now let in purple, let's think about the probability that our random variable X is equal to two. Well this is going to be equal to, and now I'll actually resort to the combinatorics. So this is, you have five flips, and you're choosing two of them to be heads over 32 equally likely possibilities. So this is the number of possibilities that result in two heads. Two of the five flips have chosen to be heads."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well this is going to be equal to, and now I'll actually resort to the combinatorics. So this is, you have five flips, and you're choosing two of them to be heads over 32 equally likely possibilities. So this is the number of possibilities that result in two heads. Two of the five flips have chosen to be heads. I guess you could think of it that way by the random gods or whatever you want to say. So this is the fraction of the 32 equally likely possibilities. So this is the probability that X equals two."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two of the five flips have chosen to be heads. I guess you could think of it that way by the random gods or whatever you want to say. So this is the fraction of the 32 equally likely possibilities. So this is the probability that X equals two. Well what's this going to be? Well I'll do it right over here, and actually no reason for me to have to keep switching colors. So five choose two is going to be equal to five factorial over two factorial times five minus two factorial."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the probability that X equals two. Well what's this going to be? Well I'll do it right over here, and actually no reason for me to have to keep switching colors. So five choose two is going to be equal to five factorial over two factorial times five minus two factorial. Five minus two factorial, so this is five factorial over two factorial times three factorial. And this is going to be equal to five times four times three times two. I could write times one, but that doesn't really do anything for us."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So five choose two is going to be equal to five factorial over two factorial times five minus two factorial. Five minus two factorial, so this is five factorial over two factorial times three factorial. And this is going to be equal to five times four times three times two. I could write times one, but that doesn't really do anything for us. And then two factorial is just going to be two. And then the three factorial is three times two. I could write times one, but once again it doesn't do anything for us."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could write times one, but that doesn't really do anything for us. And then two factorial is just going to be two. And then the three factorial is three times two. I could write times one, but once again it doesn't do anything for us. That cancels with that. Four divided by two is two. Five times two is 10."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could write times one, but once again it doesn't do anything for us. That cancels with that. Four divided by two is two. Five times two is 10. So this is equal to 10. This right over here is equal to 10 32nds. And obviously we could simplify this fraction, but I like to leave it this way because we're now thinking everything is in terms of 30 seconds."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Five times two is 10. So this is equal to 10. This right over here is equal to 10 32nds. And obviously we could simplify this fraction, but I like to leave it this way because we're now thinking everything is in terms of 30 seconds. There's a one 32nd chance, x equals zero. Five 32nds chance that x equals one. And a 10 32nds chance that x equals two."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And obviously we could simplify this fraction, but I like to leave it this way because we're now thinking everything is in terms of 30 seconds. There's a one 32nd chance, x equals zero. Five 32nds chance that x equals one. And a 10 32nds chance that x equals two. Let's keep on going. All right, I'll go in orange. So what is the probability that our random variable x is equal to three?"}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And a 10 32nds chance that x equals two. Let's keep on going. All right, I'll go in orange. So what is the probability that our random variable x is equal to three? Well this is going to be five. Out of the five flips, we're going to need to choose three of them to be heads to figure out which of the possibilities involve exactly three heads. And this is over 32 equally likely possibilities."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the probability that our random variable x is equal to three? Well this is going to be five. Out of the five flips, we're going to need to choose three of them to be heads to figure out which of the possibilities involve exactly three heads. And this is over 32 equally likely possibilities. And this is going to be equal to, so five choose three, is equal to five factorial over three factorial times five minus three factorial. Actually let me just write it down. Five minus three factorial, which is equal to five factorial over three factorial times two factorial."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is over 32 equally likely possibilities. And this is going to be equal to, so five choose three, is equal to five factorial over three factorial times five minus three factorial. Actually let me just write it down. Five minus three factorial, which is equal to five factorial over three factorial times two factorial. Well that's exactly what we had up here. We just swapped the three and the two. So this also is going to be equal to 10."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Five minus three factorial, which is equal to five factorial over three factorial times two factorial. Well that's exactly what we had up here. We just swapped the three and the two. So this also is going to be equal to 10. So this is also going to be equal to 10 32nds. All right, two more to go. And I think you're going to start seeing a little bit of a symmetry here."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this also is going to be equal to 10. So this is also going to be equal to 10 32nds. All right, two more to go. And I think you're going to start seeing a little bit of a symmetry here. One, five, 10, 10. Let's keep going. I haven't used white yet, so maybe I'll use white."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I think you're going to start seeing a little bit of a symmetry here. One, five, 10, 10. Let's keep going. I haven't used white yet, so maybe I'll use white. The probability that our random variable X is equal to four. Well, out of our five flips, we want to select four of them to be heads. Or out of the five, and we want to see, we're obviously not actively selecting."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I haven't used white yet, so maybe I'll use white. The probability that our random variable X is equal to four. Well, out of our five flips, we want to select four of them to be heads. Or out of the five, and we want to see, we're obviously not actively selecting. One way to think about it, we want to figure out the possibilities that involve out of the five flips, four of them are chosen to be heads, or four of them are heads. And this is over 32 equally likely possibilities. So five choose four is equal to five factorial over four factorial times five minus four factorial, which is equal to, well that's just going to be five factorial."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or out of the five, and we want to see, we're obviously not actively selecting. One way to think about it, we want to figure out the possibilities that involve out of the five flips, four of them are chosen to be heads, or four of them are heads. And this is over 32 equally likely possibilities. So five choose four is equal to five factorial over four factorial times five minus four factorial, which is equal to, well that's just going to be five factorial. This is going to be one factorial right over here. So that doesn't change the value. You're just going to multiply one factorial times four factorial."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So five choose four is equal to five factorial over four factorial times five minus four factorial, which is equal to, well that's just going to be five factorial. This is going to be one factorial right over here. So that doesn't change the value. You're just going to multiply one factorial times four factorial. So it's five factorial over four factorial, which is equal to five. So once again, this is five 32nds. And you could have reasoned through this, because if you're saying you want five heads, that means you have one tail."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're just going to multiply one factorial times four factorial. So it's five factorial over four factorial, which is equal to five. So once again, this is five 32nds. And you could have reasoned through this, because if you're saying you want five heads, that means you have one tail. And there's five different places you could put that one tail. There are five possibilities with one tail, five of the 32 equally likely. And then, and you could probably guess what we're going to get for x equals five, because having five heads means you have zero tails, and there's only going to be one possibility out of the 32 with zero tails, or that have all heads."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you could have reasoned through this, because if you're saying you want five heads, that means you have one tail. And there's five different places you could put that one tail. There are five possibilities with one tail, five of the 32 equally likely. And then, and you could probably guess what we're going to get for x equals five, because having five heads means you have zero tails, and there's only going to be one possibility out of the 32 with zero tails, or that have all heads. Let's write that down. So the probability that a random variable x is equal to five, so we have all five heads. And you could say this is five, and we're choosing five of them to be heads out of the 32 equally likely possibilities."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then, and you could probably guess what we're going to get for x equals five, because having five heads means you have zero tails, and there's only going to be one possibility out of the 32 with zero tails, or that have all heads. Let's write that down. So the probability that a random variable x is equal to five, so we have all five heads. And you could say this is five, and we're choosing five of them to be heads out of the 32 equally likely possibilities. Well, five choose five, that's going to be, actually let me just write it here, since I've done it for all the other ones. Five choose five is five factorial over five factorial times five minus five factorial. Well, this right over here is zero factorial, which is equal to one, and so this whole thing simplifies to one."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you could say this is five, and we're choosing five of them to be heads out of the 32 equally likely possibilities. Well, five choose five, that's going to be, actually let me just write it here, since I've done it for all the other ones. Five choose five is five factorial over five factorial times five minus five factorial. Well, this right over here is zero factorial, which is equal to one, and so this whole thing simplifies to one. So this is going to be one out of 130 seconds. And so you see the symmetry, 130 second, 130 seconds, 532nd, 532nd, 1032nd, 1032nd. And that makes sense, because the probability of getting five heads is the same as the probability of getting zero tails."}, {"video_title": "Binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this right over here is zero factorial, which is equal to one, and so this whole thing simplifies to one. So this is going to be one out of 130 seconds. And so you see the symmetry, 130 second, 130 seconds, 532nd, 532nd, 1032nd, 1032nd. And that makes sense, because the probability of getting five heads is the same as the probability of getting zero tails. And the probability of getting zero tails should be the same as the probability of getting zero heads. So I'll leave you there for this video. In the next video, we'll kind of graphically represent this and we'll see the probability distribution for this random variable."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Now the goal of this video is to think about, well what is the expected value of a geometric random variable like this? And I'll tell you the answer in future videos when we will apply this formula. But in this video we're actually going to prove it to ourselves mathematically. But the expected value of a geometric random variable is gonna be one over the probability of success on any given trial. So now let's prove it to ourselves. So the expected value of any random variable is just going to be the probability weighted outcomes that you could have. So you could say it is the probability, the probability that our random variable is equal to one times one plus the probability that our random variable is equal to two times two plus, and you get the general idea, it goes on and on and on."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "But the expected value of a geometric random variable is gonna be one over the probability of success on any given trial. So now let's prove it to ourselves. So the expected value of any random variable is just going to be the probability weighted outcomes that you could have. So you could say it is the probability, the probability that our random variable is equal to one times one plus the probability that our random variable is equal to two times two plus, and you get the general idea, it goes on and on and on. And a geometric random variable, it can only take on values one, two, three, four, so forth and so on. It will not take on the value zero because you cannot have a success if you have not had a trial yet. But what is this going to be equal to?"}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So you could say it is the probability, the probability that our random variable is equal to one times one plus the probability that our random variable is equal to two times two plus, and you get the general idea, it goes on and on and on. And a geometric random variable, it can only take on values one, two, three, four, so forth and so on. It will not take on the value zero because you cannot have a success if you have not had a trial yet. But what is this going to be equal to? Well this is going to be equal to, what's the probability that we have a success on our first trial? And actually let me just write it over here. So this is going to be P. What is this going to be?"}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "But what is this going to be equal to? Well this is going to be equal to, what's the probability that we have a success on our first trial? And actually let me just write it over here. So this is going to be P. What is this going to be? What is the probability that we don't have a success on our first trial but we have one on our second trial? Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the second trial. And actually let me do a few more terms here."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to be P. What is this going to be? What is the probability that we don't have a success on our first trial but we have one on our second trial? Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the second trial. And actually let me do a few more terms here. So let me erase this a little bit, do a few more terms. This is going to be the probability that X equals two, sorry, the probability that X equals three, times three. And we're going to keep going on and on and on."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And actually let me do a few more terms here. So let me erase this a little bit, do a few more terms. This is going to be the probability that X equals two, sorry, the probability that X equals three, times three. And we're going to keep going on and on and on. Well what's this going to be? Well the probability that X equals three is we're going to have to get two unsuccessful trials. And so the probability of two unsuccessful trials is one minus P squared."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And we're going to keep going on and on and on. Well what's this going to be? Well the probability that X equals three is we're going to have to get two unsuccessful trials. And so the probability of two unsuccessful trials is one minus P squared. And then one successful trial, just like that. So you get the general idea. So if I wanted to rewrite this, I'm just going to rewrite it to make it a little bit simpler."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And so the probability of two unsuccessful trials is one minus P squared. And then one successful trial, just like that. So you get the general idea. So if I wanted to rewrite this, I'm just going to rewrite it to make it a little bit simpler. So the expected, at least for the purposes of this proof, so the expected value of X is equal to, I'll write this as one P plus two P times one minus P plus three P times one minus P squared. And we're going to keep going on and on and on forever like that. So how do we figure out this sum?"}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So if I wanted to rewrite this, I'm just going to rewrite it to make it a little bit simpler. So the expected, at least for the purposes of this proof, so the expected value of X is equal to, I'll write this as one P plus two P times one minus P plus three P times one minus P squared. And we're going to keep going on and on and on forever like that. So how do we figure out this sum? Well now I'm going to do a little bit of mathematical trickery or gymnastics, but it's all valid. And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar technique. What I'm going to do here is I'm going to think about well what is one minus P times this expected value?"}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So how do we figure out this sum? Well now I'm going to do a little bit of mathematical trickery or gymnastics, but it's all valid. And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar technique. What I'm going to do here is I'm going to think about well what is one minus P times this expected value? So let's do that. So if I say one minus P times the expected value of X, what is that going to be equal to? Well I would multiply every one of these terms by one minus P. So one P times one minus P would be one P times one minus P. You would get that right over there."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "What I'm going to do here is I'm going to think about well what is one minus P times this expected value? So let's do that. So if I say one minus P times the expected value of X, what is that going to be equal to? Well I would multiply every one of these terms by one minus P. So one P times one minus P would be one P times one minus P. You would get that right over there. What about two P times one minus P? What would that be equal to? Well that would be two P times one minus P and now we're going to multiply it by one minus P again so you're going to get one minus P squared."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Well I would multiply every one of these terms by one minus P. So one P times one minus P would be one P times one minus P. You would get that right over there. What about two P times one minus P? What would that be equal to? Well that would be two P times one minus P and now we're going to multiply it by one minus P again so you're going to get one minus P squared. And so I think you see where this is going and we're just going to keep adding and adding and adding from there. So now we're going to do something really fun and interesting at least from a mathematical point of view. If this is equal to that, if the left hand side is equal to the right hand side, let's just subtract this value from both sides."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Well that would be two P times one minus P and now we're going to multiply it by one minus P again so you're going to get one minus P squared. And so I think you see where this is going and we're just going to keep adding and adding and adding from there. So now we're going to do something really fun and interesting at least from a mathematical point of view. If this is equal to that, if the left hand side is equal to the right hand side, let's just subtract this value from both sides. So on the left hand side I would have the expected value of X, that's that, minus this. Minus one minus P times the expected value of X. So I'm just subtracting this from that side but let me subtract this from that side."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "If this is equal to that, if the left hand side is equal to the right hand side, let's just subtract this value from both sides. So on the left hand side I would have the expected value of X, that's that, minus this. Minus one minus P times the expected value of X. So I'm just subtracting this from that side but let me subtract this from that side. Well I could subtract this expression from that but this is equivalent so I'm just going to subtract this from that. And so what do I get? Well let's see, I'm going to have one minus P and then if I subtract one P times one minus P from two P times one minus P, well I'm just going to be left with plus one P times one minus P. And then if I subtract this from that, I'm going to be left with one P times one minus P squared."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So I'm just subtracting this from that side but let me subtract this from that side. Well I could subtract this expression from that but this is equivalent so I'm just going to subtract this from that. And so what do I get? Well let's see, I'm going to have one minus P and then if I subtract one P times one minus P from two P times one minus P, well I'm just going to be left with plus one P times one minus P. And then if I subtract this from that, I'm going to be left with one P times one minus P squared. And we're just going to keep going on and on and on. And so let me simplify this a little bit. If I distribute this negative, this could be plus and then this would be P minus one."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Well let's see, I'm going to have one minus P and then if I subtract one P times one minus P from two P times one minus P, well I'm just going to be left with plus one P times one minus P. And then if I subtract this from that, I'm going to be left with one P times one minus P squared. And we're just going to keep going on and on and on. And so let me simplify this a little bit. If I distribute this negative, this could be plus and then this would be P minus one. And then if we distribute this expected value of X, we get on the left hand side, let me scroll up a little bit, I don't want to squinch it too much. So let's see, we have the expected value of X and then plus P times the expected value of X, P times the expected value of X minus the expected value of X. These cancel out."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "If I distribute this negative, this could be plus and then this would be P minus one. And then if we distribute this expected value of X, we get on the left hand side, let me scroll up a little bit, I don't want to squinch it too much. So let's see, we have the expected value of X and then plus P times the expected value of X, P times the expected value of X minus the expected value of X. These cancel out. It's going to be equal to P plus P times one minus P plus P times one minus P squared. And it's going to keep going on and on and on. Well on the left hand side, all I have is a P times the expected value of X."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "These cancel out. It's going to be equal to P plus P times one minus P plus P times one minus P squared. And it's going to keep going on and on and on. Well on the left hand side, all I have is a P times the expected value of X. If I want to solve for the expected value of X, I just divide both sides by P. So I get, and this is kind of neat, through this mathematical gymnastics I now have, I'm just dividing everything by P, both sides. On the left hand side, I just have the expected value of X. If I divide all of these terms by P, this first term becomes one, the second term becomes one minus P. This third term, if I divide by P, becomes plus one minus P squared, so forth and so on."}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Well on the left hand side, all I have is a P times the expected value of X. If I want to solve for the expected value of X, I just divide both sides by P. So I get, and this is kind of neat, through this mathematical gymnastics I now have, I'm just dividing everything by P, both sides. On the left hand side, I just have the expected value of X. If I divide all of these terms by P, this first term becomes one, the second term becomes one minus P. This third term, if I divide by P, becomes plus one minus P squared, so forth and so on. Now what's cool about this, this is a classic geometric series with a common ratio of one minus P. And if that term is completely unfamiliar to you, I encourage you, and this is why it's actually called a geometric, one of the reasons, arguments for why it's called a geometric random variable, but I encourage you to review what a geometric series is on Khan Academy if this looks completely unfamiliar. But in other places, we prove, using actually a very similar technique that we did up here, that this sum is going to be equal to one over one minus our common ratio. And our common ratio is one minus P. So what is this going to be equal to?"}, {"video_title": "Proof of expected value of geometric random variable   AP Statistics   Khan Academy.mp3", "Sentence": "If I divide all of these terms by P, this first term becomes one, the second term becomes one minus P. This third term, if I divide by P, becomes plus one minus P squared, so forth and so on. Now what's cool about this, this is a classic geometric series with a common ratio of one minus P. And if that term is completely unfamiliar to you, I encourage you, and this is why it's actually called a geometric, one of the reasons, arguments for why it's called a geometric random variable, but I encourage you to review what a geometric series is on Khan Academy if this looks completely unfamiliar. But in other places, we prove, using actually a very similar technique that we did up here, that this sum is going to be equal to one over one minus our common ratio. And our common ratio is one minus P. So what is this going to be equal to? And we are really in the home stretch right over here. This is going to be equal to one over one minus one plus P. One minus one plus P, which is indeed equal to one over P. So there you have it. We have proven to ourselves that the expected value of a geometric random variable, using some, I think, cool mathematics, is indeed equal to one over P."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, I wrote the standard deviation here in the last video, that should be the mean. And let's say it has some variance. I could write it like that, or I could write the standard deviation there. But as long as it has a well-defined mean and standard deviation, I don't care what the distribution looks like. What I can do is take samples in the last video of, say, size 4. So that means I take literally four instances of this random variable. This is one example."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "But as long as it has a well-defined mean and standard deviation, I don't care what the distribution looks like. What I can do is take samples in the last video of, say, size 4. So that means I take literally four instances of this random variable. This is one example. I take their mean, and I consider this the sample mean for my first trial. Or you could almost say for my first sample. I know it's very confusing, because you can consider that a sample, the set to be a sample, or you could consider each member of the set as a sample."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is one example. I take their mean, and I consider this the sample mean for my first trial. Or you could almost say for my first sample. I know it's very confusing, because you can consider that a sample, the set to be a sample, or you could consider each member of the set as a sample. So that can be a little bit confusing there. But I have this first sample mean, and then I keep doing that over and over. In my second sample, my sample size is 4."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I know it's very confusing, because you can consider that a sample, the set to be a sample, or you could consider each member of the set as a sample. So that can be a little bit confusing there. But I have this first sample mean, and then I keep doing that over and over. In my second sample, my sample size is 4. I got four instances of this random variable. I average them. I have another sample mean."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "In my second sample, my sample size is 4. I got four instances of this random variable. I average them. I have another sample mean. And the cool thing about the central limit theorem is as I keep plotting the frequency distribution of my sample means, it starts to approach something that approximates the normal distribution. And it's going to do a better job of approximating that normal distribution as n gets larger. And just so we have a little terminology on our belt, this frequency distribution right here that I've plotted out, or here, or up here, that I started plotting out, that is called, and it's kind of confusing because we use the word sample so much, that is called the sampling distribution of the sample mean."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I have another sample mean. And the cool thing about the central limit theorem is as I keep plotting the frequency distribution of my sample means, it starts to approach something that approximates the normal distribution. And it's going to do a better job of approximating that normal distribution as n gets larger. And just so we have a little terminology on our belt, this frequency distribution right here that I've plotted out, or here, or up here, that I started plotting out, that is called, and it's kind of confusing because we use the word sample so much, that is called the sampling distribution of the sample mean. And let's dissect this a little bit, just so that this long description of this distribution starts to make a little bit of sense. When we say it's the sampling distribution, that's telling us that it's being derived from, it's the distribution of some statistic, which in this case happens to be the sample mean, and we're deriving it from samples of an original distribution. So each of these, so this is my first sample."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just so we have a little terminology on our belt, this frequency distribution right here that I've plotted out, or here, or up here, that I started plotting out, that is called, and it's kind of confusing because we use the word sample so much, that is called the sampling distribution of the sample mean. And let's dissect this a little bit, just so that this long description of this distribution starts to make a little bit of sense. When we say it's the sampling distribution, that's telling us that it's being derived from, it's the distribution of some statistic, which in this case happens to be the sample mean, and we're deriving it from samples of an original distribution. So each of these, so this is my first sample. My sample size is 4. I'm using the statistic, the mean. I actually could have done it with other things."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So each of these, so this is my first sample. My sample size is 4. I'm using the statistic, the mean. I actually could have done it with other things. I could have done the mode, or the range, or other statistics, but the sampling distribution of the sample mean is the most common one. It's probably, in my mind, the best place to start learning about the central limit theorem, and even, frankly, sampling distribution. So that's what it's called."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I actually could have done it with other things. I could have done the mode, or the range, or other statistics, but the sampling distribution of the sample mean is the most common one. It's probably, in my mind, the best place to start learning about the central limit theorem, and even, frankly, sampling distribution. So that's what it's called. And just as a little bit of background, and I'll prove this to you experimentally, not mathematically, but I think the experimental is, on some levels, more satisfying with statistics, that this will have the same mean as your original distribution right here. So it has the same mean, but we'll see in the next video that this is actually going to start approximating a normal distribution, even though my original distribution that this is kind of generated from is completely non-normal. So let's do that with this app right here."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's what it's called. And just as a little bit of background, and I'll prove this to you experimentally, not mathematically, but I think the experimental is, on some levels, more satisfying with statistics, that this will have the same mean as your original distribution right here. So it has the same mean, but we'll see in the next video that this is actually going to start approximating a normal distribution, even though my original distribution that this is kind of generated from is completely non-normal. So let's do that with this app right here. And just to give proper credit where credit is due, I think it was developed at Rice University. This is from onlinestatbook.com. This is their app, which I think is a really neat app, because it really helps you to visualize what a sampling distribution of the sample mean is."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's do that with this app right here. And just to give proper credit where credit is due, I think it was developed at Rice University. This is from onlinestatbook.com. This is their app, which I think is a really neat app, because it really helps you to visualize what a sampling distribution of the sample mean is. So I can literally create my own custom distribution here. So let me make something kind of crazy. So you could do this, in theory, with a discrete or a continuous probability density function."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is their app, which I think is a really neat app, because it really helps you to visualize what a sampling distribution of the sample mean is. So I can literally create my own custom distribution here. So let me make something kind of crazy. So you could do this, in theory, with a discrete or a continuous probability density function. But what they have here, we could take on one of 32 values and I'm just going to set the different probabilities of getting any of those 32 values. So clearly, this right here is not a normal distribution. It looks a little bit bimodal, but it doesn't have long tails."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could do this, in theory, with a discrete or a continuous probability density function. But what they have here, we could take on one of 32 values and I'm just going to set the different probabilities of getting any of those 32 values. So clearly, this right here is not a normal distribution. It looks a little bit bimodal, but it doesn't have long tails. But what I want to do is first just use the simulation to understand, or to better understand, what the sampling distribution is all about. So what I'm going to do is I'm going to take, well, let's start with 5 at a time. So my sample size is going to be 5."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "It looks a little bit bimodal, but it doesn't have long tails. But what I want to do is first just use the simulation to understand, or to better understand, what the sampling distribution is all about. So what I'm going to do is I'm going to take, well, let's start with 5 at a time. So my sample size is going to be 5. And so when I click Animated, what it's going to do is it's going to take 5 samples from this probability distribution function, it's going to take 5 samples, and you're going to see them when I click Animated, it's going to average them and plot the average down here. And then I'm going to click it again, it's going to do it again. So there you go."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So my sample size is going to be 5. And so when I click Animated, what it's going to do is it's going to take 5 samples from this probability distribution function, it's going to take 5 samples, and you're going to see them when I click Animated, it's going to average them and plot the average down here. And then I'm going to click it again, it's going to do it again. So there you go. It got 5 samples from there, it averaged them, and it hit there. What did I just do? I clicked, oh, I wanted to clear that."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there you go. It got 5 samples from there, it averaged them, and it hit there. What did I just do? I clicked, oh, I wanted to clear that. Let me make this bottom one none. So let me do that over again. So I'm going to take 5 at a time."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I clicked, oh, I wanted to clear that. Let me make this bottom one none. So let me do that over again. So I'm going to take 5 at a time. So I took 5 samples from up here, and then it took its mean and plotted the mean there. Let me do it again. 5 samples from this probability distribution function, plotted it right there."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm going to take 5 at a time. So I took 5 samples from up here, and then it took its mean and plotted the mean there. Let me do it again. 5 samples from this probability distribution function, plotted it right there. I could keep doing it, it'll take some time, but as you can see, I plotted it right there. Now, I could do this 1,000 times, it's going to take forever. Let's say I just wanted to do it 1,000 times."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "5 samples from this probability distribution function, plotted it right there. I could keep doing it, it'll take some time, but as you can see, I plotted it right there. Now, I could do this 1,000 times, it's going to take forever. Let's say I just wanted to do it 1,000 times. So this program, just to be clear, it's actually generating the random numbers. This isn't like a rigged program. It's actually going to generate the random numbers according to this probability distribution function."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I just wanted to do it 1,000 times. So this program, just to be clear, it's actually generating the random numbers. This isn't like a rigged program. It's actually going to generate the random numbers according to this probability distribution function. It's going to take 5 at a time, find their means, and plot the mean. So if I click 10,000, it's going to do that 10,000 times. So it's going to take 5 numbers from here 10,000 times and find their means 10,000 times, and then plot the 10,000 means here, so let's do that."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually going to generate the random numbers according to this probability distribution function. It's going to take 5 at a time, find their means, and plot the mean. So if I click 10,000, it's going to do that 10,000 times. So it's going to take 5 numbers from here 10,000 times and find their means 10,000 times, and then plot the 10,000 means here, so let's do that. So there you go. And notice, it's already looking a lot like a normal distribution. And like I said, the original mean of my crazy distribution here was 14.45, and the mean of after doing 10,000 samples, or 10,000 trials, my mean here is 14.42."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to take 5 numbers from here 10,000 times and find their means 10,000 times, and then plot the 10,000 means here, so let's do that. So there you go. And notice, it's already looking a lot like a normal distribution. And like I said, the original mean of my crazy distribution here was 14.45, and the mean of after doing 10,000 samples, or 10,000 trials, my mean here is 14.42. So I'm already getting pretty close to the mean there. My standard deviation, you might notice, is less than that, we'll talk about that in a future video. And the skew and kurtosis, these are things that help us measure how normal a distribution is."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And like I said, the original mean of my crazy distribution here was 14.45, and the mean of after doing 10,000 samples, or 10,000 trials, my mean here is 14.42. So I'm already getting pretty close to the mean there. My standard deviation, you might notice, is less than that, we'll talk about that in a future video. And the skew and kurtosis, these are things that help us measure how normal a distribution is. And I've talked a little bit about it in the past, and let me actually just diverge a little bit, just so it's interesting. And they're fairly straightforward concepts. Skew literally tells, so if this is, let me do it in a different color."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the skew and kurtosis, these are things that help us measure how normal a distribution is. And I've talked a little bit about it in the past, and let me actually just diverge a little bit, just so it's interesting. And they're fairly straightforward concepts. Skew literally tells, so if this is, let me do it in a different color. If this is a perfect normal distribution, and clearly my drawing is very far from perfect, if that's a perfect distribution, this would have a skew of 0. If you have a positive skew, that means you have a larger right tail than you would have otherwise expect. So something with a positive skew might look like this."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Skew literally tells, so if this is, let me do it in a different color. If this is a perfect normal distribution, and clearly my drawing is very far from perfect, if that's a perfect distribution, this would have a skew of 0. If you have a positive skew, that means you have a larger right tail than you would have otherwise expect. So something with a positive skew might look like this. It would have a large tail to the right. So this would be a positive skew, which makes it a little less than ideal for normal distribution. And a negative skew would look like this."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So something with a positive skew might look like this. It would have a large tail to the right. So this would be a positive skew, which makes it a little less than ideal for normal distribution. And a negative skew would look like this. It has a long tail to the left. So a negative skew might look like that. So that is a negative skew."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And a negative skew would look like this. It has a long tail to the left. So a negative skew might look like that. So that is a negative skew. If you have trouble remembering it, just remember which direction the tail is going. This tail is going towards the negative direction. This tail is going to the positive direction."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that is a negative skew. If you have trouble remembering it, just remember which direction the tail is going. This tail is going towards the negative direction. This tail is going to the positive direction. So if something has no skew, that means that it's nice and symmetrical around its mean. Now kurtosis, which sounds like a very fancy word, is similarly not that fancy of an idea. Kurtosis."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "This tail is going to the positive direction. So if something has no skew, that means that it's nice and symmetrical around its mean. Now kurtosis, which sounds like a very fancy word, is similarly not that fancy of an idea. Kurtosis. So once again, if I were to draw a perfect normal distribution, remember, there's no one normal distribution. You can have different means and different standard deviations. Let's say that's a perfect normal distribution."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Kurtosis. So once again, if I were to draw a perfect normal distribution, remember, there's no one normal distribution. You can have different means and different standard deviations. Let's say that's a perfect normal distribution. If I have positive kurtosis, what's going to happen is I'm going to have fatter tails. Let me draw it a little nicer than that. I'm going to have fatter tails, but I'm going to have a more pointy peak."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that's a perfect normal distribution. If I have positive kurtosis, what's going to happen is I'm going to have fatter tails. Let me draw it a little nicer than that. I'm going to have fatter tails, but I'm going to have a more pointy peak. I didn't even have to draw it that pointy. Let me draw it like this. I'm going to have fatter tails, and I'm going to have a more pointy peak than a normal distribution."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to have fatter tails, but I'm going to have a more pointy peak. I didn't even have to draw it that pointy. Let me draw it like this. I'm going to have fatter tails, and I'm going to have a more pointy peak than a normal distribution. So this right here is positive kurtosis. So something that has positive kurtosis, depending on how positive it is, it tells you it's a little bit more pointy than a real normal distribution. Positive kurtosis."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to have fatter tails, and I'm going to have a more pointy peak than a normal distribution. So this right here is positive kurtosis. So something that has positive kurtosis, depending on how positive it is, it tells you it's a little bit more pointy than a real normal distribution. Positive kurtosis. And negative kurtosis has smaller tails, but it's smoother near the middle. So it's like this. So something like this would have negative kurtosis."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Positive kurtosis. And negative kurtosis has smaller tails, but it's smoother near the middle. So it's like this. So something like this would have negative kurtosis. And maybe in future videos we'll explore that in more detail. But in the context of this simulation, it's just telling us how normal this distribution is. So when our sample size was n equals 5 and we did 10,000 trials, we got pretty close to normal distribution."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So something like this would have negative kurtosis. And maybe in future videos we'll explore that in more detail. But in the context of this simulation, it's just telling us how normal this distribution is. So when our sample size was n equals 5 and we did 10,000 trials, we got pretty close to normal distribution. Let's do another 10,000 trials just to see what happens. It looks even more like a normal distribution. Our mean is now the exact same number, but we still have a little bit of skew and a little bit of kurtosis."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when our sample size was n equals 5 and we did 10,000 trials, we got pretty close to normal distribution. Let's do another 10,000 trials just to see what happens. It looks even more like a normal distribution. Our mean is now the exact same number, but we still have a little bit of skew and a little bit of kurtosis. Now let's see what happens if we were to do the same thing with a larger sample size. And we can actually do them simultaneously. So here's n equals 5."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our mean is now the exact same number, but we still have a little bit of skew and a little bit of kurtosis. Now let's see what happens if we were to do the same thing with a larger sample size. And we can actually do them simultaneously. So here's n equals 5. Let's do here n equals 25. Let me clear them. I'm going to do the sampling distribution of the sample mean, and I'm going to run 10,000 trials."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here's n equals 5. Let's do here n equals 25. Let me clear them. I'm going to do the sampling distribution of the sample mean, and I'm going to run 10,000 trials. So I'll do one animated trial, just so you remember what's going on. So I'm literally taking first five samples from up here, find their mean. Now I'm taking 25 samples from up here, find its mean, and then plotting it down here."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to do the sampling distribution of the sample mean, and I'm going to run 10,000 trials. So I'll do one animated trial, just so you remember what's going on. So I'm literally taking first five samples from up here, find their mean. Now I'm taking 25 samples from up here, find its mean, and then plotting it down here. So here the sample size is 25, here it's 5. I'll do it one more time. I take 5, get the mean, plot it."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now I'm taking 25 samples from up here, find its mean, and then plotting it down here. So here the sample size is 25, here it's 5. I'll do it one more time. I take 5, get the mean, plot it. Take 25, get the mean, and then plot it down there. This is a larger sample size. Now that thing that I just did, I'm going to do 10,000 times, and that's interesting."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I take 5, get the mean, plot it. Take 25, get the mean, and then plot it down there. This is a larger sample size. Now that thing that I just did, I'm going to do 10,000 times, and that's interesting. Remember, our first distribution was just this really crazy, very non-normal distribution. But once we did it, so here what's interesting. I mean, they both look a little normal, but if you look at the skew and the kurtosis when our sample size is larger, it's more normal."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now that thing that I just did, I'm going to do 10,000 times, and that's interesting. Remember, our first distribution was just this really crazy, very non-normal distribution. But once we did it, so here what's interesting. I mean, they both look a little normal, but if you look at the skew and the kurtosis when our sample size is larger, it's more normal. This has a lower skew than when our sample size was only 5, and it has a less negative kurtosis than when our sample size was 5. So this is a more normal distribution. And one thing that we're going to explore further in a future video is not only is it more normal in its shape, but it's also a tighter fit around the mean."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I mean, they both look a little normal, but if you look at the skew and the kurtosis when our sample size is larger, it's more normal. This has a lower skew than when our sample size was only 5, and it has a less negative kurtosis than when our sample size was 5. So this is a more normal distribution. And one thing that we're going to explore further in a future video is not only is it more normal in its shape, but it's also a tighter fit around the mean. And you can even think about why that kind of makes sense. When your sample size is larger, your odds of getting really far away from the mean is lower, because it's very low likelihood if you're taking 25 samples or 100 samples that you're just going to get a bunch of stuff way out here, or a bunch of stuff way out here. You're very likely to get a reasonable spread of things."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And one thing that we're going to explore further in a future video is not only is it more normal in its shape, but it's also a tighter fit around the mean. And you can even think about why that kind of makes sense. When your sample size is larger, your odds of getting really far away from the mean is lower, because it's very low likelihood if you're taking 25 samples or 100 samples that you're just going to get a bunch of stuff way out here, or a bunch of stuff way out here. You're very likely to get a reasonable spread of things. So it makes sense that your mean, your sample mean, is less likely to be far away from the mean. We're going to talk a little bit more about that in the future. But hopefully this kind of satisfies you that, at least experimentally, I haven't proven it to you with mathematical rigor, which hopefully we'll do in the future, but hopefully this satisfies you at least experimentally that the central limit theorem really does apply to any distribution."}, {"video_title": "Sampling distribution of the sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're very likely to get a reasonable spread of things. So it makes sense that your mean, your sample mean, is less likely to be far away from the mean. We're going to talk a little bit more about that in the future. But hopefully this kind of satisfies you that, at least experimentally, I haven't proven it to you with mathematical rigor, which hopefully we'll do in the future, but hopefully this satisfies you at least experimentally that the central limit theorem really does apply to any distribution. I mean, this is a crazy distribution. I encourage you to use this applet at onlinestatbook.com and experiment with other crazy distributions to believe it for yourself. But the interesting things are that we're approaching a normal distribution, but as my sample size got larger, it's a better fit for a normal distribution."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "His results are displayed in the table below. Alright, this is interesting. These columns, on time, delayed, and the total. So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed?"}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. And we do have a good number of experiments here."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. Well, let's see."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy?"}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. We have a total of 20 snowy days."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. This right over here is less than 0.1."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days."}, {"video_title": "Conditional probability and independence   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days. And so based on this data, because the experimental probability of being delayed given snowy is so much higher than the experimental probability of just being delayed, I would make the statement that these are not independent. So for these days, are the events delayed and snowy independent? No."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Here are some summary statistics for each exam. So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean?"}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close."}, {"video_title": "Comparing with z-scores   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable. He did roughly two standard deviations if we were around to the nearest standard deviation, but if you wanted to get precise, he did a little bit better, relatively speaking, on the LSAT. He did 2.1 standard deviations here, while over here he did 1.86 or 1.9 standard deviations."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "In this video, we are going to introduce ourselves to the idea of permutations, which is a fancy word for a pretty straightforward concept, which is what are the number of ways that we can arrange things? How many different possibilities are there? And to make that a little bit tangible, let's have an example with, say, a sofa. My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one. And we said, all right, how many different possibilities are, how many different people could sit in seat number one, assuming no one has sat down before? Well, three different people could sit in seat number one."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one. And we said, all right, how many different possibilities are, how many different people could sit in seat number one, assuming no one has sat down before? Well, three different people could sit in seat number one. You can see it right over here. This is where A is sitting in seat number one. This is where B is sitting in seat number one."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, three different people could sit in seat number one. You can see it right over here. This is where A is sitting in seat number one. This is where B is sitting in seat number one. And this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is where B is sitting in seat number one. And this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is tongue-twister, there's two different possibilities for seat number two. And so you're gonna have two different possibilities here."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is tongue-twister, there's two different possibilities for seat number two. And so you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways of getting that, and so there's two people left who could sit in the second seat. And we saw that right over here where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two?"}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways of getting that, and so there's two people left who could sit in the second seat. And we saw that right over here where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two? Well, you would multiply. For each of these three, you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three?"}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so how many different permutations are there for seat number one and seat number two? Well, you would multiply. For each of these three, you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial which is going to be equal to six which is exactly what we've got here. And to appreciate the power of this, let's extend our example."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial which is going to be equal to six which is exactly what we've got here. And to appreciate the power of this, let's extend our example. Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to appreciate the power of this, let's extend our example. Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. Five times four is 20. 20 times three is 60. And then 60 times two is 120."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. Five times four is 20. 20 times three is 60. And then 60 times two is 120. And then 120 times one is equal to 120. And once again, that makes a lot of sense. There's five different, if no one sat down, there's five different possibilities for seat number one."}, {"video_title": "Factorial and counting seat arrangements   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then 60 times two is 120. And then 120 times one is equal to 120. And once again, that makes a lot of sense. There's five different, if no one sat down, there's five different possibilities for seat number one. And then for each of those possibilities, there's four people who could sit in seat number two. And then for each of those 20 possibilities in seat numbers one and two, well, there's going to be three people who could sit in seat number three. And for each of these 60 possibilities, there's two people who can sit in seat number four."}, {"video_title": "Clarification of confidence interval of difference of means   Khan Academy.mp3", "Sentence": "Near the end of the last video, I wasn't as articulate as I would like to be, mainly because I think 15 minutes into a video my brain starts to really warm up too much. But what I want to do is restate what I was trying to say. We got this confidence interval. We were confident, I'll rewrite it here, we're confident that there's a 95% chance, I'll just restate the confidence interval. So there's the 95% confidence interval for the mean of this distribution, confidence interval for the mean of that distribution, we got as being 1.91 plus or minus 1.21. And near the end of the video I tried to explain why that is neat. Because here we have, it's a confidence interval for this weird mean of the difference between the sampling mean, it seems kind of confusing, but I just want to restate what we saw in previous videos."}, {"video_title": "Clarification of confidence interval of difference of means   Khan Academy.mp3", "Sentence": "We were confident, I'll rewrite it here, we're confident that there's a 95% chance, I'll just restate the confidence interval. So there's the 95% confidence interval for the mean of this distribution, confidence interval for the mean of that distribution, we got as being 1.91 plus or minus 1.21. And near the end of the video I tried to explain why that is neat. Because here we have, it's a confidence interval for this weird mean of the difference between the sampling mean, it seems kind of confusing, but I just want to restate what we saw in previous videos. This thing right over here, the mean of the difference of the sampling means, we saw two or three videos ago, is the same thing as the mean of the difference of the means of the sampling distributions. And we know that the mean of each of the sampling distributions is actually the same as the mean of the population distributions. So this is the same thing as the mean of population 1 minus the mean of population 2."}, {"video_title": "Clarification of confidence interval of difference of means   Khan Academy.mp3", "Sentence": "Because here we have, it's a confidence interval for this weird mean of the difference between the sampling mean, it seems kind of confusing, but I just want to restate what we saw in previous videos. This thing right over here, the mean of the difference of the sampling means, we saw two or three videos ago, is the same thing as the mean of the difference of the means of the sampling distributions. And we know that the mean of each of the sampling distributions is actually the same as the mean of the population distributions. So this is the same thing as the mean of population 1 minus the mean of population 2. And this was the neat result about the last video. This isn't just a 95% confidence interval for this parameter right here, it's actually a 95% confidence interval for this parameter right here. And this is the parameter that we really care about."}, {"video_title": "Clarification of confidence interval of difference of means   Khan Academy.mp3", "Sentence": "So this is the same thing as the mean of population 1 minus the mean of population 2. And this was the neat result about the last video. This isn't just a 95% confidence interval for this parameter right here, it's actually a 95% confidence interval for this parameter right here. And this is the parameter that we really care about. The true difference in weight loss between going on the diet, going on the low-fat diet, and not going on the low-fat diet. And we have a 95% confidence interval that that difference is between 0.7 and 3.12 pounds, which tells us that we have a 95% confidence interval that you're definitely going to lose some weight. We're not 100% sure, but we're confident that there's a 95% probability of that."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The first time you're exposed to permutations and combinations, it takes a little bit to get your brain around it. So I think it never hurts to do as many examples. But each incremental example, I'm gonna go, I'm gonna review what we've done before, but hopefully go a little bit further. So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that?"}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots?"}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot?"}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them?"}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations."}, {"video_title": "Combination formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations. There's 360 permutations for putting six people into four chairs, but there's only 15 combinations, because we're no longer counting all of the different arrangements for the same four people in the four chairs. We're saying, hey, if it's the same four people, that is now one combination, and you can see how many ways are there to arrange four people into four chairs? Well, that's the four factorial part right over here, the four factorial part right over here, which is four times three times two times one, which is 24, so we essentially just took the 360 divided by 24 to get 15, but once again, I don't want to, I don't think I can stress this enough."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say there are a couple of herbs that people believe help prevent the flu. So to test this, what we do is we wait for flu season, and we randomly assign people to three different groups. And over the course of flu season, we have them either in one group taking herb 1, in the second group taking herb 2, and in the third group they take a placebo. And if you don't know what a placebo is, it's something that to the patient or to the person participating, it feels like they're taking something that you've told them might help them, but it does nothing. It could be just a sugar pill, just so it feels like medicine. The reason why you even go through the effort of giving them something is because oftentimes there's something called a placebo effect, where people get better just because they're being told that they're being given something that will make them better. So this could right here just be a sugar pill."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you don't know what a placebo is, it's something that to the patient or to the person participating, it feels like they're taking something that you've told them might help them, but it does nothing. It could be just a sugar pill, just so it feels like medicine. The reason why you even go through the effort of giving them something is because oftentimes there's something called a placebo effect, where people get better just because they're being told that they're being given something that will make them better. So this could right here just be a sugar pill. And a very small amount of sugar, so it really can't affect their actual likelihood of getting the flu. So over here we have a table, and this is actually called a contingency table. And it has on it, in each group, the number that got sick, the number that didn't get sick."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this could right here just be a sugar pill. And a very small amount of sugar, so it really can't affect their actual likelihood of getting the flu. So over here we have a table, and this is actually called a contingency table. And it has on it, in each group, the number that got sick, the number that didn't get sick. And so we also can, from this, calculate the total number. So in group 1, we had a total of 120 people. In group 2, we had a total of 30 plus 110 is 140 people."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it has on it, in each group, the number that got sick, the number that didn't get sick. And so we also can, from this, calculate the total number. So in group 1, we had a total of 120 people. In group 2, we had a total of 30 plus 110 is 140 people. And in the placebo group, the group that just got the sugar pill, we had a total of 120 people. And then we could also tabulate the total number of people that got sick. So that's 20 plus 30 is 50, plus 30 is 80."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "In group 2, we had a total of 30 plus 110 is 140 people. And in the placebo group, the group that just got the sugar pill, we had a total of 120 people. And then we could also tabulate the total number of people that got sick. So that's 20 plus 30 is 50, plus 30 is 80. This is the total column right over here. And then the total people that didn't get sick over here is 100 plus 110 is 210, plus 90 is 300. And then the total people here are 380."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's 20 plus 30 is 50, plus 30 is 80. This is the total column right over here. And then the total people that didn't get sick over here is 100 plus 110 is 210, plus 90 is 300. And then the total people here are 380. Both this column and this row should add up to 380. So with that out of the way, let's think about how we can use this information in the contingency table and our knowledge of the chi-squared distribution to come up with some conclusion. So let's just make a null hypothesis."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the total people here are 380. Both this column and this row should add up to 380. So with that out of the way, let's think about how we can use this information in the contingency table and our knowledge of the chi-squared distribution to come up with some conclusion. So let's just make a null hypothesis. Our null hypothesis is that the herbs do nothing. Let me get some space here. So let's assume the null hypothesis that the herbs do nothing."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just make a null hypothesis. Our null hypothesis is that the herbs do nothing. Let me get some space here. So let's assume the null hypothesis that the herbs do nothing. And then we have our alternative hypothesis, our alternate hypothesis, that the herbs do something. Notice, I don't even care whether they actually improve. I'm just saying they do something."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's assume the null hypothesis that the herbs do nothing. And then we have our alternative hypothesis, our alternate hypothesis, that the herbs do something. Notice, I don't even care whether they actually improve. I'm just saying they do something. They might even increase your likelihood of getting the flu. We're not testing whether they're actually good. We're just saying, are they different than just doing nothing?"}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just saying they do something. They might even increase your likelihood of getting the flu. We're not testing whether they're actually good. We're just saying, are they different than just doing nothing? So like we did do with all of our hypothesis tests, let's just assume the null. We're going to assume the null. And given that assumption, figure out if the likelihood of getting data like this or more extreme is really low."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're just saying, are they different than just doing nothing? So like we did do with all of our hypothesis tests, let's just assume the null. We're going to assume the null. And given that assumption, figure out if the likelihood of getting data like this or more extreme is really low. And if it is really low, then we will reject the null hypothesis. And in this test, like every hypothesis test, we need a significance level. And let's say our significance level we care about, for whatever reason, is 10% or 0.10."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And given that assumption, figure out if the likelihood of getting data like this or more extreme is really low. And if it is really low, then we will reject the null hypothesis. And in this test, like every hypothesis test, we need a significance level. And let's say our significance level we care about, for whatever reason, is 10% or 0.10. That's the significance level that we care about. Now, to do this, we have to calculate a chi-squared statistic for this contingency table. And to do that, we do it very similar to what we did with the restaurant situation."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's say our significance level we care about, for whatever reason, is 10% or 0.10. That's the significance level that we care about. Now, to do this, we have to calculate a chi-squared statistic for this contingency table. And to do that, we do it very similar to what we did with the restaurant situation. We figure out, assuming the null hypothesis, the expected results you would have gotten in each of these cells. You can view each of these entries as a cell. That's what we do with it."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to do that, we do it very similar to what we did with the restaurant situation. We figure out, assuming the null hypothesis, the expected results you would have gotten in each of these cells. You can view each of these entries as a cell. That's what we do with it. You call each of those entries in Excel also a cell, each of the entries in a table. What we do is we figure out what the expected value would have been if you do assume the null hypothesis. Then we find the squared distance from that expected value, and we, I guess you could call it, normalize it by the expected value, take the sum of all of those differences."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what we do with it. You call each of those entries in Excel also a cell, each of the entries in a table. What we do is we figure out what the expected value would have been if you do assume the null hypothesis. Then we find the squared distance from that expected value, and we, I guess you could call it, normalize it by the expected value, take the sum of all of those differences. And if those differences, those squared differences, are really big, the probability of getting it would be really small, and maybe we'll reject the null hypothesis. So let's just figure out how we can get the expected number. So we're assuming the herbs do nothing."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we find the squared distance from that expected value, and we, I guess you could call it, normalize it by the expected value, take the sum of all of those differences. And if those differences, those squared differences, are really big, the probability of getting it would be really small, and maybe we'll reject the null hypothesis. So let's just figure out how we can get the expected number. So we're assuming the herbs do nothing. So if the herbs do nothing, then we can just figure out that this whole population, they just had nothing happen to them. These herbs were useless. And so we can use this population sample, or I shouldn't call it the population."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're assuming the herbs do nothing. So if the herbs do nothing, then we can just figure out that this whole population, they just had nothing happen to them. These herbs were useless. And so we can use this population sample, or I shouldn't call it the population. We should use this sample right here to figure out the expected number of people who would get sick or not sick. And so over here we have 80 out of 380 did not get sick. And I want to be careful."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so we can use this population sample, or I shouldn't call it the population. We should use this sample right here to figure out the expected number of people who would get sick or not sick. And so over here we have 80 out of 380 did not get sick. And I want to be careful. I just said the word population, but we haven't sampled the whole universe of all people taking this herb. This is a sample. I don't want to confuse you."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I want to be careful. I just said the word population, but we haven't sampled the whole universe of all people taking this herb. This is a sample. I don't want to confuse you. I was using the population in more of the conversational sense than the statistical sense. But anyway, of our sample, and we're using all of the data, because we're assuming there's no difference. We might as well just use the total data to figure out the expected frequency of getting sick and not getting sick."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "I don't want to confuse you. I was using the population in more of the conversational sense than the statistical sense. But anyway, of our sample, and we're using all of the data, because we're assuming there's no difference. We might as well just use the total data to figure out the expected frequency of getting sick and not getting sick. So 80 divided by 380 did not get sick, and that's 21%. So let me write that over here. So that's 21% of the total."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "We might as well just use the total data to figure out the expected frequency of getting sick and not getting sick. So 80 divided by 380 did not get sick, and that's 21%. So let me write that over here. So that's 21% of the total. And then this would be 79% if we just subtract 1 minus 21. We could divide 300 by 380. We should get 79% as well."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's 21% of the total. And then this would be 79% if we just subtract 1 minus 21. We could divide 300 by 380. We should get 79% as well. So one would expect that 21% of each of your total, based on the total sample right over here, our best guess is that 21% should be getting sick and 79% should not be getting sick. So let's look at it for each of these groups. If we assume that 21% of these 120 people should have gotten sick, what would have been the expected value right over here?"}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "We should get 79% as well. So one would expect that 21% of each of your total, based on the total sample right over here, our best guess is that 21% should be getting sick and 79% should not be getting sick. So let's look at it for each of these groups. If we assume that 21% of these 120 people should have gotten sick, what would have been the expected value right over here? So let's just multiply this 21% times 120. That gets us to 25.3 people should have gotten sick. So the expected, so let me write it over here."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we assume that 21% of these 120 people should have gotten sick, what would have been the expected value right over here? So let's just multiply this 21% times 120. That gets us to 25.3 people should have gotten sick. So the expected, so let me write it over here. I'll do expected in yellow. So the expected right over here, if you assume that 21% of each group should have gotten sick, is that you would have expected 25.3 people to get sick in group 1, in herb 1 group, and then the remainder will not get sick. So let's just subtract, or I could actually multiply 79% times 120."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the expected, so let me write it over here. I'll do expected in yellow. So the expected right over here, if you assume that 21% of each group should have gotten sick, is that you would have expected 25.3 people to get sick in group 1, in herb 1 group, and then the remainder will not get sick. So let's just subtract, or I could actually multiply 79% times 120. Either one of those will be good. But let me just take 120 minus 25.3, and then I get 94.7. So you would have expected 94.7 to not get sick."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just subtract, or I could actually multiply 79% times 120. Either one of those will be good. But let me just take 120 minus 25.3, and then I get 94.7. So you would have expected 94.7 to not get sick. So this is expected again. 94.7 to not get sick. And now let's do that for each of these groups."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you would have expected 94.7 to not get sick. So this is expected again. 94.7 to not get sick. And now let's do that for each of these groups. So once again, group 2, you would have expected 21% to get sick, 21% of the total people in that group, so that's 140. So that's 29.4. And then the remainder, 140 minus 29.4, should not have gotten sick."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now let's do that for each of these groups. So once again, group 2, you would have expected 21% to get sick, 21% of the total people in that group, so that's 140. So that's 29.4. And then the remainder, 140 minus 29.4, should not have gotten sick. So that gets us this right here. We have 20, 29.4 should have gotten sick if the herbs did nothing. And then over here, we would have 110.6 should not have gotten sick."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the remainder, 140 minus 29.4, should not have gotten sick. So that gets us this right here. We have 20, 29.4 should have gotten sick if the herbs did nothing. And then over here, we would have 110.6 should not have gotten sick. And these are pretty close. So just looking at the numbers, it looks like this herb doesn't do too much relative to all of the groups combined. And then in the placebo group, let's see what happens."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then over here, we would have 110.6 should not have gotten sick. And these are pretty close. So just looking at the numbers, it looks like this herb doesn't do too much relative to all of the groups combined. And then in the placebo group, let's see what happens. You have 30, sorry, we expect 21% to get sick, 21% of our group of 120, so it's 25.2. So this right over here. And actually, I should make this, this should be a 25 point, since we're rounding."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then in the placebo group, let's see what happens. You have 30, sorry, we expect 21% to get sick, 21% of our group of 120, so it's 25.2. So this right over here. And actually, I should make this, this should be a 25 point, since we're rounding. Actually, these will be the same number over here. So I said 21%, but it's 21 point something, something, something. The group sizes are the same."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, I should make this, this should be a 25 point, since we're rounding. Actually, these will be the same number over here. So I said 21%, but it's 21 point something, something, something. The group sizes are the same. So we should expect the same proportion to get sick. So I'll say 25.3, just to make it consistent. The reason why I got 25.2 just now is because I lost some of the trailing decimals over here."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "The group sizes are the same. So we should expect the same proportion to get sick. So I'll say 25.3, just to make it consistent. The reason why I got 25.2 just now is because I lost some of the trailing decimals over here. But since I had them over here, I'm going to use them over here as well. And then over here, in this group, you would expect 94.7 to get sick. So if you just actually relied on this data, it looks like ERB2 is actually, to some degree, even worse than the, oh, no, no, I take that back."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "The reason why I got 25.2 just now is because I lost some of the trailing decimals over here. But since I had them over here, I'm going to use them over here as well. And then over here, in this group, you would expect 94.7 to get sick. So if you just actually relied on this data, it looks like ERB2 is actually, to some degree, even worse than the, oh, no, no, I take that back. It's not worse, because you would have expected a small number, and a lot of people got sick here. So this is the placebo. Well, anyway, we don't want to make judgments just staring at the numbers."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you just actually relied on this data, it looks like ERB2 is actually, to some degree, even worse than the, oh, no, no, I take that back. It's not worse, because you would have expected a small number, and a lot of people got sick here. So this is the placebo. Well, anyway, we don't want to make judgments just staring at the numbers. Let's figure out our chi-squared statistic. And to do that, let's get our statistic, our chi-squared statistic. I'll write it like this, maybe for fun."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, anyway, we don't want to make judgments just staring at the numbers. Let's figure out our chi-squared statistic. And to do that, let's get our statistic, our chi-squared statistic. I'll write it like this, maybe for fun. Or maybe I'll write it as a big X, because this random variable's distribution is approximately a chi-squared distribution. So I'll write it like that. And we'll talk about the degrees of freedom in a second."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll write it like this, maybe for fun. Or maybe I'll write it as a big X, because this random variable's distribution is approximately a chi-squared distribution. So I'll write it like that. And we'll talk about the degrees of freedom in a second. Actually, let me write it with a curly X, just so you see that some people write it with the chi instead of the X. So our chi-squared statistic over here, we're literally just going to find the squared distance between the observed and expected, and then divide it by the expected. So this is going to be 20 minus 25.3 squared over 25.3 plus 30 minus 29.4 squared over 29.4."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we'll talk about the degrees of freedom in a second. Actually, let me write it with a curly X, just so you see that some people write it with the chi instead of the X. So our chi-squared statistic over here, we're literally just going to find the squared distance between the observed and expected, and then divide it by the expected. So this is going to be 20 minus 25.3 squared over 25.3 plus 30 minus 29.4 squared over 29.4. I'm going to run out of space. Plus 30 minus 25.3 squared over 25.3. And then I'm going to have to do these over here."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be 20 minus 25.3 squared over 25.3 plus 30 minus 29.4 squared over 29.4. I'm going to run out of space. Plus 30 minus 25.3 squared over 25.3. And then I'm going to have to do these over here. So let me just continue it. You can ignore this h1 over here. So plus 100 minus 94.7 squared over 94.7 plus, I think you see where this is going, 110 minus 110.6 squared over 110.6."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then I'm going to have to do these over here. So let me just continue it. You can ignore this h1 over here. So plus 100 minus 94.7 squared over 94.7 plus, I think you see where this is going, 110 minus 110.6 squared over 110.6. And then finally, plus 90 over 94. Sorry, 90 minus 94.7 squared. All of that over 94.7."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So plus 100 minus 94.7 squared over 94.7 plus, I think you see where this is going, 110 minus 110.6 squared over 110.6. And then finally, plus 90 over 94. Sorry, 90 minus 94.7 squared. All of that over 94.7. So let me just get the calculator out to calculate this, take a little bit of time. So we have, I'll have to type it on the calculator for these parentheses. So we have 20 minus 25.3 squared divided by 25.3 plus, open parentheses, 30 minus 29.4 squared divided by 29.4 plus, open parentheses, 30 minus 25.3 squared divided by 25.3, halfway there, plus 100, open parentheses, this is the tedious part, 100 minus 94.7 squared divided by 94.7 plus, 110 minus, well this will, I'll actually type it out, we can do a lot of these in our head, but let me just do it."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of that over 94.7. So let me just get the calculator out to calculate this, take a little bit of time. So we have, I'll have to type it on the calculator for these parentheses. So we have 20 minus 25.3 squared divided by 25.3 plus, open parentheses, 30 minus 29.4 squared divided by 29.4 plus, open parentheses, 30 minus 25.3 squared divided by 25.3, halfway there, plus 100, open parentheses, this is the tedious part, 100 minus 94.7 squared divided by 94.7 plus, 110 minus, well this will, I'll actually type it out, we can do a lot of these in our head, but let me just do it. 110 minus 110.6 squared divided by 110.6. And then last one, home stretch, assuming we haven't made any mistakes. We have 90 minus 94.7 squared divided by 94.7."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have 20 minus 25.3 squared divided by 25.3 plus, open parentheses, 30 minus 29.4 squared divided by 29.4 plus, open parentheses, 30 minus 25.3 squared divided by 25.3, halfway there, plus 100, open parentheses, this is the tedious part, 100 minus 94.7 squared divided by 94.7 plus, 110 minus, well this will, I'll actually type it out, we can do a lot of these in our head, but let me just do it. 110 minus 110.6 squared divided by 110.6. And then last one, home stretch, assuming we haven't made any mistakes. We have 90 minus 94.7 squared divided by 94.7. And let's see what we get. We get 2.528. So let's just say it's 2.53."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 90 minus 94.7 squared divided by 94.7. And let's see what we get. We get 2.528. So let's just say it's 2.53. So our chi-square statistic, I always have trouble saying that, our chi-square statistic, assuming the null hypothesis is correct, is equal to 2.53. Now, the next thing we have to do is figure out the degrees of freedom that we had in calculating this chi-square statistic. And I'll give you the rule of thumb, and I'll give you a little bit of a sense of why this is the rule of thumb for contingency table like this."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just say it's 2.53. So our chi-square statistic, I always have trouble saying that, our chi-square statistic, assuming the null hypothesis is correct, is equal to 2.53. Now, the next thing we have to do is figure out the degrees of freedom that we had in calculating this chi-square statistic. And I'll give you the rule of thumb, and I'll give you a little bit of a sense of why this is the rule of thumb for contingency table like this. And in the future, we'll talk a little bit more deeply about degrees of freedom. So when you do, the rule of thumb for contingency table is you have the number of rows, so you have rows, and then you have your number of columns. So here we have two rows and we have three columns, you don't count the totals."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll give you the rule of thumb, and I'll give you a little bit of a sense of why this is the rule of thumb for contingency table like this. And in the future, we'll talk a little bit more deeply about degrees of freedom. So when you do, the rule of thumb for contingency table is you have the number of rows, so you have rows, and then you have your number of columns. So here we have two rows and we have three columns, you don't count the totals. So you have three columns over here. And the degrees of freedom, and this is the rule of thumb, the degrees of freedom for your contingency table is going to be the number of rows minus 1 times the number of columns minus 1. In our situation, we have two rows and three columns, so it's going to be 2 minus 1 times 3 minus 1."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here we have two rows and we have three columns, you don't count the totals. So you have three columns over here. And the degrees of freedom, and this is the rule of thumb, the degrees of freedom for your contingency table is going to be the number of rows minus 1 times the number of columns minus 1. In our situation, we have two rows and three columns, so it's going to be 2 minus 1 times 3 minus 1. So it's going to be 2 minus 1 times 3 minus 1, which is just 1 times 2, which is 2. We have two degrees of freedom. Now, the reason that that should make a little bit of intuitive sense, we'll talk about this in more depth in the future, is that if you assume that you know the totals, so let's just assume that you know the totals."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "In our situation, we have two rows and three columns, so it's going to be 2 minus 1 times 3 minus 1. So it's going to be 2 minus 1 times 3 minus 1, which is just 1 times 2, which is 2. We have two degrees of freedom. Now, the reason that that should make a little bit of intuitive sense, we'll talk about this in more depth in the future, is that if you assume that you know the totals, so let's just assume that you know the totals. So if you know all of this information over here, if you know the total information, or actually if you knew the parameters of the population as well, but if you know the total information and if you know this information, or if you know r minus 1 of the information in the rows, the last one can be figured out just by subtracting from the total. So for example, in this situation, if you know this, you can easily figure out this. This is not new information."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, the reason that that should make a little bit of intuitive sense, we'll talk about this in more depth in the future, is that if you assume that you know the totals, so let's just assume that you know the totals. So if you know all of this information over here, if you know the total information, or actually if you knew the parameters of the population as well, but if you know the total information and if you know this information, or if you know r minus 1 of the information in the rows, the last one can be figured out just by subtracting from the total. So for example, in this situation, if you know this, you can easily figure out this. This is not new information. It's just the total minus 20. Same thing. If you know this one right over here, this one over here is not new information."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is not new information. It's just the total minus 20. Same thing. If you know this one right over here, this one over here is not new information. And similarly, if you know these two, this guy over here isn't new information. You can always just calculate him based on the total and everything else. So that's the sense of why our degrees of freedom are the columns minus 1 times the rows minus 1."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you know this one right over here, this one over here is not new information. And similarly, if you know these two, this guy over here isn't new information. You can always just calculate him based on the total and everything else. So that's the sense of why our degrees of freedom are the columns minus 1 times the rows minus 1. But anyway, so our chi-square statistic has 2 degrees of freedom. So what we have to do is, remember, our alpha value, let me get it up here. We had it right over here."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's the sense of why our degrees of freedom are the columns minus 1 times the rows minus 1. But anyway, so our chi-square statistic has 2 degrees of freedom. So what we have to do is, remember, our alpha value, let me get it up here. We had it right over here. Our significance level that we care about, our alpha value is 10%. Let me rewrite it over here. So our alpha is 10%."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "We had it right over here. Our significance level that we care about, our alpha value is 10%. Let me rewrite it over here. So our alpha is 10%. So what we're going to do is figure out what is our critical chi-square statistic that gives us an alpha of 10%. If this is more extreme than that, if the probability of getting this is even less than that critical statistic, it will be less than 10% and we'll reject the null hypothesis. If it's not more extreme, then we won't reject the null hypothesis."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our alpha is 10%. So what we're going to do is figure out what is our critical chi-square statistic that gives us an alpha of 10%. If this is more extreme than that, if the probability of getting this is even less than that critical statistic, it will be less than 10% and we'll reject the null hypothesis. If it's not more extreme, then we won't reject the null hypothesis. So what we need to do is to figure out with a chi-square distribution and 2 degrees of freedom, what is our critical chi-square statistic? So let's just go back. So we have 2 degrees of freedom here and we care about a significance level of 10%."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's not more extreme, then we won't reject the null hypothesis. So what we need to do is to figure out with a chi-square distribution and 2 degrees of freedom, what is our critical chi-square statistic? So let's just go back. So we have 2 degrees of freedom here and we care about a significance level of 10%. So our critical chi-square value is 4.60. So another way to visualize this, if we look at the chi- square distribution with 2 degrees of freedom, that's this blue one over here. I'm trying to pick a nice blue to use."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have 2 degrees of freedom here and we care about a significance level of 10%. So our critical chi-square value is 4.60. So another way to visualize this, if we look at the chi- square distribution with 2 degrees of freedom, that's this blue one over here. I'm trying to pick a nice blue to use. At a value of a critical value of 4.60, so 4.60, this is 5, so 4.60 will be right around here. At a critical value of 4.60, so this is 4.60, the probability of getting something at least that extreme, so that extreme or more extreme, is 10%. This is what we care about."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm trying to pick a nice blue to use. At a value of a critical value of 4.60, so 4.60, this is 5, so 4.60 will be right around here. At a critical value of 4.60, so this is 4.60, the probability of getting something at least that extreme, so that extreme or more extreme, is 10%. This is what we care about. Now, if the chi-square distribution is less than 10%, the chi-square statistic that we calculated falls into this rejection region, then we're going to reject the null hypothesis. But our chi-square statistic is only 2.53. So it's sitting someplace right over here."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is what we care about. Now, if the chi-square distribution is less than 10%, the chi-square statistic that we calculated falls into this rejection region, then we're going to reject the null hypothesis. But our chi-square statistic is only 2.53. So it's sitting someplace right over here. It's actually ours. So it's not that crazy to get it if you assume the null hypothesis. So based on our data that we have right now, we cannot reject the null hypothesis."}, {"video_title": "Contingency table chi-square test   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's sitting someplace right over here. It's actually ours. So it's not that crazy to get it if you assume the null hypothesis. So based on our data that we have right now, we cannot reject the null hypothesis. So we don't know for a fact that the herbs do nothing, but we can't say that they do something based on this. So we're not going to reject it. We won't say 100% that it's true, but we can't say that we're rejecting it."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the last video, we figured out the probability of getting exactly 3 heads when we have 5 flips of a fair coin. What I want to do in this video is think about it in a slightly more general way. We're still going to assume we have a fair coin, although we'll shortly see we don't have to make that assumption. What I want to do is figure out the probability of getting k heads in n flips of the fair coin. The first thing to think about is how many possibilities there are. There's going to be n flips, so literally there's first flip, second flip, third flip, all the way to the nth flip. This is a fair coin."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I want to do is figure out the probability of getting k heads in n flips of the fair coin. The first thing to think about is how many possibilities there are. There's going to be n flips, so literally there's first flip, second flip, third flip, all the way to the nth flip. This is a fair coin. Each of these has two equally likely possibilities. The total number of possibilities is going to be 2 times 2 times 2 n times. This is going to be equal to 2 to the nth possibilities."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is a fair coin. Each of these has two equally likely possibilities. The total number of possibilities is going to be 2 times 2 times 2 n times. This is going to be equal to 2 to the nth possibilities. Let's think about how many of those equally likely possibilities involve k heads. Just like we did for the case where we're thinking about 3 heads, we say, well look, the first of those k heads, how many different buckets could it fall into? The first of the k heads could fall into n different buckets."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be equal to 2 to the nth possibilities. Let's think about how many of those equally likely possibilities involve k heads. Just like we did for the case where we're thinking about 3 heads, we say, well look, the first of those k heads, how many different buckets could it fall into? The first of the k heads could fall into n different buckets. It could be the first flip, second flip, all the way to the nth flip. Then the second of those k heads, we don't know exactly how many k is, will have n minus 1 possibilities. The third of those k heads will have n minus 2 possibilities, since two of the spots are already taken up."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "The first of the k heads could fall into n different buckets. It could be the first flip, second flip, all the way to the nth flip. Then the second of those k heads, we don't know exactly how many k is, will have n minus 1 possibilities. The third of those k heads will have n minus 2 possibilities, since two of the spots are already taken up. We would do this until we have essentially accounted for all of the k heads. This will go down all the way to, we will multiply the number of things we're multiplying is going to be k. One for each of the k heads. This is 1, 2, 3, and then you're going to go all the way to n minus k minus 1."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "The third of those k heads will have n minus 2 possibilities, since two of the spots are already taken up. We would do this until we have essentially accounted for all of the k heads. This will go down all the way to, we will multiply the number of things we're multiplying is going to be k. One for each of the k heads. This is 1, 2, 3, and then you're going to go all the way to n minus k minus 1. You could try this out in the case of 5. When n was 5 and k was 3, this resulted in 5 times 4 times 3. That was this term right over here."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 1, 2, 3, and then you're going to go all the way to n minus k minus 1. You could try this out in the case of 5. When n was 5 and k was 3, this resulted in 5 times 4 times 3. That was this term right over here. I'm doing a case that is a little bit longer, where k is a slightly larger number, because this right over here is 5 minus 2. That is this one over here. Just not to confuse you, let me write it like this."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That was this term right over here. I'm doing a case that is a little bit longer, where k is a slightly larger number, because this right over here is 5 minus 2. That is this one over here. Just not to confuse you, let me write it like this. You'll have n spots for that first head, n minus 1 spots for that second head. Then you keep going and you're going to have a total of these k things you're multiplying. That last one is going to have n minus k minus 1 possibilities."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just not to confuse you, let me write it like this. You'll have n spots for that first head, n minus 1 spots for that second head. Then you keep going and you're going to have a total of these k things you're multiplying. That last one is going to have n minus k minus 1 possibilities. Now hopefully it will map a little bit better to the one where we had 5 flips and 3 heads. Here there was 5 possibilities for the first head, 4 possibilities for the second head. Then n is 5, 5 minus 2, you had 3 possibilities for the last head."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That last one is going to have n minus k minus 1 possibilities. Now hopefully it will map a little bit better to the one where we had 5 flips and 3 heads. Here there was 5 possibilities for the first head, 4 possibilities for the second head. Then n is 5, 5 minus 2, you had 3 possibilities for the last head. This actually works. This is the number of spots or the number of ways that we can put 3 heads into 5 different possible buckets. Just like the last video, we don't want to over count things, because we don't care about the order."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then n is 5, 5 minus 2, you had 3 possibilities for the last head. This actually works. This is the number of spots or the number of ways that we can put 3 heads into 5 different possible buckets. Just like the last video, we don't want to over count things, because we don't care about the order. We don't want to differentiate one ordering of heads. I'm just going to use these letters to differentiate the different heads. We don't want to differentiate this from this."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just like the last video, we don't want to over count things, because we don't care about the order. We don't want to differentiate one ordering of heads. I'm just going to use these letters to differentiate the different heads. We don't want to differentiate this from this. Heads A, heads B, or any of the other orderings of this. What we need to do is divide this number so that we don't count all of those different orderings. We need to divide this by the different ways that you can order k things."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't want to differentiate this from this. Heads A, heads B, or any of the other orderings of this. What we need to do is divide this number so that we don't count all of those different orderings. We need to divide this by the different ways that you can order k things. If you have k things, how many ways can you order it? The first thing can be in k different positions. Maybe I'll do it like this."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We need to divide this by the different ways that you can order k things. If you have k things, how many ways can you order it? The first thing can be in k different positions. Maybe I'll do it like this. Maybe I'll do it T for thing. Thing 1, thing 2, thing 3, all the way to thing k. How many different ways can you order it? Thing 1 can be in k different positions."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe I'll do it like this. Maybe I'll do it T for thing. Thing 1, thing 2, thing 3, all the way to thing k. How many different ways can you order it? Thing 1 can be in k different positions. Thing 2 will be in k minus 1 positions. Then all the way down to the last one is only going to have one position. This is going to be k times k minus 1 times k minus 2, all the way down to 1."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Thing 1 can be in k different positions. Thing 2 will be in k minus 1 positions. Then all the way down to the last one is only going to have one position. This is going to be k times k minus 1 times k minus 2, all the way down to 1. In the example where we had 3 heads in 5 flips, this was 3 times 2 all the way down to 1. 3 times 2 times 1. Is there a simpler way that we can write this?"}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be k times k minus 1 times k minus 2, all the way down to 1. In the example where we had 3 heads in 5 flips, this was 3 times 2 all the way down to 1. 3 times 2 times 1. Is there a simpler way that we can write this? This expression right over here is the same thing as k factorial. If you haven't ever heard of what a factorial is, it's exactly this thing right over here. k factorial literally means k times k minus 1 times k minus 2, all the way down to 1."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is there a simpler way that we can write this? This expression right over here is the same thing as k factorial. If you haven't ever heard of what a factorial is, it's exactly this thing right over here. k factorial literally means k times k minus 1 times k minus 2, all the way down to 1. For example, 2 factorial is equal to 2 times 1. 3 factorial is equal to 3 times 2 times 1. 4 factorial is equal to 4 times 3 times 2 times 1."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "k factorial literally means k times k minus 1 times k minus 2, all the way down to 1. For example, 2 factorial is equal to 2 times 1. 3 factorial is equal to 3 times 2 times 1. 4 factorial is equal to 4 times 3 times 2 times 1. It's actually a fun thing to play with. Factorials get large very, very, very, very fast. Anyway, this denominator right over here can be rewritten as k factorial."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "4 factorial is equal to 4 times 3 times 2 times 1. It's actually a fun thing to play with. Factorials get large very, very, very, very fast. Anyway, this denominator right over here can be rewritten as k factorial. This right over here can be rewritten as k factorial. Is there any way to rewrite this numerator in terms of factorials? If we were to write n factorial, let me see how we can write this."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Anyway, this denominator right over here can be rewritten as k factorial. This right over here can be rewritten as k factorial. Is there any way to rewrite this numerator in terms of factorials? If we were to write n factorial, let me see how we can write this. If we were to write n factorial, n factorial would be equal to n times n minus 1 times n minus 2, all the way down to 1. That's kind of what we want. We just want the first k terms of it."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we were to write n factorial, let me see how we can write this. If we were to write n factorial, n factorial would be equal to n times n minus 1 times n minus 2, all the way down to 1. That's kind of what we want. We just want the first k terms of it. What if we divide this by n minus k factorial? Let's think about what that is going to do. If we have n minus k factorial, that is the same thing as n minus k times n minus k minus 1 times n minus k times n minus k minus 2, all the way down to 1."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We just want the first k terms of it. What if we divide this by n minus k factorial? Let's think about what that is going to do. If we have n minus k factorial, that is the same thing as n minus k times n minus k minus 1 times n minus k times n minus k minus 2, all the way down to 1. When you divide these, the 1's are going to cancel out. What you may or may not realize, and you can work out the math, is everything is going to cancel out here until you're just left with up here, everything from n times n minus 1 to n minus k minus 1. If you expand this out or if you distribute this negative number, this is the same thing as n minus k plus 1. n minus k plus 1 is the integer that's 1 larger than this right over here."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we have n minus k factorial, that is the same thing as n minus k times n minus k minus 1 times n minus k times n minus k minus 2, all the way down to 1. When you divide these, the 1's are going to cancel out. What you may or may not realize, and you can work out the math, is everything is going to cancel out here until you're just left with up here, everything from n times n minus 1 to n minus k minus 1. If you expand this out or if you distribute this negative number, this is the same thing as n minus k plus 1. n minus k plus 1 is the integer that's 1 larger than this right over here. If you were to divide it out, this would cancel with something up here, this would cancel with something up here, this would cancel with something up here. What you're going to be left with is exactly this thing over here. If you don't believe me, we can actually try it out."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you expand this out or if you distribute this negative number, this is the same thing as n minus k plus 1. n minus k plus 1 is the integer that's 1 larger than this right over here. If you were to divide it out, this would cancel with something up here, this would cancel with something up here, this would cancel with something up here. What you're going to be left with is exactly this thing over here. If you don't believe me, we can actually try it out. Let's think about what 5 factorial over 5 minus 3 factorial is going to be. This is going to be 5 times 4 times 3 times 2 times 1. All of that stuff up there, all the way down to 1."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you don't believe me, we can actually try it out. Let's think about what 5 factorial over 5 minus 3 factorial is going to be. This is going to be 5 times 4 times 3 times 2 times 1. All of that stuff up there, all the way down to 1. 5 minus 3 is 2 over 2 factorial, 2 times 1. 2 cancels with 2, 1 cancels with 1. You don't have to worry about that."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of that stuff up there, all the way down to 1. 5 minus 3 is 2 over 2 factorial, 2 times 1. 2 cancels with 2, 1 cancels with 1. You don't have to worry about that. You're just left with 5 times 4 times 3. Exactly what we had up here, 5 times 4 times 3. In general, if you wanted to figure out the number of ways to stick 2 things in 5 chairs and you don't care about differentiating between those 2 things, you're going to have this expression right over here, which is the same thing as this right over here."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "You don't have to worry about that. You're just left with 5 times 4 times 3. Exactly what we had up here, 5 times 4 times 3. In general, if you wanted to figure out the number of ways to stick 2 things in 5 chairs and you don't care about differentiating between those 2 things, you're going to have this expression right over here, which is the same thing as this right over here. You're going to have n factorial over n minus k factorial. Then you're going to divide it by this expression right over here, which we already said is the same thing as k factorial. You're also going to divide it by k factorial."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "In general, if you wanted to figure out the number of ways to stick 2 things in 5 chairs and you don't care about differentiating between those 2 things, you're going to have this expression right over here, which is the same thing as this right over here. You're going to have n factorial over n minus k factorial. Then you're going to divide it by this expression right over here, which we already said is the same thing as k factorial. You're also going to divide it by k factorial. Then you have a generalized way of figuring out the number of ways you can stick 2 things, or the number of ways you can stick k things in n different buckets, k heads in n different flips. This is actually a generalized formula for binomial coefficients. Another way to write this is the number of ways, given that you have n buckets, you can put k things in them without having to differentiate it."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're also going to divide it by k factorial. Then you have a generalized way of figuring out the number of ways you can stick 2 things, or the number of ways you can stick k things in n different buckets, k heads in n different flips. This is actually a generalized formula for binomial coefficients. Another way to write this is the number of ways, given that you have n buckets, you can put k things in them without having to differentiate it. Another way to think about it is if you have n buckets or n flips and you want to choose k of them to be heads, or you want to choose k of them in some way but you don't want to differentiate. All of these are generalized ways for binomial coefficients. Going back to the original problem, what is the probability of getting k heads in n flips of the fair coin?"}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Another way to write this is the number of ways, given that you have n buckets, you can put k things in them without having to differentiate it. Another way to think about it is if you have n buckets or n flips and you want to choose k of them to be heads, or you want to choose k of them in some way but you don't want to differentiate. All of these are generalized ways for binomial coefficients. Going back to the original problem, what is the probability of getting k heads in n flips of the fair coin? There's 2 to the n equally likely possibilities. Let's write this down. The probability of 2 to the n equally likely possibilities, and how many of those possibilities result in exactly k heads?"}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Going back to the original problem, what is the probability of getting k heads in n flips of the fair coin? There's 2 to the n equally likely possibilities. Let's write this down. The probability of 2 to the n equally likely possibilities, and how many of those possibilities result in exactly k heads? We just figured that out during this video. That's the number of possibilities. n factorial over k factorial times n minus k factorial."}, {"video_title": "Generalizing with binomial coefficients (bit advanced)   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability of 2 to the n equally likely possibilities, and how many of those possibilities result in exactly k heads? We just figured that out during this video. That's the number of possibilities. n factorial over k factorial times n minus k factorial. It probably is an okay idea to memorize this, but I'll just tell you frankly, the only reason why I still know how to do this 20 years after first seeing it, or whenever I first saw it, is that I actually just like to reason through it every time. I like to just reason through, okay, I've got 5 flips, 3 of them need to be heads, the first of those heads can be in 5 different buckets, the next in 4 different buckets, the next one in 3 different buckets, and then of course I don't want to differentiate between all of the different ways that I can rearrange 3 different things, so I have to make sure that I divide it by 3 factorial, by 3 times 2 times 1. I want to make sure that I divide it by all of the different ways that I can arrange 3 different things."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "All right, so here we have the different names of the drinks, and then here we have the type of the drink, and it looks like they're either hot or cold. Here we have the calories for each of those drinks. Here we have the sugar content in grams for each of those drinks, and here we have the caffeine in milligrams for each of those drinks. And then we are asked, the individuals in this data set are, and we have three choices, Ben's Beans customers, Ben's Beans drinks, or the caffeine contents. Now, we have to be careful. When someone says the individuals in a data set, they don't necessarily mean that they have to be people. They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "And then we are asked, the individuals in this data set are, and we have three choices, Ben's Beans customers, Ben's Beans drinks, or the caffeine contents. Now, we have to be careful. When someone says the individuals in a data set, they don't necessarily mean that they have to be people. They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. So the different types of drinks that Ben's Beans offers, those are the individuals in this data set. So they're Ben's Beans drinks. Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. So the different types of drinks that Ben's Beans offers, those are the individuals in this data set. So they're Ben's Beans drinks. Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. So if we look up here, let's look at the variables. So this first column that is essentially giving us the type of drink, this wouldn't be a variable. This would be more of an identifier, but all of these other columns are representing variables."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. So if we look up here, let's look at the variables. So this first column that is essentially giving us the type of drink, this wouldn't be a variable. This would be more of an identifier, but all of these other columns are representing variables. So for example, type is a variable. It can either be hot or cold. And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "This would be more of an identifier, but all of these other columns are representing variables. So for example, type is a variable. It can either be hot or cold. And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. And you don't just have two categories. You could have more than two categories. But it isn't just some type of variable number that can take on a bunch of different values."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. And you don't just have two categories. You could have more than two categories. But it isn't just some type of variable number that can take on a bunch of different values. So this right over here is a categorical variable. Calories is not a categorical variable. You could have something with 4.1 calories."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set   Khan Academy.mp3", "Sentence": "But it isn't just some type of variable number that can take on a bunch of different values. So this right over here is a categorical variable. Calories is not a categorical variable. You could have something with 4.1 calories. You could have something with 178. Things aren't fitting into nice buckets. Same thing for sugars and for the caffeine, that these are quantitative variables that don't just fit into a category."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "What we are going to do in this video is talk about the idea of power when we are dealing with significance tests. And power is an idea that you might encounter in a first-year statistics course. It turns out that it's fairly difficult to calculate, but it's interesting to know what it means and what are the levers that might increase the power or decrease the power in a significance test. So just to cut to the chase, power is a probability. You can view it as the probability that you're doing the right thing when the null hypothesis is not true. And the right thing is you should reject the null hypothesis if it's not true. So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So just to cut to the chase, power is a probability. You can view it as the probability that you're doing the right thing when the null hypothesis is not true. And the right thing is you should reject the null hypothesis if it's not true. So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So you could view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to type two errors."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So you could view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to type two errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a type, type two error. So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "We can connect it to type two errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a type, type two error. So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making, not making a type, type two error."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making, not making a type, type two error. So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions. One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So you could say it's the probability of not making, not making a type, type two error. So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions. One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one. So if you assumed a world where the null hypothesis is true, so I'll do that in blue. So if we assume the null hypothesis is true, what would be our sampling distribution?"}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one. So if you assumed a world where the null hypothesis is true, so I'll do that in blue. So if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population, let me draw that. You have a population right over here, and our hypotheses are making some statement about a parameter in that population, and to test it, we take a sample of a certain size, we calculate a statistic, in this case, we would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population, let me draw that. You have a population right over here, and our hypotheses are making some statement about a parameter in that population, and to test it, we take a sample of a certain size, we calculate a statistic, in this case, we would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. The null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. The null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error. So your significance level is some area, and so let's say it's this area that I'm shading in orange right over here, that would be your significance level. So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error. So your significance level is some area, and so let's say it's this area that I'm shading in orange right over here, that would be your significance level. So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it. But for power, we are concerned with a type two error. So in this one, it's a conditional probability that our null hypothesis is false."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it. But for power, we are concerned with a type two error. So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. And let's say that mu two is right over here. And in this reality, our sampling distribution might look something like this. Once again, it'll be for a given sample size."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. And let's say that mu two is right over here. And in this reality, our sampling distribution might look something like this. Once again, it'll be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation?"}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, it'll be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation? So in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So in which situation? So in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. A sample where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a type two error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. A sample where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a type two error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, remember the significance level is an area."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, remember the significance level is an area. So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it. Now you might say, oh, well, hey, if we wanna increase the power, power sounds like a good thing. Why don't we just always increase alpha?"}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "If we increase our significance level, say from that, remember the significance level is an area. So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it. Now you might say, oh, well, hey, if we wanna increase the power, power sounds like a good thing. Why don't we just always increase alpha? Well, the problem with that is if you increase alpha, so let me write this down. So if you take alpha, your significance level, and you increase it, that will increase the power, that will increase the power, but it's also going to increase your probability of a type one error. Because remember, that's one way to conceptualize what alpha is, what your significance level is."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Why don't we just always increase alpha? Well, the problem with that is if you increase alpha, so let me write this down. So if you take alpha, your significance level, and you increase it, that will increase the power, that will increase the power, but it's also going to increase your probability of a type one error. Because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a type one error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a type one error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power. And this, in general, is always a good thing if you can do it. Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power. And this, in general, is always a good thing if you can do it. Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. So these two are not typically under your control, but the sample size is, and the significance level is. Significance level, there's a trade-off, though. If you increase the power through that, you're also increasing the probability of a type one error."}, {"video_title": "Introduction to power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. So these two are not typically under your control, but the sample size is, and the significance level is. Significance level, there's a trade-off, though. If you increase the power through that, you're also increasing the probability of a type one error. So for a lot of researchers, they might say, hey, if a type two error is worse, I'm willing to make this trade-off. I'll increase the significance level. But if a type one error is actually what I'm afraid of, then I wouldn't wanna use this lever."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "And then each of these pairs of bar charts give us for an individual, where the blue bar is, for example, how Ishan did on the midterm. The yellow is how he did on the final. For Emily, the blue is how she did on the midterm. Yellow is how she did on the final. And we have a bunch of interesting questions here. The first question, what was the median score for the final exam? So just as a review, median literally means what was the middle score."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Yellow is how she did on the final. And we have a bunch of interesting questions here. The first question, what was the median score for the final exam? So just as a review, median literally means what was the middle score. So really, we should list all the scores for the final exam and sort them in order, and then figure out what the middle score actually was. So let's look at all the scores on final exams. So scores on the final exam."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So just as a review, median literally means what was the middle score. So really, we should list all the scores for the final exam and sort them in order, and then figure out what the middle score actually was. So let's look at all the scores on final exams. So scores on the final exam. So you have 100 here. Ishan got 100 on the final exam. Remember, this yellow bar is the final exam."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So scores on the final exam. So you have 100 here. Ishan got 100 on the final exam. Remember, this yellow bar is the final exam. So there's 100. Emily also got 100 on the final exam. Looks like it was an easy final exam."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Remember, this yellow bar is the final exam. So there's 100. Emily also got 100 on the final exam. Looks like it was an easy final exam. Daniel also got 100 on the final exam. And then let's see. Jessica looks like she got a 75."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Looks like it was an easy final exam. Daniel also got 100 on the final exam. And then let's see. Jessica looks like she got a 75. And then William got an 80. So if we were to sort these in order, and let's say we did it in increasing order, you could write, well, the lowest score was a 75. Then you have an 80."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Jessica looks like she got a 75. And then William got an 80. So if we were to sort these in order, and let's say we did it in increasing order, you could write, well, the lowest score was a 75. Then you have an 80. And then you have three 100s. 100, another 100, and another 100. So there's five scores right over here."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Then you have an 80. And then you have three 100s. 100, another 100, and another 100. So there's five scores right over here. So you will have a middle. If you had an even number, then you would take the mean of the two center values. But here you have one center value."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So there's five scores right over here. So you will have a middle. If you had an even number, then you would take the mean of the two center values. But here you have one center value. And when you order it like this, it's pretty clear that your center value, your middle value, is 100. So the median score for the final exam is 100. And that's because you had so many 100s here that the median, the middle score, was still 100."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "But here you have one center value. And when you order it like this, it's pretty clear that your center value, your middle value, is 100. So the median score for the final exam is 100. And that's because you had so many 100s here that the median, the middle score, was still 100. What is the midrange of the midterm scores? I'll do it in blue in honor of the color of the bars for the midterm. So the midrange is the mean of your highest and lowest scores."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "And that's because you had so many 100s here that the median, the middle score, was still 100. What is the midrange of the midterm scores? I'll do it in blue in honor of the color of the bars for the midterm. So the midrange is the mean of your highest and lowest scores. So let's calculate this. So let's go to the midrange. The midrange, you could view it as the arithmetic mean or the average of your highest and lowest scores."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So the midrange is the mean of your highest and lowest scores. So let's calculate this. So let's go to the midrange. The midrange, you could view it as the arithmetic mean or the average of your highest and lowest scores. So the midrange of midterm. So let's see. The highest midterm score, looking at the blue, the highest one is right here."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "The midrange, you could view it as the arithmetic mean or the average of your highest and lowest scores. So the midrange of midterm. So let's see. The highest midterm score, looking at the blue, the highest one is right here. So Jessica got 100 on the midterm. So that's your highest score. Your lowest score on the midterm looks like this one right over here."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "The highest midterm score, looking at the blue, the highest one is right here. So Jessica got 100 on the midterm. So that's your highest score. Your lowest score on the midterm looks like this one right over here. Daniel got a 60. And so the midrange is going to be the arithmetic mean of these two numbers. So you add 100 plus 60, divide by 2, you get 160 over 2, or 80."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Your lowest score on the midterm looks like this one right over here. Daniel got a 60. And so the midrange is going to be the arithmetic mean of these two numbers. So you add 100 plus 60, divide by 2, you get 160 over 2, or 80. So this right over here is going to be 80. What was the average student score for the final exam? Well, for that, we just have to add up the scores on the final exams and then divide by the number of scores we have."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So you add 100 plus 60, divide by 2, you get 160 over 2, or 80. So this right over here is going to be 80. What was the average student score for the final exam? Well, for that, we just have to add up the scores on the final exams and then divide by the number of scores we have. So we might be able to do that in our heads. Let's see. Well, let me just write it over here."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Well, for that, we just have to add up the scores on the final exams and then divide by the number of scores we have. So we might be able to do that in our heads. Let's see. Well, let me just write it over here. So we have 100 plus 100 plus 100 plus 75 plus 80. So all of that divided by 5, that'll give us our average. If someone tells you average without giving more information, they're probably talking about the arithmetic mean."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "Well, let me just write it over here. So we have 100 plus 100 plus 100 plus 75 plus 80. So all of that divided by 5, that'll give us our average. If someone tells you average without giving more information, they're probably talking about the arithmetic mean. So this is going to be 300 plus another 155. So it's going to be 455 over 5. This is equal to 455 over 5."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "If someone tells you average without giving more information, they're probably talking about the arithmetic mean. So this is going to be 300 plus another 155. So it's going to be 455 over 5. This is equal to 455 over 5. And let's see. 5, this is going to be equal to 5 goes into 450 90 times and into 5 once. So this is going to be equal to 91."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "This is equal to 455 over 5. And let's see. 5, this is going to be equal to 5 goes into 450 90 times and into 5 once. So this is going to be equal to 91. So the average score for the final exam is a 91. What was the mode for the final exam scores? So the mode is the most common score."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So this is going to be equal to 91. So the average score for the final exam is a 91. What was the mode for the final exam scores? So the mode is the most common score. So once again, we've listed all of them. And it's pretty clear that the most common score here is 100. 100 shows up three times, while a 75 only shows up once and an 80 only shows up once."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "So the mode is the most common score. So once again, we've listed all of them. And it's pretty clear that the most common score here is 100. 100 shows up three times, while a 75 only shows up once and an 80 only shows up once. So here, the most common score is 100. It shows up three times. What is the range of the midterm scores?"}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "100 shows up three times, while a 75 only shows up once and an 80 only shows up once. So here, the most common score is 100. It shows up three times. What is the range of the midterm scores? So the range is literally the difference between the highest score and the lowest score. So the highest score, we already figured up. And this is for the midterm."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "What is the range of the midterm scores? So the range is literally the difference between the highest score and the lowest score. So the highest score, we already figured up. And this is for the midterm. So the highest score is 100. And we're going to subtract from that the lowest score, which was a 60. So the range is equal to 40."}, {"video_title": "Reading bar charts putting it together with central tendency   Pre-Algebra   Khan Academy.mp3", "Sentence": "And this is for the midterm. So the highest score is 100. And we're going to subtract from that the lowest score, which was a 60. So the range is equal to 40. The midrange was the average of these two things. The range is just the difference between the two. So the range of the midterm scores is 40."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "Let C be the number of cake orders Liliana receives in a month until she first gets an order over the telephone. Assume the method of placing each cake order is independent. So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it. So there's a couple of ways to approach it."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. What is the probability that C equals one?"}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two?"}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders?"}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that. Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy (2).mp3", "Sentence": "So this is a lot easier to calculate. So let's do that. Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439. So this is equal to 0.3439."}, {"video_title": "Independent or dependent probability event   Precalculus   Khan Academy.mp3", "Sentence": "The marching band is holding a raffle at a football game with two prizes. After the first ticket is pulled out and the winner determined, the ticket is taped to the prize. The next ticket is pulled out to determine the winner of the second prize. Are the two events independent? Explain. Now before we even think about this exact case, let's think about what it means for events to be independent. It means that the outcome of one event doesn't affect the outcome of the other event."}, {"video_title": "Independent or dependent probability event   Precalculus   Khan Academy.mp3", "Sentence": "Are the two events independent? Explain. Now before we even think about this exact case, let's think about what it means for events to be independent. It means that the outcome of one event doesn't affect the outcome of the other event. Now in this situation, the first event, after the first ticket is pulled out and the winner determined, the ticket is taped to the prize. Then the next ticket is pulled out to determine the winner of the second prize. Now, the winner of the second prize, the possible winners, the possible outcomes for the second prize, is dependent on who was pulled out for the first prize."}, {"video_title": "Independent or dependent probability event   Precalculus   Khan Academy.mp3", "Sentence": "It means that the outcome of one event doesn't affect the outcome of the other event. Now in this situation, the first event, after the first ticket is pulled out and the winner determined, the ticket is taped to the prize. Then the next ticket is pulled out to determine the winner of the second prize. Now, the winner of the second prize, the possible winners, the possible outcomes for the second prize, is dependent on who was pulled out for the first prize. You can imagine, if there's three tickets, let's say there's tickets A, B, and C in the bag, and for the first prize, they pull out ticket A. That's for the first prize. Now, when we think about who could be pulled out for the second prize, it's only going to be tickets B or C. Now, the first prize could have gone the other way."}, {"video_title": "Independent or dependent probability event   Precalculus   Khan Academy.mp3", "Sentence": "Now, the winner of the second prize, the possible winners, the possible outcomes for the second prize, is dependent on who was pulled out for the first prize. You can imagine, if there's three tickets, let's say there's tickets A, B, and C in the bag, and for the first prize, they pull out ticket A. That's for the first prize. Now, when we think about who could be pulled out for the second prize, it's only going to be tickets B or C. Now, the first prize could have gone the other way. It could have been A, B, and C. The first prize could have gone to ticket B. And then the possible outcomes for the second prize would be A or C. So the possible outcomes for the second event, for the second prize, are completely dependent, are completely dependent on what happened, or what ticket was pulled out for the first prize. So these are not independent events."}, {"video_title": "Independent or dependent probability event   Precalculus   Khan Academy.mp3", "Sentence": "Now, when we think about who could be pulled out for the second prize, it's only going to be tickets B or C. Now, the first prize could have gone the other way. It could have been A, B, and C. The first prize could have gone to ticket B. And then the possible outcomes for the second prize would be A or C. So the possible outcomes for the second event, for the second prize, are completely dependent, are completely dependent on what happened, or what ticket was pulled out for the first prize. So these are not independent events. The second event, the outcomes for it, are dependent on what happened in the first event. So they are not independent. The way that we could have made them independent is after the first ticket was pulled out, if they just wrote down the name or something, and then put that ticket back in."}, {"video_title": "Independent or dependent probability event   Precalculus   Khan Academy.mp3", "Sentence": "So these are not independent events. The second event, the outcomes for it, are dependent on what happened in the first event. So they are not independent. The way that we could have made them independent is after the first ticket was pulled out, if they just wrote down the name or something, and then put that ticket back in. Instead, they taped it to the prize. But if they put that ticket back in, then the second prize, it would have still had all the tickets there. It wouldn't have mattered who was picked out in the first time, because their name was just written down, but their ticket was put back in, and then you would have been independent."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do a couple of exercises from our probability one module. So we have a bag with nine red marbles, two blue marbles, and three green marbles in it. What is the probability of randomly selecting a non-blue marble from the bag? So let's draw this bag here. So that's my bag, and we're going to assume that it's a transparent bag. That looks like a vase. But we have nine red marbles, so let me draw nine red marbles."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's draw this bag here. So that's my bag, and we're going to assume that it's a transparent bag. That looks like a vase. But we have nine red marbles, so let me draw nine red marbles. One, two, three, four, five, six, seven, eight, nine red marbles. So that looks kind of orange-ish, but it does the job. Two blue marbles."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we have nine red marbles, so let me draw nine red marbles. One, two, three, four, five, six, seven, eight, nine red marbles. So that looks kind of orange-ish, but it does the job. Two blue marbles. So we have one blue marble, two blue marbles. And then we have three green marbles. Let me draw those three."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two blue marbles. So we have one blue marble, two blue marbles. And then we have three green marbles. Let me draw those three. So one, two, three. What is the probability of randomly selecting a non-blue marble from the bag? So maybe we mix them all up, and we have an equal probability of selecting any one of these."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me draw those three. So one, two, three. What is the probability of randomly selecting a non-blue marble from the bag? So maybe we mix them all up, and we have an equal probability of selecting any one of these. And the way you just think about it is, is what fraction of all of the possible events meet our constraints? So let's just think about all of the possible events first. How many different possible marbles can we take out?"}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So maybe we mix them all up, and we have an equal probability of selecting any one of these. And the way you just think about it is, is what fraction of all of the possible events meet our constraints? So let's just think about all of the possible events first. How many different possible marbles can we take out? Well, that's just the total number of marbles there are. So there are one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen possible marbles. So this is the number of possibilities."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "How many different possible marbles can we take out? Well, that's just the total number of marbles there are. So there are one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen possible marbles. So this is the number of possibilities. And then we just have to think, what fraction of those possibilities meet our constraints? And the other way you could have gotten fourteen is just taking nine plus two plus three. So what number of those possibilities meet our constraints?"}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the number of possibilities. And then we just have to think, what fraction of those possibilities meet our constraints? And the other way you could have gotten fourteen is just taking nine plus two plus three. So what number of those possibilities meet our constraints? And remember, our constraint is selecting a non-blue marble from the bag. Another way to think about it is a red or a green marble, because the only non-blue ones, the only other two colors we have are red and green. So how many non-blue marbles are there?"}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what number of those possibilities meet our constraints? And remember, our constraint is selecting a non-blue marble from the bag. Another way to think about it is a red or a green marble, because the only non-blue ones, the only other two colors we have are red and green. So how many non-blue marbles are there? Well, there's a couple of ways to think about it. You could say there's fourteen total marbles. Two are blue, so they're going to be fourteen minus two, which is twelve non-blue marbles."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many non-blue marbles are there? Well, there's a couple of ways to think about it. You could say there's fourteen total marbles. Two are blue, so they're going to be fourteen minus two, which is twelve non-blue marbles. Or you could just count them. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. So there are twelve non-blue marbles."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two are blue, so they're going to be fourteen minus two, which is twelve non-blue marbles. Or you could just count them. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. So there are twelve non-blue marbles. So that's the number of non-blue. So these are the possibilities that meet our constraints over all of the possibilities. And then if we want to, this isn't in simplified form right here, since both twelve and fourteen are divisible by two."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there are twelve non-blue marbles. So that's the number of non-blue. So these are the possibilities that meet our constraints over all of the possibilities. And then if we want to, this isn't in simplified form right here, since both twelve and fourteen are divisible by two. So let's divide both the numerator and the denominator by two, and you get six over seven. So we have a six-seventh chance of selecting a non-blue marble from the bag. Let's do another one."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then if we want to, this isn't in simplified form right here, since both twelve and fourteen are divisible by two. So let's divide both the numerator and the denominator by two, and you get six over seven. So we have a six-seventh chance of selecting a non-blue marble from the bag. Let's do another one. If a number is randomly chosen from the following list, what is the probability that the number is a multiple of five? So once again, we want to find the fraction of the total possibilities that meet our constraint. And our constraint is being a multiple of five."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do another one. If a number is randomly chosen from the following list, what is the probability that the number is a multiple of five? So once again, we want to find the fraction of the total possibilities that meet our constraint. And our constraint is being a multiple of five. So how many total possibilities are there? Let's think about that. Total possibilities."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And our constraint is being a multiple of five. So how many total possibilities are there? Let's think about that. Total possibilities. How many do we have? Well, that's just the total number of numbers we have to pick from. So one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Total possibilities. How many do we have? Well, that's just the total number of numbers we have to pick from. So one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. So there are twelve possibilities. We have an equal chance of picking any one of these twelve. Now, which of these twelve are a multiple of five?"}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. So there are twelve possibilities. We have an equal chance of picking any one of these twelve. Now, which of these twelve are a multiple of five? So let's do this in a different color. So let me pick out the multiples of five. Thirty-two is not a multiple of five."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, which of these twelve are a multiple of five? So let's do this in a different color. So let me pick out the multiples of five. Thirty-two is not a multiple of five. Forty-nine is not a multiple of five. Fifty-five is a multiple of five. Really, we're just looking for the numbers that, in the ones place, either have a five or a zero."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Thirty-two is not a multiple of five. Forty-nine is not a multiple of five. Fifty-five is a multiple of five. Really, we're just looking for the numbers that, in the ones place, either have a five or a zero. Fifty-five is a multiple of five. Thirty is a multiple of five. That's six times five."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Really, we're just looking for the numbers that, in the ones place, either have a five or a zero. Fifty-five is a multiple of five. Thirty is a multiple of five. That's six times five. Fifty-five is eleven times five. Not fifty-six, not twenty-eight. This is clearly five times ten."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's six times five. Fifty-five is eleven times five. Not fifty-six, not twenty-eight. This is clearly five times ten. This is eight times five. This is the same number again. Also eight times five."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is clearly five times ten. This is eight times five. This is the same number again. Also eight times five. So all of these are multiples of five. Forty-five, that's a nine times five. Three is not a multiple of five."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Also eight times five. So all of these are multiples of five. Forty-five, that's a nine times five. Three is not a multiple of five. Twenty-five, clearly five times five. So I've circled all of the multiples of five. So of all the possibilities, the ones that meet our constraint of being a multiple of five, there are one, two, three, four, five, six, seven possibilities."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Three is not a multiple of five. Twenty-five, clearly five times five. So I've circled all of the multiples of five. So of all the possibilities, the ones that meet our constraint of being a multiple of five, there are one, two, three, four, five, six, seven possibilities. So seven meet our constraint. So in this example, the probability of selecting a number that is a multiple of five is 7 twelfths. Let's do another one."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So of all the possibilities, the ones that meet our constraint of being a multiple of five, there are one, two, three, four, five, six, seven possibilities. So seven meet our constraint. So in this example, the probability of selecting a number that is a multiple of five is 7 twelfths. Let's do another one. The circumference of a circle is 36 pi. Let's draw this circle. The circumference of a circle is 36 pi."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do another one. The circumference of a circle is 36 pi. Let's draw this circle. The circumference of a circle is 36 pi. So let's say the circle looks something like that. And the circumference, we have to be careful here. They're giving us interesting."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "The circumference of a circle is 36 pi. So let's say the circle looks something like that. And the circumference, we have to be careful here. They're giving us interesting. So the circumference is 36 pi. Then they tell us that contained in that circle is a smaller circle with area 16 pi. So inside the bigger circle, we have a smaller circle that has this guy right over here has an area of 16 pi."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "They're giving us interesting. So the circumference is 36 pi. Then they tell us that contained in that circle is a smaller circle with area 16 pi. So inside the bigger circle, we have a smaller circle that has this guy right over here has an area of 16 pi. A point is selected at random from inside the larger circle. So we're going to randomly select some point in this larger circle. What is the probability that the point also lies in this smaller circle?"}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So inside the bigger circle, we have a smaller circle that has this guy right over here has an area of 16 pi. A point is selected at random from inside the larger circle. So we're going to randomly select some point in this larger circle. What is the probability that the point also lies in this smaller circle? So here's a little bit interesting because you actually have an infinite number of points in both of these circles. It's not kind of separate balls or marbles like we saw in the first example or separate numbers. There's actually an infinite number of points you can pick here."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability that the point also lies in this smaller circle? So here's a little bit interesting because you actually have an infinite number of points in both of these circles. It's not kind of separate balls or marbles like we saw in the first example or separate numbers. There's actually an infinite number of points you can pick here. So when we talk about the probability that the point also lies in the smaller circle, we're really thinking about the percentage of the points in the larger circle that are also in the smaller circle. Or another way to think about it is the probability that if we were to pick a point from this larger circle, the probability that it's also in the smaller circle is really just going to be the percentage of the larger circle that is the smaller circle. I know that might sound confusing, but we really just have to figure out the areas for both of them."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's actually an infinite number of points you can pick here. So when we talk about the probability that the point also lies in the smaller circle, we're really thinking about the percentage of the points in the larger circle that are also in the smaller circle. Or another way to think about it is the probability that if we were to pick a point from this larger circle, the probability that it's also in the smaller circle is really just going to be the percentage of the larger circle that is the smaller circle. I know that might sound confusing, but we really just have to figure out the areas for both of them. It's really just going to be the ratios. So let's think about that. So there's a temptation to just use this 36 pi up here."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "I know that might sound confusing, but we really just have to figure out the areas for both of them. It's really just going to be the ratios. So let's think about that. So there's a temptation to just use this 36 pi up here. We have to remember this was the circumference and we need to figure out the area of both of these circles. And so for area, we need to know the radius because area is pi r squared. So we can figure out the radius from the circumference by saying, well, circumference is equal to 2 times pi times the radius of the circle."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a temptation to just use this 36 pi up here. We have to remember this was the circumference and we need to figure out the area of both of these circles. And so for area, we need to know the radius because area is pi r squared. So we can figure out the radius from the circumference by saying, well, circumference is equal to 2 times pi times the radius of the circle. Or if you say 36 pi, which we're told is a circumference, is equal to 2 times pi times the radius. We can divide both sides by 2 pi. And on the left-hand side, 36 divided by 2 is 18."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can figure out the radius from the circumference by saying, well, circumference is equal to 2 times pi times the radius of the circle. Or if you say 36 pi, which we're told is a circumference, is equal to 2 times pi times the radius. We can divide both sides by 2 pi. And on the left-hand side, 36 divided by 2 is 18. The pi's cancel out. We get our radius as being equal to 18 for this larger circle. This larger circle has a radius of 18."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And on the left-hand side, 36 divided by 2 is 18. The pi's cancel out. We get our radius as being equal to 18 for this larger circle. This larger circle has a radius of 18. So if we want to know its area, its area is going to be pi r squared, which is equal to pi times 18 squared. Now let's figure out what 18 squared is. 18 times 18."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "This larger circle has a radius of 18. So if we want to know its area, its area is going to be pi r squared, which is equal to pi times 18 squared. Now let's figure out what 18 squared is. 18 times 18. 8 times 8 is 64. 8 times 1 is 8. Plus 6 is 14."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "18 times 18. 8 times 8 is 64. 8 times 1 is 8. Plus 6 is 14. And then we have, we put that 0 there because we're now in the tens place. 1 times 8 is 8. 1 times 1 is 1."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 6 is 14. And then we have, we put that 0 there because we're now in the tens place. 1 times 8 is 8. 1 times 1 is 1. And really this is a 10 times a 10. That's why it gives us 100. But anyway, 4 plus 0 is a 4."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 times 1 is 1. And really this is a 10 times a 10. That's why it gives us 100. But anyway, 4 plus 0 is a 4. 4 plus 8 is a 12. 1 plus 1 plus 1 is a 3. So it's 324."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "But anyway, 4 plus 0 is a 4. 4 plus 8 is a 12. 1 plus 1 plus 1 is a 3. So it's 324. So the area here is equal to pi times 324, or we could say 324 pi. So the area of the entire larger circle, the part that I've shaded in yellow, including what's kind of under this orange circle, if you want to view it that way, this area right over here is equal to 324 pi. So the probability that a point that we select from this larger circle is also in the smaller circle is really just a percentage of the larger circle that is the smaller circle."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 324. So the area here is equal to pi times 324, or we could say 324 pi. So the area of the entire larger circle, the part that I've shaded in yellow, including what's kind of under this orange circle, if you want to view it that way, this area right over here is equal to 324 pi. So the probability that a point that we select from this larger circle is also in the smaller circle is really just a percentage of the larger circle that is the smaller circle. So our probability, I'll just write it like this, the probability that the point also lies in the smaller circle, so all of that stuff I'll put in, the probability of that is going to be equal to the percentage of this larger circle that is the smaller one. So that's going to be, or we could say the fraction of the larger circle's area that is the smaller circle's area. So it's going to be 16 pi over 324 pi."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability that a point that we select from this larger circle is also in the smaller circle is really just a percentage of the larger circle that is the smaller circle. So our probability, I'll just write it like this, the probability that the point also lies in the smaller circle, so all of that stuff I'll put in, the probability of that is going to be equal to the percentage of this larger circle that is the smaller one. So that's going to be, or we could say the fraction of the larger circle's area that is the smaller circle's area. So it's going to be 16 pi over 324 pi. And the pi's cancel out. And let's see, it looks like both of them are divisible by 4. If we divide the numerator by 4, we get 4."}, {"video_title": "Finding probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be 16 pi over 324 pi. And the pi's cancel out. And let's see, it looks like both of them are divisible by 4. If we divide the numerator by 4, we get 4. If we divide the denominator by 4, what do we get? 4 goes into 320, 80 times, it goes into 4 once, so we get an 81. So the probability, so I didn't even draw this to scale, this area is actually much smaller when you do it to scale."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "He's going to choose a volunteer, and he wants each kid to have the same chance of getting chosen. Fair enough. Now we have three methods by which he can do it, and let's just think about whether each of these methods are fair, where each kid does have the same chance of getting chosen, and if they're not, if they don't lead to each kid having the same chance of getting chosen, think about why that is the case. So method one, the magician starts with the birthday boy and moves clockwise, passing out 100 pieces of paper numbered one through 100. He cycles around the circle until all the pieces are distributed. He then uses a random number generator to pick an integer one through 100 and chooses the volunteer with that number. So let's just think about what's happening."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So method one, the magician starts with the birthday boy and moves clockwise, passing out 100 pieces of paper numbered one through 100. He cycles around the circle until all the pieces are distributed. He then uses a random number generator to pick an integer one through 100 and chooses the volunteer with that number. So let's just think about what's happening. So there's 15 kids in a circle. So there's one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. I planned that out amazingly well."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just think about what's happening. So there's 15 kids in a circle. So there's one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. I planned that out amazingly well. I didn't think I would be able to fit exactly 15, but it worked out. So 15 kids in a circle, and then he's gonna hand out pieces of paper. So he's going to, let's give one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I planned that out amazingly well. I didn't think I would be able to fit exactly 15, but it worked out. So 15 kids in a circle, and then he's gonna hand out pieces of paper. So he's going to, let's give one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. And now this person's gonna get pieces of paper one and 16. Now this person's gonna get two and 17. You're gonna keep going around and around and around until all 100 pieces of paper are going to get distributed."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So he's going to, let's give one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. And now this person's gonna get pieces of paper one and 16. Now this person's gonna get two and 17. You're gonna keep going around and around and around until all 100 pieces of paper are going to get distributed. Now, something to think about is whether every child here is going to get the same number of pieces of paper. And I encourage you to pause this video and think about that. If we just keep cycling around all the way to 100, does each child get the same number of pieces of paper?"}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're gonna keep going around and around and around until all 100 pieces of paper are going to get distributed. Now, something to think about is whether every child here is going to get the same number of pieces of paper. And I encourage you to pause this video and think about that. If we just keep cycling around all the way to 100, does each child get the same number of pieces of paper? Well, just think about it. In order to get the same number of pieces of paper, 100 has to be divisible by 15. And we know 100 isn't divisible by 15."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we just keep cycling around all the way to 100, does each child get the same number of pieces of paper? Well, just think about it. In order to get the same number of pieces of paper, 100 has to be divisible by 15. And we know 100 isn't divisible by 15. 15 goes into 100 six times. Six times 15 is 90. And you have a remainder of 10."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we know 100 isn't divisible by 15. 15 goes into 100 six times. Six times 15 is 90. And you have a remainder of 10. So what's going to happen is all 15 kids are going to get six pieces of paper. And then another 10 of the 15 are going to get a seventh piece of paper. So they're not all getting an equal number of pieces of paper."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you have a remainder of 10. So what's going to happen is all 15 kids are going to get six pieces of paper. And then another 10 of the 15 are going to get a seventh piece of paper. So they're not all getting an equal number of pieces of paper. So even though he's randomly picking an integer between one and 100, some of the students are going to have a higher chance than the other ones. The 10 that have seven pieces of paper are going to have a higher chance than the other five who only have six pieces, who only have six pieces of paper. And so I would say method one is not fair."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they're not all getting an equal number of pieces of paper. So even though he's randomly picking an integer between one and 100, some of the students are going to have a higher chance than the other ones. The 10 that have seven pieces of paper are going to have a higher chance than the other five who only have six pieces, who only have six pieces of paper. And so I would say method one is not fair. Let me write this down. Not fair. Sometimes life isn't fair."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so I would say method one is not fair. Let me write this down. Not fair. Sometimes life isn't fair. But in this case, it's not fair. Where we define fair is the same chance of getting chosen. And that's because they all have different numbers of pieces of paper."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sometimes life isn't fair. But in this case, it's not fair. Where we define fair is the same chance of getting chosen. And that's because they all have different numbers of pieces of paper. All of the students are not equally likely to get picked. Let's look at method two. The magician starts with the birthday boy and moves counterclockwise, passing up 75 pieces of paper, numbered one through 75."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's because they all have different numbers of pieces of paper. All of the students are not equally likely to get picked. Let's look at method two. The magician starts with the birthday boy and moves counterclockwise, passing up 75 pieces of paper, numbered one through 75. He cycles around the circle until all the pieces are distributed. He then uses a random number generator to pick an integer between one through 75 and chooses to volunteer with that number. So I encourage you to pause this video and think about whether this one, method two, whether that one is fair."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The magician starts with the birthday boy and moves counterclockwise, passing up 75 pieces of paper, numbered one through 75. He cycles around the circle until all the pieces are distributed. He then uses a random number generator to pick an integer between one through 75 and chooses to volunteer with that number. So I encourage you to pause this video and think about whether this one, method two, whether that one is fair. Well, method two is the same as method one, except for instead of using 100 pieces of paper, we're using 75 pieces of paper. And so we have to think about is 75 divisible by 15? And 75 is."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I encourage you to pause this video and think about whether this one, method two, whether that one is fair. Well, method two is the same as method one, except for instead of using 100 pieces of paper, we're using 75 pieces of paper. And so we have to think about is 75 divisible by 15? And 75 is. 5 times 15 is 75. So in this situation, each student is going to get five pieces of paper. Each gets five pieces of paper."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And 75 is. 5 times 15 is 75. So in this situation, each student is going to get five pieces of paper. Each gets five pieces of paper. So they all have an equally likely chance of getting picked. And then he's using a random number generator to pick them. So they all have an equally likely chance."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Each gets five pieces of paper. So they all have an equally likely chance of getting picked. And then he's using a random number generator to pick them. So they all have an equally likely chance. I would say method two is indeed fair. They all have the same chance of getting chosen. Now let's think about method three."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they all have an equally likely chance. I would say method two is indeed fair. They all have the same chance of getting chosen. Now let's think about method three. The magician starts with the birthday boy and moves clockwise, passing out 30 pieces of paper numbered one through 30. So they're all going to get the same number of pieces of paper. They're all going to get two pieces of paper each."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's think about method three. The magician starts with the birthday boy and moves clockwise, passing out 30 pieces of paper numbered one through 30. So they're all going to get the same number of pieces of paper. They're all going to get two pieces of paper each. 15 children getting two pieces of paper each would be 30 pieces of paper. So that looks reasonable so far. He cycles around the circle until all the pieces are distributed."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "They're all going to get two pieces of paper each. 15 children getting two pieces of paper each would be 30 pieces of paper. So that looks reasonable so far. He cycles around the circle until all the pieces are distributed. So everyone gets two pieces. He gives number one to the birthday boy, number two to the next kid, and so on. So that all seems reasonable, kind of consistent with method two, except now instead of 75, it's 30."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "He cycles around the circle until all the pieces are distributed. So everyone gets two pieces. He gives number one to the birthday boy, number two to the next kid, and so on. So that all seems reasonable, kind of consistent with method two, except now instead of 75, it's 30. And obviously, 75 was overkill. Even here, this is overkill. He just really needs 15 pieces of paper."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that all seems reasonable, kind of consistent with method two, except now instead of 75, it's 30. And obviously, 75 was overkill. Even here, this is overkill. He just really needs 15 pieces of paper. And he then counts the number of windows in the room and chooses the volunteer with that number. So the question here is, is the number of windows in the room, is it random, and is it evenly distributed? So maybe you could make a case that depending on what building it's in and someone's house, it's somewhat random on how many windows that house happens to have, the house that's happening to host the birthday party."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "He just really needs 15 pieces of paper. And he then counts the number of windows in the room and chooses the volunteer with that number. So the question here is, is the number of windows in the room, is it random, and is it evenly distributed? So maybe you could make a case that depending on what building it's in and someone's house, it's somewhat random on how many windows that house happens to have, the house that's happening to host the birthday party. But it's not going to be evenly distributed. Most, I don't know. There's probably some, if you were to sit and plot all of the houses that had a birthday party, you'd probably see that it's more likely that they have 10 windows than one window."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So maybe you could make a case that depending on what building it's in and someone's house, it's somewhat random on how many windows that house happens to have, the house that's happening to host the birthday party. But it's not going to be evenly distributed. Most, I don't know. There's probably some, if you were to sit and plot all of the houses that had a birthday party, you'd probably see that it's more likely that they have 10 windows than one window. And definitely more likely that they have 10 windows than, let's say, 30 windows, or even maybe 15 windows even. And so it's not going to be evenly distributed. Every house, I guess, has a somewhat different number of windows."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's probably some, if you were to sit and plot all of the houses that had a birthday party, you'd probably see that it's more likely that they have 10 windows than one window. And definitely more likely that they have 10 windows than, let's say, 30 windows, or even maybe 15 windows even. And so it's not going to be evenly distributed. Every house, I guess, has a somewhat different number of windows. And the house that is happening to host the party seems to be somewhat random. But it's not going to be evenly distributed here. And I would say it's not a really good random number generator."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Every house, I guess, has a somewhat different number of windows. And the house that is happening to host the party seems to be somewhat random. But it's not going to be evenly distributed here. And I would say it's not a really good random number generator. Because it's not evenly generated. And so I would say method three is not fair. The number of windows is not a really good random number generator."}, {"video_title": "Picking fairly   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I would say it's not a really good random number generator. Because it's not evenly generated. And so I would say method three is not fair. The number of windows is not a really good random number generator. A good random number generator, we would want, say, a number 1 through 75, where any of these have an equally likely chance of getting picked. It's somewhat random, the number of windows that building has. But they're not all equally likely."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And let me just show you how to figure out a histogram for some data, and I think you're going to get the point pretty easily. So I have some data here, and I want to represent it with a histogram. And what we're going to see is how frequent are each of these numbers. And in order to figure that out, let me just write the numbers down, and let me just categorize them in their respective buckets. So I have a 1 here. So I have 1 1 right there. I have a 4."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And in order to figure that out, let me just write the numbers down, and let me just categorize them in their respective buckets. So I have a 1 here. So I have 1 1 right there. I have a 4. Let me put, I want to leave space for the 2, the 3, and put a 4 there. I have a 2. That 2, I have a 1."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "I have a 4. Let me put, I want to leave space for the 2, the 3, and put a 4 there. I have a 2. That 2, I have a 1. Let me put that 1 on a stack right above that 1. I have a 0. Let me put the 0 to the left of the 1."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "That 2, I have a 1. Let me put that 1 on a stack right above that 1. I have a 0. Let me put the 0 to the left of the 1. I want to put them in order. I have a 2, another 2. Let me stack that above my first 2."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let me put the 0 to the left of the 1. I want to put them in order. I have a 2, another 2. Let me stack that above my first 2. I have another 1. Let me stack that above my other 2 1's. I have another 0."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let me stack that above my first 2. I have another 1. Let me stack that above my other 2 1's. I have another 0. Let me stack it there. I have another 1, another 1 right there. Then I have another 2, another 1."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "I have another 0. Let me stack it there. I have another 1, another 1 right there. Then I have another 2, another 1. I have 2 more 0's. I have 2 more 2's. I have a 3."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Then I have another 2, another 1. I have 2 more 0's. I have 2 more 2's. I have a 3. I have 2 more 1's, another 3. And then I have a 6. So no 5's, and then I have a 6."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "I have a 3. I have 2 more 1's, another 3. And then I have a 6. So no 5's, and then I have a 6. And that space right there was unnecessary. But right here, I've just rewritten these numbers, and I've essentially categorized them. Now what I want to do is calculate how many of each of these numbers we have."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So no 5's, and then I have a 6. And that space right there was unnecessary. But right here, I've just rewritten these numbers, and I've essentially categorized them. Now what I want to do is calculate how many of each of these numbers we have. So let me go down here. So I want to look at the frequency of each of these numbers. So I have 1, 2, 3, 4 0's."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Now what I want to do is calculate how many of each of these numbers we have. So let me go down here. So I want to look at the frequency of each of these numbers. So I have 1, 2, 3, 4 0's. I have 1, 2, 3, 4, 5, 6, 7 1's. I have 1, 2, 3, 4, 5 2's. I have 1, 2 3's."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So I have 1, 2, 3, 4 0's. I have 1, 2, 3, 4, 5, 6, 7 1's. I have 1, 2, 3, 4, 5 2's. I have 1, 2 3's. I have 1, 4, and 1, 6. So we could write it this way. We could write the number, and then we can have the frequency."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "I have 1, 2 3's. I have 1, 4, and 1, 6. So we could write it this way. We could write the number, and then we can have the frequency. And so I have the numbers 0, 1, 2, 3, 4. We could even throw 5 in there. Although 5 has a frequency of 0."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "We could write the number, and then we can have the frequency. And so I have the numbers 0, 1, 2, 3, 4. We could even throw 5 in there. Although 5 has a frequency of 0. And we have a 6. So 0 showed up 4 times in this data set. 1 showed up 7 times in this data set."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Although 5 has a frequency of 0. And we have a 6. So 0 showed up 4 times in this data set. 1 showed up 7 times in this data set. 2 showed up 5 times. 3 showed up 2 times. 4 showed up 1 time."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "1 showed up 7 times in this data set. 2 showed up 5 times. 3 showed up 2 times. 4 showed up 1 time. 5 didn't show up. And 6 showed up 1 time. All I did is I counted this data set."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "4 showed up 1 time. 5 didn't show up. And 6 showed up 1 time. All I did is I counted this data set. I mean, I did this first, but you could say, how many times do I see a 0? Or I see it 1, 2, 3, 4 times. How many times do I see a 1?"}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "All I did is I counted this data set. I mean, I did this first, but you could say, how many times do I see a 0? Or I see it 1, 2, 3, 4 times. How many times do I see a 1? 1, 2, 3, 4, 5, 6, 7 times. That's what we mean by frequency. Now a histogram is really just a plot, a kind of a bar graph, plotting the frequency of each of these numbers."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "How many times do I see a 1? 1, 2, 3, 4, 5, 6, 7 times. That's what we mean by frequency. Now a histogram is really just a plot, a kind of a bar graph, plotting the frequency of each of these numbers. It's going to look a lot like this original thing that I drew. So let me draw some axes here. So the different buckets here are the numbers."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Now a histogram is really just a plot, a kind of a bar graph, plotting the frequency of each of these numbers. It's going to look a lot like this original thing that I drew. So let me draw some axes here. So the different buckets here are the numbers. And that worked out because we're dealing with very clean integers that tend to repeat. If you're dealing with things that are more, that aren't just, you know, the exact number doesn't repeat, oftentimes people will put the numbers into buckets or ranges. But here they fit into nice little buckets."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So the different buckets here are the numbers. And that worked out because we're dealing with very clean integers that tend to repeat. If you're dealing with things that are more, that aren't just, you know, the exact number doesn't repeat, oftentimes people will put the numbers into buckets or ranges. But here they fit into nice little buckets. So you have the numbers 0, 1, 2, 3, 4, 5, and 6. This is the actual numbers. And then on the vertical axis, we're going to plot the frequency."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "But here they fit into nice little buckets. So you have the numbers 0, 1, 2, 3, 4, 5, and 6. This is the actual numbers. And then on the vertical axis, we're going to plot the frequency. So 1, 2, 3, 4, 5, 6, 7. So that's 7, 6, 5, 4, 3, 2, 1. So 0 shows up four times."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And then on the vertical axis, we're going to plot the frequency. So 1, 2, 3, 4, 5, 6, 7. So that's 7, 6, 5, 4, 3, 2, 1. So 0 shows up four times. So we'll draw a little bar graph here. 0 shows up four times. Draw it just like that."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So 0 shows up four times. So we'll draw a little bar graph here. 0 shows up four times. Draw it just like that. 0 shows up four times. That is that information right there. 1 shows up seven times."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Draw it just like that. 0 shows up four times. That is that information right there. 1 shows up seven times. So I'll do a little bar graph. 1 shows up seven times, just like that. 1 shows up."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "1 shows up seven times. So I'll do a little bar graph. 1 shows up seven times, just like that. 1 shows up. I want to make it a little bit straighter than that. 1 shows up seven times. 2, I'll do it in a different color."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "1 shows up. I want to make it a little bit straighter than that. 1 shows up seven times. 2, I'll do it in a different color. 2 shows up five times. So a bar graph, go all the way up to 5. 2 shows up five times."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "2, I'll do it in a different color. 2 shows up five times. So a bar graph, go all the way up to 5. 2 shows up five times. 3 shows up two times. We have 1, 3, 2, 3's. 4 shows up one time here."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "2 shows up five times. 3 shows up two times. We have 1, 3, 2, 3's. 4 shows up one time here. 5 doesn't show up at all. So it doesn't even get any height there. And then finally, 6 shows up one time."}, {"video_title": "Histograms   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "4 shows up one time here. 5 doesn't show up at all. So it doesn't even get any height there. And then finally, 6 shows up one time. So I'll do 6 showing up one time. What I just plotted here, this is a histogram. This right here is a histogram."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Represent the following data using a box and whiskers plot. Once again, exclude the median when computing the quartiles. And they gave us a bunch of data points. And it says if it helps, you might drag the numbers around, which I will do, because that will be useful. And they say the order isn't checked, and that's because I'm doing this on Khan Academy exercises up here in the top right where you can't see there's actually a check answer. So I encourage you to use the exercises yourself. But let's just use this as an example."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And it says if it helps, you might drag the numbers around, which I will do, because that will be useful. And they say the order isn't checked, and that's because I'm doing this on Khan Academy exercises up here in the top right where you can't see there's actually a check answer. So I encourage you to use the exercises yourself. But let's just use this as an example. So the first thing, if I'm going to do a box and whiskers, I'm going to order these numbers. So let me order these numbers from least to greatest. So let's see, there is a 1 here."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But let's just use this as an example. So the first thing, if I'm going to do a box and whiskers, I'm going to order these numbers. So let me order these numbers from least to greatest. So let's see, there is a 1 here. And we've got some 2's here. And some 3's. I have one 4."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's see, there is a 1 here. And we've got some 2's here. And some 3's. I have one 4. Then 5's. I have a 6. I have a 7."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I have one 4. Then 5's. I have a 6. I have a 7. I have a couple of 8's. And I have a 10. So there you go."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I have a 7. I have a couple of 8's. And I have a 10. So there you go. I have ordered these numbers from least to greatest. And now, well, just like that I can plot the whiskers because I see the range. My lowest number is 1."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So there you go. I have ordered these numbers from least to greatest. And now, well, just like that I can plot the whiskers because I see the range. My lowest number is 1. So my lowest number is 1. My largest number is 10. So the whiskers help me visualize the range."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "My lowest number is 1. So my lowest number is 1. My largest number is 10. So the whiskers help me visualize the range. Now let me think about what the median of my data set is. So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers. Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the whiskers help me visualize the range. Now let me think about what the median of my data set is. So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers. Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. I might say this number right over here, this 4, but notice there's 1, 2, 3, 4, 5, 6, 7 above it, and there's only 1, 2, 3, 4, 5, 6 below it. The same thing would have been true for this 5. So this 4 and 5, the middle is actually in between these 2."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. I might say this number right over here, this 4, but notice there's 1, 2, 3, 4, 5, 6, 7 above it, and there's only 1, 2, 3, 4, 5, 6 below it. The same thing would have been true for this 5. So this 4 and 5, the middle is actually in between these 2. So when you have an even number of numbers like this, you take the middle 2 numbers, this 4 and this 5, and you take the mean of the 2. So the mean of 4 and 5 is going to be 4 and 1 half. So that's going to be the median of our entire data set, 4 and 1 half."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this 4 and 5, the middle is actually in between these 2. So when you have an even number of numbers like this, you take the middle 2 numbers, this 4 and this 5, and you take the mean of the 2. So the mean of 4 and 5 is going to be 4 and 1 half. So that's going to be the median of our entire data set, 4 and 1 half. Now I want to figure out the median of the bottom half of numbers and the top half of numbers. Here they say exclude the median. Of course I'm going to exclude the median."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So that's going to be the median of our entire data set, 4 and 1 half. Now I want to figure out the median of the bottom half of numbers and the top half of numbers. Here they say exclude the median. Of course I'm going to exclude the median. It's not even included in our data points right here because our median is 4.5. So now let's take this bottom half of numbers over here and find the middle. So this is the bottom 7 numbers."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Of course I'm going to exclude the median. It's not even included in our data points right here because our median is 4.5. So now let's take this bottom half of numbers over here and find the middle. So this is the bottom 7 numbers. So the median of those is going to be the one that has 3 on either side. So it's going to be this 2 right over here. So that right over there is kind of the left boundary of our box."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is the bottom 7 numbers. So the median of those is going to be the one that has 3 on either side. So it's going to be this 2 right over here. So that right over there is kind of the left boundary of our box. And then for the right boundary, we need to figure out the middle of our top half of numbers. Remember, 4 and 5 were our middle 2 numbers. Our median is right in between at 4 and 1 half."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So that right over there is kind of the left boundary of our box. And then for the right boundary, we need to figure out the middle of our top half of numbers. Remember, 4 and 5 were our middle 2 numbers. Our median is right in between at 4 and 1 half. So our top half of numbers starts at this 5 and goes to this 10, 7 numbers. The middle one is going to have 3 on both sides. The 7 has 3 to the left, remember, of the top half, and 3 to the right."}, {"video_title": "Another example constructing box plot   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Our median is right in between at 4 and 1 half. So our top half of numbers starts at this 5 and goes to this 10, 7 numbers. The middle one is going to have 3 on both sides. The 7 has 3 to the left, remember, of the top half, and 3 to the right. And so the 7 is, I guess you could say, the right side of our box. And we're done. We've constructed our box and whiskers plot, which helps us visualize the entire range, but also you could say the middle, roughly the middle half of our numbers."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "All right, now let's do this together. A group of doctors was interested in comparing the effectiveness of placebo pills and real pills in treating migraines. And placebo pills are pills that look just like the regular pill, and from a patient's point of view, they don't know which one they're getting. And the reason why we do this in studies is because there's something called the placebo effect. Oftentimes, just by taking something that you think is good for you, some medicine that you think might help you, it actually does help you. And we can think about why the placebo effect happens, but this is well-documented. And so when people are trying to test medicine, they wanna say, well, does this have an effect above and beyond the placebo effect?"}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And the reason why we do this in studies is because there's something called the placebo effect. Oftentimes, just by taking something that you think is good for you, some medicine that you think might help you, it actually does help you. And we can think about why the placebo effect happens, but this is well-documented. And so when people are trying to test medicine, they wanna say, well, does this have an effect above and beyond the placebo effect? And so that's why they are putting them in these two, that's why they're comparing the real pills to the placebo pills. So they randomly assigned a group of 300 patients suffering from migraines into two groups. So they have their 300, they have their 300 patients right over here, and they're randomly putting them into the placebo group, placebo group, or the real, or the real pill group."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so when people are trying to test medicine, they wanna say, well, does this have an effect above and beyond the placebo effect? And so that's why they are putting them in these two, that's why they're comparing the real pills to the placebo pills. So they randomly assigned a group of 300 patients suffering from migraines into two groups. So they have their 300, they have their 300 patients right over here, and they're randomly putting them into the placebo group, placebo group, or the real, or the real pill group. One group was given a real pill, oh, we already read that. Both groups were told which kind of pill they got. That is sketchy, because the whole point about a placebo is that you think you got the real thing, or you think you might have gotten the real thing, and if you're told you got a placebo, that tends to undermine the placebo effect."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So they have their 300, they have their 300 patients right over here, and they're randomly putting them into the placebo group, placebo group, or the real, or the real pill group. One group was given a real pill, oh, we already read that. Both groups were told which kind of pill they got. That is sketchy, because the whole point about a placebo is that you think you got the real thing, or you think you might have gotten the real thing, and if you're told you got a placebo, that tends to undermine the placebo effect. If you're just told you're given a sugar pill, well, then the impact of, well, anyway. This is, that right over there is definitely bad practice. So let's keep going here."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "That is sketchy, because the whole point about a placebo is that you think you got the real thing, or you think you might have gotten the real thing, and if you're told you got a placebo, that tends to undermine the placebo effect. If you're just told you're given a sugar pill, well, then the impact of, well, anyway. This is, that right over there is definitely bad practice. So let's keep going here. Before taking the pills, and a day afterwards, the patients were asked to fill out questionnaires regarding their condition. Then the doctors analyzed the overall changes in questionnaires for each group and compared them. All right, so first of all, what type of study is this?"}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's keep going here. Before taking the pills, and a day afterwards, the patients were asked to fill out questionnaires regarding their condition. Then the doctors analyzed the overall changes in questionnaires for each group and compared them. All right, so first of all, what type of study is this? Well, we're taking our groups, we are randomly putting them into two different groups. You could call the placebo group, maybe the control group, and then you're putting the real pill as your actual experimental group. And so this is a classic experiment."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "All right, so first of all, what type of study is this? Well, we're taking our groups, we are randomly putting them into two different groups. You could call the placebo group, maybe the control group, and then you're putting the real pill as your actual experimental group. And so this is a classic experiment. This is a classic experiment. You're trying to establish a causal relationship. You want to see whether this real pill actually makes migraines better."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is a classic experiment. This is a classic experiment. You're trying to establish a causal relationship. You want to see whether this real pill actually makes migraines better. Migraines better. So does it actually do it? And does it actually do it better than a placebo?"}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You want to see whether this real pill actually makes migraines better. Migraines better. So does it actually do it? And does it actually do it better than a placebo? And you're randomly putting the people in both groups to try to distribute any confounding variables that there might be there. So this is clearly an experiment. Now the other options, you might say, well, is this maybe an observational study?"}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And does it actually do it better than a placebo? And you're randomly putting the people in both groups to try to distribute any confounding variables that there might be there. So this is clearly an experiment. Now the other options, you might say, well, is this maybe an observational study? And remember, an observational study, observational, that's more of where you look at a population, you look at a group, you ask them a bunch of questions often, or you make a bunch of observations, and you see if there's correlations between two variables. So variable one and variable two, and you're able to make some type of correlational statement. And you're not trying to get at causality."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now the other options, you might say, well, is this maybe an observational study? And remember, an observational study, observational, that's more of where you look at a population, you look at a group, you ask them a bunch of questions often, or you make a bunch of observations, and you see if there's correlations between two variables. So variable one and variable two, and you're able to make some type of correlational statement. And you're not trying to get at causality. And in a sample study, a sample study, this is just you trying to estimate a parameter for the entire population. So a sample study might have been of the entire population, what percentage gets migraines? And you can't talk to the entire population, maybe the entire population is millions of people, so you take a sample of, say, 100 people, and you ask them, do you get migraines?"}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you're not trying to get at causality. And in a sample study, a sample study, this is just you trying to estimate a parameter for the entire population. So a sample study might have been of the entire population, what percentage gets migraines? And you can't talk to the entire population, maybe the entire population is millions of people, so you take a sample of, say, 100 people, and you ask them, do you get migraines? And then you say, okay, that percentage of our sample that get migraines, that's a good estimate for the parameter of what percentage of my entire population actually gets migraines. So this was clearly an experiment. Now the next question is, was this experiment conducted well?"}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you can't talk to the entire population, maybe the entire population is millions of people, so you take a sample of, say, 100 people, and you ask them, do you get migraines? And then you say, okay, that percentage of our sample that get migraines, that's a good estimate for the parameter of what percentage of my entire population actually gets migraines. So this was clearly an experiment. Now the next question is, was this experiment conducted well? Well, even when I read it, I was bothered by both groups were told which kind of pill they got. That completely defeats the purpose of a placebo. The placebo effect is, hey, I think I'm taking something good for me, and it's been documented that when you think you're taking a pill that helps you, it oftentimes does help you."}, {"video_title": "Worked example identifying experiment   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now the next question is, was this experiment conducted well? Well, even when I read it, I was bothered by both groups were told which kind of pill they got. That completely defeats the purpose of a placebo. The placebo effect is, hey, I think I'm taking something good for me, and it's been documented that when you think you're taking a pill that helps you, it oftentimes does help you. And so if someone's coming up with a new medicine, it better perform, or it better be more effective than just that placebo. So I don't like the fact, and this is a very bad study, to tell both groups what kind of pill they got. You actually should tell neither group which type of pill they got."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So let's just think about this a little bit. Let's say I have negative 10, 0, 10, 20, and 30. Let's say that's one data set right there. And let's say the other data set is 8, 9, 10, 11, and 12. Now, let's calculate the arithmetic mean for both of these data sets. So let's calculate the mean. And when you go further on in statistics, you're going to understand the difference between a population and a sample."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "And let's say the other data set is 8, 9, 10, 11, and 12. Now, let's calculate the arithmetic mean for both of these data sets. So let's calculate the mean. And when you go further on in statistics, you're going to understand the difference between a population and a sample. We're assuming that this is the entire population of our data. This is the entire population of our data. So we're going to be dealing with the population mean."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "And when you go further on in statistics, you're going to understand the difference between a population and a sample. We're assuming that this is the entire population of our data. This is the entire population of our data. So we're going to be dealing with the population mean. We're going to be dealing with, as you see, the population measures of dispersion. I know these are all fancy words. In the future, you're not going to have all of the data."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So we're going to be dealing with the population mean. We're going to be dealing with, as you see, the population measures of dispersion. I know these are all fancy words. In the future, you're not going to have all of the data. You're just going to have some samples of it. You're going to try to estimate things for the entire population. So I don't want you to worry too much about that just now."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "In the future, you're not going to have all of the data. You're just going to have some samples of it. You're going to try to estimate things for the entire population. So I don't want you to worry too much about that just now. But if you are going to go further in statistics, I just want to make that clarification. Now, the population mean, or the arithmetic mean, of this data set right here, it is negative 10 plus 0 plus 10 plus 20 plus 30 over, we have 5 data points, over 5. And what is this equal to?"}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So I don't want you to worry too much about that just now. But if you are going to go further in statistics, I just want to make that clarification. Now, the population mean, or the arithmetic mean, of this data set right here, it is negative 10 plus 0 plus 10 plus 20 plus 30 over, we have 5 data points, over 5. And what is this equal to? That negative 10 cancels out with that 10. 20 plus 30 is 50. Divided by 5, it's equal to 10."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "And what is this equal to? That negative 10 cancels out with that 10. 20 plus 30 is 50. Divided by 5, it's equal to 10. Now, what's the mean of this data set? 8 plus 9 plus 10 plus 11 plus 12. All of that over 5."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Divided by 5, it's equal to 10. Now, what's the mean of this data set? 8 plus 9 plus 10 plus 11 plus 12. All of that over 5. And the way we could think about it, 8 plus 12 is 20. 9 plus 11 is another 20, so that's 40. And then we have a 50 there, add another 10."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "All of that over 5. And the way we could think about it, 8 plus 12 is 20. 9 plus 11 is another 20, so that's 40. And then we have a 50 there, add another 10. So this is, once again, going to be 50 over 5. So this has the exact same population means. Or if you don't want to worry about the word population or sample and all of that, both of these data sets have the exact same arithmetic mean."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "And then we have a 50 there, add another 10. So this is, once again, going to be 50 over 5. So this has the exact same population means. Or if you don't want to worry about the word population or sample and all of that, both of these data sets have the exact same arithmetic mean. When you average all these numbers and divide by 5, or when you take the sum of these numbers, divide by 5, you get 10, sum of these numbers, divide by 5, you get 10 as well, but clearly these sets of numbers are different. If you just looked at this number, you'd say, oh, maybe these sets are very similar to each other. But when you look at these two data sets, one thing might pop out at you."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Or if you don't want to worry about the word population or sample and all of that, both of these data sets have the exact same arithmetic mean. When you average all these numbers and divide by 5, or when you take the sum of these numbers, divide by 5, you get 10, sum of these numbers, divide by 5, you get 10 as well, but clearly these sets of numbers are different. If you just looked at this number, you'd say, oh, maybe these sets are very similar to each other. But when you look at these two data sets, one thing might pop out at you. All of these numbers are very close to 10. I mean, the furthest number here is 2 away from 10. 12 is only 2 away from 10."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "But when you look at these two data sets, one thing might pop out at you. All of these numbers are very close to 10. I mean, the furthest number here is 2 away from 10. 12 is only 2 away from 10. Here, these numbers are further away from 10. Even the closer ones are still 10 away, and then these guys are 20 away from 10. So this right here, this data set right here, is more dispersed."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "12 is only 2 away from 10. Here, these numbers are further away from 10. Even the closer ones are still 10 away, and then these guys are 20 away from 10. So this right here, this data set right here, is more dispersed. These guys are further away from our mean than these guys are from this mean. So let's think about different ways we can measure dispersion, or how far away we are from the center on average. Now, one way, this is kind of the most simple way, is the range, and you won't see it used too often, but it's kind of a very simple way of understanding how far is the spread between the largest and the smallest number."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So this right here, this data set right here, is more dispersed. These guys are further away from our mean than these guys are from this mean. So let's think about different ways we can measure dispersion, or how far away we are from the center on average. Now, one way, this is kind of the most simple way, is the range, and you won't see it used too often, but it's kind of a very simple way of understanding how far is the spread between the largest and the smallest number. And literally, take the largest number, which is 30 in our example, and from that you subtract the smallest number. So 30 minus negative 10, which is equal to 40, which tells us that the difference between the largest and the smallest number is 40. So we have a range of 40 for this data set."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Now, one way, this is kind of the most simple way, is the range, and you won't see it used too often, but it's kind of a very simple way of understanding how far is the spread between the largest and the smallest number. And literally, take the largest number, which is 30 in our example, and from that you subtract the smallest number. So 30 minus negative 10, which is equal to 40, which tells us that the difference between the largest and the smallest number is 40. So we have a range of 40 for this data set. Here, the range is the largest number, 12, minus the smallest number, which is 8, which is equal to 4. So here, range is actually a pretty good measure of dispersion. We say, OK, both of these guys have a mean of 10, but when I look at the range, this guy has a much larger range."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So we have a range of 40 for this data set. Here, the range is the largest number, 12, minus the smallest number, which is 8, which is equal to 4. So here, range is actually a pretty good measure of dispersion. We say, OK, both of these guys have a mean of 10, but when I look at the range, this guy has a much larger range. So that tells me this is a more dispersed set. But range is always not going to tell you the whole picture. You might have two data sets with the exact same range, where still, based on how things are bunched up, it could still have very different distributions of where the numbers lie."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "We say, OK, both of these guys have a mean of 10, but when I look at the range, this guy has a much larger range. So that tells me this is a more dispersed set. But range is always not going to tell you the whole picture. You might have two data sets with the exact same range, where still, based on how things are bunched up, it could still have very different distributions of where the numbers lie. Now, the one that you'll see used most often is called the variance. Actually, you're going to see the standard deviation in this video. That's probably what's used most often, but it has a very close relationship to the variance."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "You might have two data sets with the exact same range, where still, based on how things are bunched up, it could still have very different distributions of where the numbers lie. Now, the one that you'll see used most often is called the variance. Actually, you're going to see the standard deviation in this video. That's probably what's used most often, but it has a very close relationship to the variance. So the symbol for the variance, and we're going to deal with the population variance. Once again, we're assuming that this is all of the data for our whole population, that we're not just sampling, taking a subset of the data. So the variance, its symbol is literally this sigma, this Greek letter squared."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "That's probably what's used most often, but it has a very close relationship to the variance. So the symbol for the variance, and we're going to deal with the population variance. Once again, we're assuming that this is all of the data for our whole population, that we're not just sampling, taking a subset of the data. So the variance, its symbol is literally this sigma, this Greek letter squared. That is the symbol for variance. And we'll see that the sigma letter actually is the symbol for standard deviation. And that is for a reason."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So the variance, its symbol is literally this sigma, this Greek letter squared. That is the symbol for variance. And we'll see that the sigma letter actually is the symbol for standard deviation. And that is for a reason. But anyway, the definition of variance is you literally take each of these data points, find the difference between those data points and your mean, square them, and then take the average of those squares. I know that sounds very complicated, but when I actually calculate it, you're going to see it's not too bad. So remember, the mean here is 10."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "And that is for a reason. But anyway, the definition of variance is you literally take each of these data points, find the difference between those data points and your mean, square them, and then take the average of those squares. I know that sounds very complicated, but when I actually calculate it, you're going to see it's not too bad. So remember, the mean here is 10. So I take the first data point. I say, let me do it over here. Let me scroll down a little bit."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So remember, the mean here is 10. So I take the first data point. I say, let me do it over here. Let me scroll down a little bit. So I take the first data point, negative 10. From that, I'm going to subtract our mean, and I'm going to square that. So I just found the difference from that first data point to the mean and squared it."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Let me scroll down a little bit. So I take the first data point, negative 10. From that, I'm going to subtract our mean, and I'm going to square that. So I just found the difference from that first data point to the mean and squared it. And that's essentially to make it positive. Plus the second data point, 0 minus 10 minus the mean. This is the mean."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So I just found the difference from that first data point to the mean and squared it. And that's essentially to make it positive. Plus the second data point, 0 minus 10 minus the mean. This is the mean. This is that 10 right there. Squared. Plus 10 minus 10 squared."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "This is the mean. This is that 10 right there. Squared. Plus 10 minus 10 squared. That's the middle 10 right there. Plus 20 minus 10. That's the 20."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Plus 10 minus 10 squared. That's the middle 10 right there. Plus 20 minus 10. That's the 20. Squared plus 30 minus 10. Squared. So this is the squared differences between each number and the mean."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "That's the 20. Squared plus 30 minus 10. Squared. So this is the squared differences between each number and the mean. This is the mean. This is the mean right there. That is the mean."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So this is the squared differences between each number and the mean. This is the mean. This is the mean right there. That is the mean. I'm finding the difference between every data point and the mean, squaring them, summing them up, and then dividing by that number of data points. I'm taking the average of these numbers, of the squared distances. So when you say it kind of verbally, it sounds very complicated."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "That is the mean. I'm finding the difference between every data point and the mean, squaring them, summing them up, and then dividing by that number of data points. I'm taking the average of these numbers, of the squared distances. So when you say it kind of verbally, it sounds very complicated. But you're just taking each number, what's the difference between that, the mean, square it, take the average of those. So I have 1, 2, 3, 4, 5. Divide by 5."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So when you say it kind of verbally, it sounds very complicated. But you're just taking each number, what's the difference between that, the mean, square it, take the average of those. So I have 1, 2, 3, 4, 5. Divide by 5. So what is this going to be equal to? Negative 10 minus 10 is negative 20. Negative 20 squared is 400."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Divide by 5. So what is this going to be equal to? Negative 10 minus 10 is negative 20. Negative 20 squared is 400. 0 minus 10 is negative 10 squared is 100. So plus 100. 10 minus 10 squared, that's just 0 squared, which is 0."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Negative 20 squared is 400. 0 minus 10 is negative 10 squared is 100. So plus 100. 10 minus 10 squared, that's just 0 squared, which is 0. Plus 20 minus 10 is 10 squared is 100. Plus 30 minus 10, which is 20, squared is 400. All of that over 5."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "10 minus 10 squared, that's just 0 squared, which is 0. Plus 20 minus 10 is 10 squared is 100. Plus 30 minus 10, which is 20, squared is 400. All of that over 5. And what do we have here? 400 plus 100 is 500. Plus another 500 is 1,000."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "All of that over 5. And what do we have here? 400 plus 100 is 500. Plus another 500 is 1,000. It's equal to 1,000 over 5, which is equal to 200. So in this situation, our variance is going to be 200. That's our measure of dispersion there."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Plus another 500 is 1,000. It's equal to 1,000 over 5, which is equal to 200. So in this situation, our variance is going to be 200. That's our measure of dispersion there. And let's compare it to this data set over here. Let's compare it to the variance of this less dispersed data set. So let me scroll over a little bit."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "That's our measure of dispersion there. And let's compare it to this data set over here. Let's compare it to the variance of this less dispersed data set. So let me scroll over a little bit. So we have some real estate, although I'm running out. Maybe I could scroll up here. There you go."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So let me scroll over a little bit. So we have some real estate, although I'm running out. Maybe I could scroll up here. There you go. So let me calculate the variance of this data set. So we already know its mean. So its variance of this data set is going to be equal to 8 minus 10 squared plus 9 minus 10 squared plus 10 minus 10 squared plus 11 minus 10 squared plus 12 minus 10 squared."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "There you go. So let me calculate the variance of this data set. So we already know its mean. So its variance of this data set is going to be equal to 8 minus 10 squared plus 9 minus 10 squared plus 10 minus 10 squared plus 11 minus 10 squared plus 12 minus 10 squared. Remember that 10 is just the mean that we calculated. You have to calculate the mean first. Divided by, we have 1, 2, 3, 4, 5 squared differences."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So its variance of this data set is going to be equal to 8 minus 10 squared plus 9 minus 10 squared plus 10 minus 10 squared plus 11 minus 10 squared plus 12 minus 10 squared. Remember that 10 is just the mean that we calculated. You have to calculate the mean first. Divided by, we have 1, 2, 3, 4, 5 squared differences. So this is going to be equal to 8 minus 10 is negative 2 squared is positive 4. 9 minus 10 is negative 1 squared is positive 1. 10 minus 10 is 0 squared."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Divided by, we have 1, 2, 3, 4, 5 squared differences. So this is going to be equal to 8 minus 10 is negative 2 squared is positive 4. 9 minus 10 is negative 1 squared is positive 1. 10 minus 10 is 0 squared. You still get 0. 11 minus 10 is 1 squared. You get 1."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "10 minus 10 is 0 squared. You still get 0. 11 minus 10 is 1 squared. You get 1. 12 minus 10 is 2 squared. You get 4. Now what is this equal to?"}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "You get 1. 12 minus 10 is 2 squared. You get 4. Now what is this equal to? All of that over 5. This is 10 over 5. So this is going to be, all right, this is 10 over 5, which is equal to 2."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Now what is this equal to? All of that over 5. This is 10 over 5. So this is going to be, all right, this is 10 over 5, which is equal to 2. So the variance here, let me make sure I got that right. Yes, we have 10 over 5. So the variance of this less dispersed data set is a lot smaller, the variance here."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So this is going to be, all right, this is 10 over 5, which is equal to 2. So the variance here, let me make sure I got that right. Yes, we have 10 over 5. So the variance of this less dispersed data set is a lot smaller, the variance here. The variance of this data set right here is only 2. So that gave you a sense. That tells you, look, this is definitely a less dispersed data set than that there."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So the variance of this less dispersed data set is a lot smaller, the variance here. The variance of this data set right here is only 2. So that gave you a sense. That tells you, look, this is definitely a less dispersed data set than that there. Now the problem with the variance is you're taking these numbers, you're taking the difference between them and the mean, then you're squaring it. It kind of gives you a bit of an arbitrary number. And if you're dealing with units, let's say if these are each negative, well, let's say they're distances."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "That tells you, look, this is definitely a less dispersed data set than that there. Now the problem with the variance is you're taking these numbers, you're taking the difference between them and the mean, then you're squaring it. It kind of gives you a bit of an arbitrary number. And if you're dealing with units, let's say if these are each negative, well, let's say they're distances. So this is negative 10 meters, 0 meters, 10 meters, this is 8 meters, so on and so forth. Then when you square it, you get your variance in terms of meters squared. It's kind of an odd set of units."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "And if you're dealing with units, let's say if these are each negative, well, let's say they're distances. So this is negative 10 meters, 0 meters, 10 meters, this is 8 meters, so on and so forth. Then when you square it, you get your variance in terms of meters squared. It's kind of an odd set of units. So what people like to do is talk in terms of standard deviation. Which is just the square root of the variance. It's just the square root of the variance."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "It's kind of an odd set of units. So what people like to do is talk in terms of standard deviation. Which is just the square root of the variance. It's just the square root of the variance. Or the square root of sigma squared. And the symbol for the standard deviation is just sigma. So now that we've figured out the variance, it's very easy to figure out the standard deviation of both of these characters."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "It's just the square root of the variance. Or the square root of sigma squared. And the symbol for the standard deviation is just sigma. So now that we've figured out the variance, it's very easy to figure out the standard deviation of both of these characters. The standard deviation of this first one up here, of this first data set, is going to be the square root of 200. Square root of 200 is what? The square root of 2 times 100, this is equal to 10 square roots of 2."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So now that we've figured out the variance, it's very easy to figure out the standard deviation of both of these characters. The standard deviation of this first one up here, of this first data set, is going to be the square root of 200. Square root of 200 is what? The square root of 2 times 100, this is equal to 10 square roots of 2. That's that first data set. Now the variance of the second data set is just going to be the square root of 2. Sorry, the standard deviation of the second data set is going to be the square root of its variance, which is 2."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "The square root of 2 times 100, this is equal to 10 square roots of 2. That's that first data set. Now the variance of the second data set is just going to be the square root of 2. Sorry, the standard deviation of the second data set is going to be the square root of its variance, which is 2. Which is just 2. So the second data set has 1 tenth the standard deviation as this first data set. This is 10 roots of 2, this is just the root of 2."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Sorry, the standard deviation of the second data set is going to be the square root of its variance, which is 2. Which is just 2. So the second data set has 1 tenth the standard deviation as this first data set. This is 10 roots of 2, this is just the root of 2. So this is 10 times the standard deviation. And this, hopefully, will make a little bit more sense. Let's think about it."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "This is 10 roots of 2, this is just the root of 2. So this is 10 times the standard deviation. And this, hopefully, will make a little bit more sense. Let's think about it. This has 10 times more the standard deviation than this. And let's remember how we calculated it. Variance, we just took each data point, how far away from the mean, squared that, took the average of those."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Let's think about it. This has 10 times more the standard deviation than this. And let's remember how we calculated it. Variance, we just took each data point, how far away from the mean, squared that, took the average of those. Then we took the square root, really just to make the units look nice, but the end result is we said that that first data set has 10 times the standard deviation. As the second data set. So let's look at the two data sets."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "Variance, we just took each data point, how far away from the mean, squared that, took the average of those. Then we took the square root, really just to make the units look nice, but the end result is we said that that first data set has 10 times the standard deviation. As the second data set. So let's look at the two data sets. This has 10 times the standard deviation. Which makes sense intuitively. I mean, they both have a 10 in here, but each of these guys, 9 is only 1 away from the 10, 0 is 1 away from the 10."}, {"video_title": "Range, variance and standard deviation as measures of dispersion   Khan Academy.mp3", "Sentence": "So let's look at the two data sets. This has 10 times the standard deviation. Which makes sense intuitively. I mean, they both have a 10 in here, but each of these guys, 9 is only 1 away from the 10, 0 is 1 away from the 10. So the standard deviation is 10 times the standard deviation. So the standard deviation, at least in my sense, is giving a much better sense of how far away, on average, we are from the mean. Anyway, hopefully you found that useful."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Or if we want to speak generally, into m different groups. What I want to do in this video is to figure out how much of this total sum of squares, how much of this is due to variation within each group versus variation between the actual groups. So first, let's figure out the total variation within the group. So let's call that the sum of squares within. So let's calculate the sum of squares within. And I'll do that in yellow. I actually already used yellow."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So let's call that the sum of squares within. So let's calculate the sum of squares within. And I'll do that in yellow. I actually already used yellow. So let's do this. Let me do blue. So the sum of squares within."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "I actually already used yellow. So let's do this. Let me do blue. So the sum of squares within. Let me make it clear. That stands for within. So we want to see how much of the variation is due to how far each of these data points are from their central tendency, from their respective means."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So the sum of squares within. Let me make it clear. That stands for within. So we want to see how much of the variation is due to how far each of these data points are from their central tendency, from their respective means. So this is going to be equal to, let's start with these guys. So instead of taking the distance between each data point and the mean of means, I'm going to find the distance between each data point and that group's mean. Because we want to square the total sum of squares between each data point and their respective means."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So we want to see how much of the variation is due to how far each of these data points are from their central tendency, from their respective means. So this is going to be equal to, let's start with these guys. So instead of taking the distance between each data point and the mean of means, I'm going to find the distance between each data point and that group's mean. Because we want to square the total sum of squares between each data point and their respective means. So let's do that. So it's 3 minus, the mean here is 2 squared, plus 2 minus 2 squared, plus 1 minus 2 squared. Plus, I'm going to do this for all of the groups, but for each group, the distance between each data point and its mean."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Because we want to square the total sum of squares between each data point and their respective means. So let's do that. So it's 3 minus, the mean here is 2 squared, plus 2 minus 2 squared, plus 1 minus 2 squared. Plus, I'm going to do this for all of the groups, but for each group, the distance between each data point and its mean. So plus 5 minus 4 squared. Plus 4 minus 4 squared. And then finally we have the third group."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Plus, I'm going to do this for all of the groups, but for each group, the distance between each data point and its mean. So plus 5 minus 4 squared. Plus 4 minus 4 squared. And then finally we have the third group. We're finding all of the sum of squares from each point to its central tendency within that. But we're going to add them all up. And then we find the third group."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And then finally we have the third group. We're finding all of the sum of squares from each point to its central tendency within that. But we're going to add them all up. And then we find the third group. So we have 5 minus 6 squared, plus 6 minus 6 squared, plus 7 minus 6 squared. And what is this going to equal? So this is going to be equal to, up here it's going to be 1 plus 0 plus 1."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And then we find the third group. So we have 5 minus 6 squared, plus 6 minus 6 squared, plus 7 minus 6 squared. And what is this going to equal? So this is going to be equal to, up here it's going to be 1 plus 0 plus 1. So that's going to be equal to 2 plus, and this is going to be equal to 1 plus 1 plus 0, so another 2. Plus, this is going to be equal to 1 plus 0 plus 1. 7 minus 6 is 1, squared is 1."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So this is going to be equal to, up here it's going to be 1 plus 0 plus 1. So that's going to be equal to 2 plus, and this is going to be equal to 1 plus 1 plus 0, so another 2. Plus, this is going to be equal to 1 plus 0 plus 1. 7 minus 6 is 1, squared is 1. So that's 2 over here. So this is going to be equal to, our sum of squares within, I should say, is 6. So one way to think about it, our total variation was 30."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "7 minus 6 is 1, squared is 1. So that's 2 over here. So this is going to be equal to, our sum of squares within, I should say, is 6. So one way to think about it, our total variation was 30. And based on this calculation, 6 of that 30 comes from variation within these samples. Now the next thing I want to think about is, how many degrees of freedom do we have in this calculation? How many kind of independent data points do we actually have?"}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So one way to think about it, our total variation was 30. And based on this calculation, 6 of that 30 comes from variation within these samples. Now the next thing I want to think about is, how many degrees of freedom do we have in this calculation? How many kind of independent data points do we actually have? Well, for each of these, so over here, if you know, we have n data points in each one. In particular, n is 3 here. But if you know n minus 1 of them, you can always figure out the nth one if you know the actual sample mean."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "How many kind of independent data points do we actually have? Well, for each of these, so over here, if you know, we have n data points in each one. In particular, n is 3 here. But if you know n minus 1 of them, you can always figure out the nth one if you know the actual sample mean. So in this case, for any of these groups, if you know 2 of these data points, you can always figure out the third. If you know these 2, you can always figure out the third if you know the sample mean. So in general, let's figure out the degrees of freedom here."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "But if you know n minus 1 of them, you can always figure out the nth one if you know the actual sample mean. So in this case, for any of these groups, if you know 2 of these data points, you can always figure out the third. If you know these 2, you can always figure out the third if you know the sample mean. So in general, let's figure out the degrees of freedom here. You have, for each group, when you did this, you had n minus 1 degrees of freedom. Remember, n is the number of data points you had in each group. So you have n minus 1 degrees of freedom for each of these groups, so it's n minus 1, n minus 1, n minus 1."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So in general, let's figure out the degrees of freedom here. You have, for each group, when you did this, you had n minus 1 degrees of freedom. Remember, n is the number of data points you had in each group. So you have n minus 1 degrees of freedom for each of these groups, so it's n minus 1, n minus 1, n minus 1. Or you have, let me put it this way, you have n minus 1 for each of these groups. And there are m groups. So there's m times n minus 1 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So you have n minus 1 degrees of freedom for each of these groups, so it's n minus 1, n minus 1, n minus 1. Or you have, let me put it this way, you have n minus 1 for each of these groups. And there are m groups. So there's m times n minus 1 degrees of freedom. And in this case, in particular, each group, n minus 1 is 2, or in each case, you had 2 degrees of freedom. And there's 3 groups of that, so there are 6 degrees of freedom. There are 6 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So there's m times n minus 1 degrees of freedom. And in this case, in particular, each group, n minus 1 is 2, or in each case, you had 2 degrees of freedom. And there's 3 groups of that, so there are 6 degrees of freedom. There are 6 degrees of freedom. Let me write 6 degrees of freedom. In the future, we might do a more detailed discussion of what degrees of freedom mean and how to mathematically think about it. But the simplest way to think about it is really, truly independent data points, assuming you knew, in this case, the central statistic that we use to calculate the squared distance in each of them."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "There are 6 degrees of freedom. Let me write 6 degrees of freedom. In the future, we might do a more detailed discussion of what degrees of freedom mean and how to mathematically think about it. But the simplest way to think about it is really, truly independent data points, assuming you knew, in this case, the central statistic that we use to calculate the squared distance in each of them. If you know them already, the third data point could actually be calculated from the other two. So we have 6 degrees of freedom over here. Now, that was how much of the total variation is due to variation within each sample."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "But the simplest way to think about it is really, truly independent data points, assuming you knew, in this case, the central statistic that we use to calculate the squared distance in each of them. If you know them already, the third data point could actually be calculated from the other two. So we have 6 degrees of freedom over here. Now, that was how much of the total variation is due to variation within each sample. Now let's think about how much of the variation is due to variation between the samples. And to do that, we're going to calculate. Let me get a nice color here."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Now, that was how much of the total variation is due to variation within each sample. Now let's think about how much of the variation is due to variation between the samples. And to do that, we're going to calculate. Let me get a nice color here. I think I've run out of all the colors. We'll call this sum of squares between. The b stands for between."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Let me get a nice color here. I think I've run out of all the colors. We'll call this sum of squares between. The b stands for between. So another way to think about it, how much of this total variation is due to the variation between the means, between the central tendency? That's what we're going to calculate right now. And how much is due to variation from each data point to its mean?"}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "The b stands for between. So another way to think about it, how much of this total variation is due to the variation between the means, between the central tendency? That's what we're going to calculate right now. And how much is due to variation from each data point to its mean? So let's figure out how much is due to variation between these guys over here. So one way to think about it, for each of these data points, actually, let's think about just this first group. For this first group, how much variation for each of these guys is due to the variation between this mean and the mean of means?"}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And how much is due to variation from each data point to its mean? So let's figure out how much is due to variation between these guys over here. So one way to think about it, for each of these data points, actually, let's think about just this first group. For this first group, how much variation for each of these guys is due to the variation between this mean and the mean of means? Well, it's going to be, so for this first guy up here, I'll just write it all out explicitly, the variation is going to be its sample mean. So it's going to be 2 minus the mean of means squared. And then for this guy, it's going to be the same thing."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "For this first group, how much variation for each of these guys is due to the variation between this mean and the mean of means? Well, it's going to be, so for this first guy up here, I'll just write it all out explicitly, the variation is going to be its sample mean. So it's going to be 2 minus the mean of means squared. And then for this guy, it's going to be the same thing. His sample mean, 2 minus the mean of means squared plus, same thing for this guy, 2 minus the mean of means squared. Or another way to think about it, this is equal to 3 times 2 minus 4 squared, which is the same thing as 3. This is equal to 3 times 4 is equal to 12."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And then for this guy, it's going to be the same thing. His sample mean, 2 minus the mean of means squared plus, same thing for this guy, 2 minus the mean of means squared. Or another way to think about it, this is equal to 3 times 2 minus 4 squared, which is the same thing as 3. This is equal to 3 times 4 is equal to 12. And then we could do it for each of them. And actually, I want to find the total sum. So let me just write it all out, actually."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "This is equal to 3 times 4 is equal to 12. And then we could do it for each of them. And actually, I want to find the total sum. So let me just write it all out, actually. I think that might be an easier thing to do. Because I want to find, for all of these guys combined, the sum of squares due to the differences between the samples. So that's from the first sample, the contribution from the first sample."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So let me just write it all out, actually. I think that might be an easier thing to do. Because I want to find, for all of these guys combined, the sum of squares due to the differences between the samples. So that's from the first sample, the contribution from the first sample. And then from the second sample, you have this guy over here, 5, oh sorry, you don't want to calculate him. For this data point, the amount of variation due to the difference between the means is going to be 4 minus 4. It's going to be 4 minus 4 squared."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So that's from the first sample, the contribution from the first sample. And then from the second sample, you have this guy over here, 5, oh sorry, you don't want to calculate him. For this data point, the amount of variation due to the difference between the means is going to be 4 minus 4. It's going to be 4 minus 4 squared. Same thing for this guy. It's going to be 4 minus 4 squared. We're not taking it into consideration."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "It's going to be 4 minus 4 squared. Same thing for this guy. It's going to be 4 minus 4 squared. We're not taking it into consideration. We're only taking its sample mean into consideration. And then finally, plus 4 minus 4 squared. We're taking this minus this squared for each of these data points."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "We're not taking it into consideration. We're only taking its sample mean into consideration. And then finally, plus 4 minus 4 squared. We're taking this minus this squared for each of these data points. And then finally, we'll do that with the last group. Sample mean is 6. So it's going to be 6 minus 4 squared plus 6 minus 4 squared plus 6 minus 4 squared."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "We're taking this minus this squared for each of these data points. And then finally, we'll do that with the last group. Sample mean is 6. So it's going to be 6 minus 4 squared plus 6 minus 4 squared plus 6 minus 4 squared. Now let's think about how many degrees of freedom we had in this calculation right over here. How many degrees of freedom? Well, in general, I guess the easiest way to think about it is how much information did we have, assuming that we knew the mean of means."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So it's going to be 6 minus 4 squared plus 6 minus 4 squared plus 6 minus 4 squared. Now let's think about how many degrees of freedom we had in this calculation right over here. How many degrees of freedom? Well, in general, I guess the easiest way to think about it is how much information did we have, assuming that we knew the mean of means. If we know the mean of means, how much here is new information? Well, if you know the mean of the mean and you know two of these sample means, you can always figure out the third. If you know this one and this one, you can figure out that one."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Well, in general, I guess the easiest way to think about it is how much information did we have, assuming that we knew the mean of means. If we know the mean of means, how much here is new information? Well, if you know the mean of the mean and you know two of these sample means, you can always figure out the third. If you know this one and this one, you can figure out that one. If you know that one and that one, you can figure out that one. And that's because this is the mean of these means over here. So in general, if you have m groups or if you have m means, there are m minus 1 degrees of freedom here."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "If you know this one and this one, you can figure out that one. If you know that one and that one, you can figure out that one. And that's because this is the mean of these means over here. So in general, if you have m groups or if you have m means, there are m minus 1 degrees of freedom here. So there's m minus 1 degrees of freedom here. Let me write that. There are m minus 1 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So in general, if you have m groups or if you have m means, there are m minus 1 degrees of freedom here. So there's m minus 1 degrees of freedom here. Let me write that. There are m minus 1 degrees of freedom. But with that said, well, in this case, m is 3. So we could say there's 2 degrees of freedom for this exact example. But actually, let's calculate the sum of squares between."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "There are m minus 1 degrees of freedom. But with that said, well, in this case, m is 3. So we could say there's 2 degrees of freedom for this exact example. But actually, let's calculate the sum of squares between. So what is this going to be? This is going to be equal to, I'll just scroll down, running out of space. This is going to be equal to, this right here is 2 minus 4 is negative 2 squared is 4."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "But actually, let's calculate the sum of squares between. So what is this going to be? This is going to be equal to, I'll just scroll down, running out of space. This is going to be equal to, this right here is 2 minus 4 is negative 2 squared is 4. And then we have 3 4's over here. So it's 3 times 4 plus 3 times 0 plus, what is this? The difference between each of these, 6 minus 4 is 2, squared is 4."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "This is going to be equal to, this right here is 2 minus 4 is negative 2 squared is 4. And then we have 3 4's over here. So it's 3 times 4 plus 3 times 0 plus, what is this? The difference between each of these, 6 minus 4 is 2, squared is 4. So we have 3 times 4 plus 3 times 4. And we get 3 times 4 is 12 plus 0 plus 12 is equal to 24. So the sum of squares, or we could say the variation due to what's the difference between the groups, between the means, is 24."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "The difference between each of these, 6 minus 4 is 2, squared is 4. So we have 3 times 4 plus 3 times 4. And we get 3 times 4 is 12 plus 0 plus 12 is equal to 24. So the sum of squares, or we could say the variation due to what's the difference between the groups, between the means, is 24. Now, let's put it all together. We said that the total variation, that if you look at all nine data points, is 30. Let me write that over here."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "So the sum of squares, or we could say the variation due to what's the difference between the groups, between the means, is 24. Now, let's put it all together. We said that the total variation, that if you look at all nine data points, is 30. Let me write that over here. So the sum of squares, the total sum of squares, is equal to 30. We figured out the sum of squares between each data point and its central tendency, its sample mean. We figured out it and we totaled it all up."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "Let me write that over here. So the sum of squares, the total sum of squares, is equal to 30. We figured out the sum of squares between each data point and its central tendency, its sample mean. We figured out it and we totaled it all up. We got 6. So the sum of squares within was equal to 6. And in this case, it was 6 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "We figured out it and we totaled it all up. We got 6. So the sum of squares within was equal to 6. And in this case, it was 6 degrees of freedom. And we also had 6 degrees of freedom. Or if we wanted to write it generally, there were m times n minus 1 degrees of freedom. And actually, for the total, we figured out we had m times n minus 1 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And in this case, it was 6 degrees of freedom. And we also had 6 degrees of freedom. Or if we wanted to write it generally, there were m times n minus 1 degrees of freedom. And actually, for the total, we figured out we had m times n minus 1 degrees of freedom. Actually, let me just write degrees of freedom in this column right over here. In this case, the number turned out to be 8. And then just now, we calculated the sum of squares between the samples is equal to 24."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And actually, for the total, we figured out we had m times n minus 1 degrees of freedom. Actually, let me just write degrees of freedom in this column right over here. In this case, the number turned out to be 8. And then just now, we calculated the sum of squares between the samples is equal to 24. And we figured out that it had m minus 1 degrees of freedom, which ended up being 2. Now, the interesting thing here, and this is why this kind of analysis of variance fits nicely together. And in future videos, we'll think about how we can actually test hypotheses using some of the tools that we're thinking about right now, is that the sum of squares within plus the sum of squares between is equal to the total sum of squares."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And then just now, we calculated the sum of squares between the samples is equal to 24. And we figured out that it had m minus 1 degrees of freedom, which ended up being 2. Now, the interesting thing here, and this is why this kind of analysis of variance fits nicely together. And in future videos, we'll think about how we can actually test hypotheses using some of the tools that we're thinking about right now, is that the sum of squares within plus the sum of squares between is equal to the total sum of squares. So a way to think about it is that the total variation in this data right here can be described as the sum of the variation within each of these groups, when you take that total, plus the sum of the variation between the groups. And even the degrees of freedom work out. The sum of squares between had 2 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "And in future videos, we'll think about how we can actually test hypotheses using some of the tools that we're thinking about right now, is that the sum of squares within plus the sum of squares between is equal to the total sum of squares. So a way to think about it is that the total variation in this data right here can be described as the sum of the variation within each of these groups, when you take that total, plus the sum of the variation between the groups. And even the degrees of freedom work out. The sum of squares between had 2 degrees of freedom. The sum of squares within each of the groups had 6 degrees of freedom. 2 plus 6 is 8. That's the total degrees of freedom we had for all of the data combined."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "The sum of squares between had 2 degrees of freedom. The sum of squares within each of the groups had 6 degrees of freedom. 2 plus 6 is 8. That's the total degrees of freedom we had for all of the data combined. It even works if you look at the more general. So our sum of squares between had m minus 1 degrees of freedom. Our sum of squares within had m times n minus 1 degrees of freedom."}, {"video_title": "ANOVA 2 Calculating SSW and SSB (total sum of squares within and between)   Khan Academy.mp3", "Sentence": "That's the total degrees of freedom we had for all of the data combined. It even works if you look at the more general. So our sum of squares between had m minus 1 degrees of freedom. Our sum of squares within had m times n minus 1 degrees of freedom. So this is equal to m minus 1 plus mn minus m. These guys cancel out. This is equal to mn minus 1 degrees of freedom, which is exactly the total degrees of freedom we had for the total sum of squares. So the whole point of the calculations that we did in the last video and in this video is just to appreciate that this total variation over here, this total variation that we first calculated, can be viewed as the sum of these kind of two component variations."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The normal distribution is arguably the most important concept in statistics. Everything we do, or almost everything we do in inferential statistics, which is essentially making inferences based on data points, is to some degree based on the normal distribution. And so what I want to do in this video and in this spreadsheet is to essentially give you as deep an understanding of the normal distribution as possible. And the rest of your life, if someone says, oh, we're assuming a normal distribution, you're like, oh, I know what that is. This is the formula, and I understand how to use it, et cetera, et cetera. So this spreadsheet, just so you know, is downloadable at www.khanacademy.org slash downloads slash, and if you just type that part in, you'll see everything that's downloadable. But then downloads slash normalintro.xls."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the rest of your life, if someone says, oh, we're assuming a normal distribution, you're like, oh, I know what that is. This is the formula, and I understand how to use it, et cetera, et cetera. So this spreadsheet, just so you know, is downloadable at www.khanacademy.org slash downloads slash, and if you just type that part in, you'll see everything that's downloadable. But then downloads slash normalintro.xls. And then you'll get this spreadsheet right here. And I think I did this in the right standard. But anyway, if you go on to Wikipedia, and if you were to type in normal distribution, or you were to do a search for normal distribution, let me actually get my pen tool going, this is what you would see."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "But then downloads slash normalintro.xls. And then you'll get this spreadsheet right here. And I think I did this in the right standard. But anyway, if you go on to Wikipedia, and if you were to type in normal distribution, or you were to do a search for normal distribution, let me actually get my pen tool going, this is what you would see. I literally copied and pasted this right here from Wikipedia. And I know it looks daunting. You have all these Greek letters there."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "But anyway, if you go on to Wikipedia, and if you were to type in normal distribution, or you were to do a search for normal distribution, let me actually get my pen tool going, this is what you would see. I literally copied and pasted this right here from Wikipedia. And I know it looks daunting. You have all these Greek letters there. But this is just the sigma right here. That is just the standard deviation of the distribution. We'll play with that a little bit in this chart and see what that means."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have all these Greek letters there. But this is just the sigma right here. That is just the standard deviation of the distribution. We'll play with that a little bit in this chart and see what that means. And well, I mean, you know what the standard deviation is in general, but this is the standard deviation of this distribution, which is a probability density function. And I encourage you to re-watch the video on probability density functions, because it's a little bit of a transition going from the binomial distribution, which is discrete, right? In the binomial distribution, say, oh, what is the probability of getting a 5?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "We'll play with that a little bit in this chart and see what that means. And well, I mean, you know what the standard deviation is in general, but this is the standard deviation of this distribution, which is a probability density function. And I encourage you to re-watch the video on probability density functions, because it's a little bit of a transition going from the binomial distribution, which is discrete, right? In the binomial distribution, say, oh, what is the probability of getting a 5? And you just kind of look at that histogram or that bar chart, and you say, oh, that's the probability. But in a continuous probability distribution, or a continuous probability density function, you can't just say, what is the probability of me getting a 5? You have to say, what is the probability of me getting between, let's say, a 4.5 and a 5.5?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the binomial distribution, say, oh, what is the probability of getting a 5? And you just kind of look at that histogram or that bar chart, and you say, oh, that's the probability. But in a continuous probability distribution, or a continuous probability density function, you can't just say, what is the probability of me getting a 5? You have to say, what is the probability of me getting between, let's say, a 4.5 and a 5.5? You have to give it some range. And then your probability isn't given by just reading this graph. The probability is given by the area under that curve."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have to say, what is the probability of me getting between, let's say, a 4.5 and a 5.5? You have to give it some range. And then your probability isn't given by just reading this graph. The probability is given by the area under that curve. It would be given by this area. And for those of you all who know calculus, if P of x is our probability density function, it doesn't have to be a normal distribution, although it almost always, well, it often is a normal distribution. The way you actually figure out the probability of, let's say, between 4.5 and 5.5, what is the probability, this is whatever, the odds of me getting between 4.5 and 5.5 inches of rain tomorrow."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability is given by the area under that curve. It would be given by this area. And for those of you all who know calculus, if P of x is our probability density function, it doesn't have to be a normal distribution, although it almost always, well, it often is a normal distribution. The way you actually figure out the probability of, let's say, between 4.5 and 5.5, what is the probability, this is whatever, the odds of me getting between 4.5 and 5.5 inches of rain tomorrow. It'll actually be the integral from 4.5 to 5.5 of this probability density function, or of this probability density function, dx. So that's just the area of the curve. For those of you who don't know calculus yet, I encourage you to watch that playlist."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The way you actually figure out the probability of, let's say, between 4.5 and 5.5, what is the probability, this is whatever, the odds of me getting between 4.5 and 5.5 inches of rain tomorrow. It'll actually be the integral from 4.5 to 5.5 of this probability density function, or of this probability density function, dx. So that's just the area of the curve. For those of you who don't know calculus yet, I encourage you to watch that playlist. But all this is, is saying the area of the curve from here to here. And actually, it turns out for the normal distribution, this isn't an easy thing to evaluate analytically. And so you do it numerically."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "For those of you who don't know calculus yet, I encourage you to watch that playlist. But all this is, is saying the area of the curve from here to here. And actually, it turns out for the normal distribution, this isn't an easy thing to evaluate analytically. And so you do it numerically. And you don't have to feel bad about doing it numerically, because, oh, how do I take the integral of this? There's actually functions for it. And you can even approximate it."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so you do it numerically. And you don't have to feel bad about doing it numerically, because, oh, how do I take the integral of this? There's actually functions for it. And you can even approximate it. I mean, one way you could approximate it is you could use it the way you approximate integrals in general, where you could say, well, what is the area of this? What's roughly the area of this trapezoid? So you could figure out the area of that trapezoid, taking the average of that point and that point and multiplying it by the base."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you can even approximate it. I mean, one way you could approximate it is you could use it the way you approximate integrals in general, where you could say, well, what is the area of this? What's roughly the area of this trapezoid? So you could figure out the area of that trapezoid, taking the average of that point and that point and multiplying it by the base. Or you could just take the level of the, let me change colors, just because I think I'm overdoing it with the green, or you could just take the height of this line right here and multiply it by the base. And you'll get the area of this rectangle, which might be a pretty good approximation for the area under the curve, right, because you'll have a little bit extra over here, but you're going to miss a little bit over there. So it might be a pretty good approximation."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could figure out the area of that trapezoid, taking the average of that point and that point and multiplying it by the base. Or you could just take the level of the, let me change colors, just because I think I'm overdoing it with the green, or you could just take the height of this line right here and multiply it by the base. And you'll get the area of this rectangle, which might be a pretty good approximation for the area under the curve, right, because you'll have a little bit extra over here, but you're going to miss a little bit over there. So it might be a pretty good approximation. That's actually what I do in the other video, just to approximate the area under the curve and give you a good sense that the normal distribution is what the binomial distribution becomes, essentially, if you have many, many, many, many trials. And what's interesting about the normal distribution, just so you know, I don't know if I already mentioned this, already, this right here, this is the graph. And this is just another word."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it might be a pretty good approximation. That's actually what I do in the other video, just to approximate the area under the curve and give you a good sense that the normal distribution is what the binomial distribution becomes, essentially, if you have many, many, many, many trials. And what's interesting about the normal distribution, just so you know, I don't know if I already mentioned this, already, this right here, this is the graph. And this is just another word. People might talk about the central limit theorem, but this is really kind of one of the most important or interesting things about our universe, central limit theorem. I won't prove it here, but it essentially tells us, and you could kind of understand it by looking at the other video where we talk about flipping coins, and if we were to do many, many, many flips of coins, right, those are independent trials of each other. And if you take the sum of all of your flips, if you were to give yourself one point if you got ahead every time, and if you would take the sum of them, as you approach an infinite number of flips, you approach the normal distribution."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is just another word. People might talk about the central limit theorem, but this is really kind of one of the most important or interesting things about our universe, central limit theorem. I won't prove it here, but it essentially tells us, and you could kind of understand it by looking at the other video where we talk about flipping coins, and if we were to do many, many, many flips of coins, right, those are independent trials of each other. And if you take the sum of all of your flips, if you were to give yourself one point if you got ahead every time, and if you would take the sum of them, as you approach an infinite number of flips, you approach the normal distribution. And what's interesting about that is each of those trials, in the case of flipping a coin, each trial is a flip of the coin, each of those trials don't have to have a normal distribution. So we could be talking about molecular interactions. And every time compound x interacts with compound y, what might result doesn't have to be normally distributed."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you take the sum of all of your flips, if you were to give yourself one point if you got ahead every time, and if you would take the sum of them, as you approach an infinite number of flips, you approach the normal distribution. And what's interesting about that is each of those trials, in the case of flipping a coin, each trial is a flip of the coin, each of those trials don't have to have a normal distribution. So we could be talking about molecular interactions. And every time compound x interacts with compound y, what might result doesn't have to be normally distributed. But what happens is if you take a sum of a ton of those interactions, then all of a sudden the end result will be normally distributed. And this is why this is such an important distribution. It shows up in nature all of the time."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And every time compound x interacts with compound y, what might result doesn't have to be normally distributed. But what happens is if you take a sum of a ton of those interactions, then all of a sudden the end result will be normally distributed. And this is why this is such an important distribution. It shows up in nature all of the time. And if people are trying to kind of, if you do take data points from something that is very, very complex, and that it is the sum of arguably many, many, almost infinite individual independent trials, it's a pretty good assumption to assume the normal distribution. We'll do other videos where we talk about when it is a good assumption and when it isn't a good assumption. But anyway, just to digest this a little bit, and let me actually rewrite it."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "It shows up in nature all of the time. And if people are trying to kind of, if you do take data points from something that is very, very complex, and that it is the sum of arguably many, many, almost infinite individual independent trials, it's a pretty good assumption to assume the normal distribution. We'll do other videos where we talk about when it is a good assumption and when it isn't a good assumption. But anyway, just to digest this a little bit, and let me actually rewrite it. This is what you'll see on Wikipedia, but this could be rewritten as 1 over sigma times the square root of 2 pi times, exp is just e to that power, so it's just e to this whole thing over here, minus x minus the mean squared over 2 sigma squared. This is a standard deviation. Standard deviation squared is just the variance."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "But anyway, just to digest this a little bit, and let me actually rewrite it. This is what you'll see on Wikipedia, but this could be rewritten as 1 over sigma times the square root of 2 pi times, exp is just e to that power, so it's just e to this whole thing over here, minus x minus the mean squared over 2 sigma squared. This is a standard deviation. Standard deviation squared is just the variance. And just so you know how to use this, you're like, oh wow, there's so many Greek letters here, what do I do? This tells you the height of the normal distribution function. Let's say that this is the distribution of people's, I don't know, how far north they live from my house or something."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Standard deviation squared is just the variance. And just so you know how to use this, you're like, oh wow, there's so many Greek letters here, what do I do? This tells you the height of the normal distribution function. Let's say that this is the distribution of people's, I don't know, how far north they live from my house or something. I don't know. Well, no, that's not a good one. Let's say it's people's heights above 5' 9\"."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that this is the distribution of people's, I don't know, how far north they live from my house or something. I don't know. Well, no, that's not a good one. Let's say it's people's heights above 5' 9\". Let's say that this was 5' 9\", and not 0. Right? What this tells you is, if you were to say, what percentage of people, or I guess, what is the probability, if you wanted to figure out what is the probability of finding someone who is roughly 5 inches taller than the average right here, what you would do is you would put in this number here, this 5, into x, and then you know the standard deviation, because you've taken a bunch of samples, you know the variance, which is the standard deviation squared, you know the mean, and you just put your x in there, and it'll tell you the height of the function."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say it's people's heights above 5' 9\". Let's say that this was 5' 9\", and not 0. Right? What this tells you is, if you were to say, what percentage of people, or I guess, what is the probability, if you wanted to figure out what is the probability of finding someone who is roughly 5 inches taller than the average right here, what you would do is you would put in this number here, this 5, into x, and then you know the standard deviation, because you've taken a bunch of samples, you know the variance, which is the standard deviation squared, you know the mean, and you just put your x in there, and it'll tell you the height of the function. And then you have to give it a range. You can't just say, how many people are exactly 5 inches taller than average. You would actually say, how many people are between 5.1 inches and 4.9 inches taller than the average?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "What this tells you is, if you were to say, what percentage of people, or I guess, what is the probability, if you wanted to figure out what is the probability of finding someone who is roughly 5 inches taller than the average right here, what you would do is you would put in this number here, this 5, into x, and then you know the standard deviation, because you've taken a bunch of samples, you know the variance, which is the standard deviation squared, you know the mean, and you just put your x in there, and it'll tell you the height of the function. And then you have to give it a range. You can't just say, how many people are exactly 5 inches taller than average. You would actually say, how many people are between 5.1 inches and 4.9 inches taller than the average? You have to give it a little bit of range, because no one is exactly, or it's almost infinitely impossible to the atom to be exactly 5' 9\". Even the definition of an inch isn't defined that particularly. So that's how you use this function."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You would actually say, how many people are between 5.1 inches and 4.9 inches taller than the average? You have to give it a little bit of range, because no one is exactly, or it's almost infinitely impossible to the atom to be exactly 5' 9\". Even the definition of an inch isn't defined that particularly. So that's how you use this function. I think this is so heavily used in, one, it shows up in nature, but in all of inferential statistics, I think it behooves you to become as familiar with this formula as possible. And I guess to make that happen, let me play around a little bit with this formula, just to kind of give you an intuition of how everything works out, et cetera, et cetera. So if I were to take this, and I like to just maybe help you memorize it, this could be rewritten as, if we take the sigma into the square root sign, if we take the standard deviation in there, it becomes 1 over the square root of 2 pi sigma squared."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's how you use this function. I think this is so heavily used in, one, it shows up in nature, but in all of inferential statistics, I think it behooves you to become as familiar with this formula as possible. And I guess to make that happen, let me play around a little bit with this formula, just to kind of give you an intuition of how everything works out, et cetera, et cetera. So if I were to take this, and I like to just maybe help you memorize it, this could be rewritten as, if we take the sigma into the square root sign, if we take the standard deviation in there, it becomes 1 over the square root of 2 pi sigma squared. I've never seen it written this way, but it gives me a little intuition. That sigma squared, it's always written as sigma squared, but it's really just the variance. And the variance is what you calculate before you calculate the standard deviation."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I were to take this, and I like to just maybe help you memorize it, this could be rewritten as, if we take the sigma into the square root sign, if we take the standard deviation in there, it becomes 1 over the square root of 2 pi sigma squared. I've never seen it written this way, but it gives me a little intuition. That sigma squared, it's always written as sigma squared, but it's really just the variance. And the variance is what you calculate before you calculate the standard deviation. So that's interesting. And then this top right here, this could be written as e to the minus 1 half times, and if we were to just take this, both of these things here are squared, so we could just say x minus the mean over sigma squared. And this kind of clarifies a little bit what's going on here a little bit better."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the variance is what you calculate before you calculate the standard deviation. So that's interesting. And then this top right here, this could be written as e to the minus 1 half times, and if we were to just take this, both of these things here are squared, so we could just say x minus the mean over sigma squared. And this kind of clarifies a little bit what's going on here a little bit better. Because what's this? x minus sigma is the distance between whatever point we want to find, let's say we're here, x minus mu, mu is the mean, so that's here, so that's this distance. And this is the standard deviation, which is this distance."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this kind of clarifies a little bit what's going on here a little bit better. Because what's this? x minus sigma is the distance between whatever point we want to find, let's say we're here, x minus mu, mu is the mean, so that's here, so that's this distance. And this is the standard deviation, which is this distance. So this in here tells me how many standard deviations I am away from the mean, and that's actually called the standard z-score I talked about in the other video. And then we square that, and then we take this to the minus 1 half, well, let me rewrite that. If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half power, right?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is the standard deviation, which is this distance. So this in here tells me how many standard deviations I am away from the mean, and that's actually called the standard z-score I talked about in the other video. And then we square that, and then we take this to the minus 1 half, well, let me rewrite that. If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half power, right? If you take something to an exponent and then take that to an exponent, you can just multiply these exponents. So likewise, this could be rewritten as, this is equal to 1 over the square root of 2 pi sigma squared, which is just the variance. And I'm just playing around with the formula, because I really want you to see all the ways that it, you know, maybe you'll get a little intuition."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half power, right? If you take something to an exponent and then take that to an exponent, you can just multiply these exponents. So likewise, this could be rewritten as, this is equal to 1 over the square root of 2 pi sigma squared, which is just the variance. And I'm just playing around with the formula, because I really want you to see all the ways that it, you know, maybe you'll get a little intuition. And I encourage you to email me if you see some insight on why this exists and all of that. But once again, I think it is cool that all of a sudden we have this other formula that has pi and e in it, and this is really just, you know, this is what the central, you know, so many phenomenon are described by this. And once again, pi and e show up together, right?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm just playing around with the formula, because I really want you to see all the ways that it, you know, maybe you'll get a little intuition. And I encourage you to email me if you see some insight on why this exists and all of that. But once again, I think it is cool that all of a sudden we have this other formula that has pi and e in it, and this is really just, you know, this is what the central, you know, so many phenomenon are described by this. And once again, pi and e show up together, right? Just like e to the i pi is equal to negative 1 tells you something about our universe. But anyway, I could rewrite this as e to the x minus mu over sigma squared, and all of that to the minus 1 half. Something to the minus 1 half power, that's just 1 over the square root, which is already going on here."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And once again, pi and e show up together, right? Just like e to the i pi is equal to negative 1 tells you something about our universe. But anyway, I could rewrite this as e to the x minus mu over sigma squared, and all of that to the minus 1 half. Something to the minus 1 half power, that's just 1 over the square root, which is already going on here. So we could just rewrite this over here as 1 over the square root of 2 pi times the variance times e to essentially our z-score squared, right? If we say z is this thing in here, z is how many standard deviations we are from the mean, z-score squared. And all of a sudden this kind of becomes a very clean, you know, we just say 2 pi times our variance times e to the number of standard deviations we are away from the mean."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Something to the minus 1 half power, that's just 1 over the square root, which is already going on here. So we could just rewrite this over here as 1 over the square root of 2 pi times the variance times e to essentially our z-score squared, right? If we say z is this thing in here, z is how many standard deviations we are from the mean, z-score squared. And all of a sudden this kind of becomes a very clean, you know, we just say 2 pi times our variance times e to the number of standard deviations we are away from the mean. You square that. You take the square root of that thing and invert it, and that's also the normal distribution. So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it, and that way if you see it in any of these other forms in the rest of your life, you won't say, what's that?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And all of a sudden this kind of becomes a very clean, you know, we just say 2 pi times our variance times e to the number of standard deviations we are away from the mean. You square that. You take the square root of that thing and invert it, and that's also the normal distribution. So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it, and that way if you see it in any of these other forms in the rest of your life, you won't say, what's that? I thought the normal distribution was this or was this, and now you know. With that said, let's play around a little bit with this normal distribution. So in this spreadsheet, I've plotted normal distribution."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it, and that way if you see it in any of these other forms in the rest of your life, you won't say, what's that? I thought the normal distribution was this or was this, and now you know. With that said, let's play around a little bit with this normal distribution. So in this spreadsheet, I've plotted normal distribution. You can change the assumptions that are in this kind of a green-blue color. So right now it's plotting it with a mean of 0 and a standard deviation of 4. And I just write the variance here just for your information."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in this spreadsheet, I've plotted normal distribution. You can change the assumptions that are in this kind of a green-blue color. So right now it's plotting it with a mean of 0 and a standard deviation of 4. And I just write the variance here just for your information. The variance is just the standard deviation squared. And so what happens when you change the mean? So if the mean goes from 0 to let's say it goes to 5."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I just write the variance here just for your information. The variance is just the standard deviation squared. And so what happens when you change the mean? So if the mean goes from 0 to let's say it goes to 5. Notice this graph just shifted to the right by 5, right? It was centered here. Now it's centered over here."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if the mean goes from 0 to let's say it goes to 5. Notice this graph just shifted to the right by 5, right? It was centered here. Now it's centered over here. If we make it minus 5, what happens? The whole bell curve just shifts 5 to the left from the center. Now what happens when you change the standard deviation?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now it's centered over here. If we make it minus 5, what happens? The whole bell curve just shifts 5 to the left from the center. Now what happens when you change the standard deviation? The standard deviation is a measure of the average squared distance from the mean. The standard deviation is the square root of that. So it's kind of, not exactly, but kind of the average distance from the mean."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what happens when you change the standard deviation? The standard deviation is a measure of the average squared distance from the mean. The standard deviation is the square root of that. So it's kind of, not exactly, but kind of the average distance from the mean. So the smaller the standard deviation, the closer a lot of the points are going to be to the mean. So we should get kind of a narrower graph. And let's see that happens."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's kind of, not exactly, but kind of the average distance from the mean. So the smaller the standard deviation, the closer a lot of the points are going to be to the mean. So we should get kind of a narrower graph. And let's see that happens. So when the standard deviation is 2, we see that. The graph, you're more likely to be really close to the mean than further away. And if you make the standard deviation, I don't know, if you make it 10, all of a sudden you get a really flat graph."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's see that happens. So when the standard deviation is 2, we see that. The graph, you're more likely to be really close to the mean than further away. And if you make the standard deviation, I don't know, if you make it 10, all of a sudden you get a really flat graph. And this thing keeps going on forever. And that's a key difference. The binomial distribution is always finite."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you make the standard deviation, I don't know, if you make it 10, all of a sudden you get a really flat graph. And this thing keeps going on forever. And that's a key difference. The binomial distribution is always finite. You can only have a finite number of values. While the normal distribution is defined over the entire real number line. So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1,000 here is very, very low."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The binomial distribution is always finite. You can only have a finite number of values. While the normal distribution is defined over the entire real number line. So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1,000 here is very, very low. But there is some probability that all of the atoms in my body just arrange perfectly that I fall through the seat I'm sitting on. It's very unlikely. And it probably won't happen in the life of the universe."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1,000 here is very, very low. But there is some probability that all of the atoms in my body just arrange perfectly that I fall through the seat I'm sitting on. It's very unlikely. And it probably won't happen in the life of the universe. But it can happen. And that could be described by a normal distribution. Because it says anything can happen, although it could be very, very, very unprobable."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it probably won't happen in the life of the universe. But it can happen. And that could be described by a normal distribution. Because it says anything can happen, although it could be very, very, very unprobable. So the thing I talked about at the beginning of the video is when you figure out a normal distribution, you can't just look at this point on the graph. Let me get the pen tool back. You have to figure out the area under the curve between two points."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because it says anything can happen, although it could be very, very, very unprobable. So the thing I talked about at the beginning of the video is when you figure out a normal distribution, you can't just look at this point on the graph. Let me get the pen tool back. You have to figure out the area under the curve between two points. So if I wanted to say, let's say this was our distribution. I said, what is the probability that I get 0? I don't know what phenomena this is describing, but that 0 happened."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have to figure out the area under the curve between two points. So if I wanted to say, let's say this was our distribution. I said, what is the probability that I get 0? I don't know what phenomena this is describing, but that 0 happened. If I say exactly 0, the probability is 0. Because I shouldn't use 0 too much. Because the area under the curve, just under 0, there's no area."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I don't know what phenomena this is describing, but that 0 happened. If I say exactly 0, the probability is 0. Because I shouldn't use 0 too much. Because the area under the curve, just under 0, there's no area. It's just a line. You have to say between a range. So you have to say the probability between, let's say minus, and actually I can type it in here."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because the area under the curve, just under 0, there's no area. It's just a line. You have to say between a range. So you have to say the probability between, let's say minus, and actually I can type it in here. I can say the probability between, let's say, minus 0.005 and plus 0.05 is, well, it rounded. So it says they're close to 0. Let me do it."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have to say the probability between, let's say minus, and actually I can type it in here. I can say the probability between, let's say, minus 0.005 and plus 0.05 is, well, it rounded. So it says they're close to 0. Let me do it. Between minus 1 and between 1. It calculated it at 7%. And I'll show you how I calculated this in a second."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do it. Between minus 1 and between 1. It calculated it at 7%. And I'll show you how I calculated this in a second. So let me get the screen drawn to a log. So what did I just do? This, between minus 1 and 1, and I'll show you behind the scenes what Excel is doing."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll show you how I calculated this in a second. So let me get the screen drawn to a log. So what did I just do? This, between minus 1 and 1, and I'll show you behind the scenes what Excel is doing. We're going from minus 1, which is roughly right here, to 1, and we're calculating the area under the curve. We're calculating this area. Or for those of you who know calculus, we're calculating the integral from minus 1 to 1 of this function, where the standard deviation is right here, is 10, and the mean is minus 5."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This, between minus 1 and 1, and I'll show you behind the scenes what Excel is doing. We're going from minus 1, which is roughly right here, to 1, and we're calculating the area under the curve. We're calculating this area. Or for those of you who know calculus, we're calculating the integral from minus 1 to 1 of this function, where the standard deviation is right here, is 10, and the mean is minus 5. Actually, let me put that in. So we're calculating, for this example, the way it's drawn right here, the normal distribution function, let's see, our standard deviation is 10 times the square root of 2 pi times e to the minus 1 half times x minus our mean. Our mean is negative right now."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or for those of you who know calculus, we're calculating the integral from minus 1 to 1 of this function, where the standard deviation is right here, is 10, and the mean is minus 5. Actually, let me put that in. So we're calculating, for this example, the way it's drawn right here, the normal distribution function, let's see, our standard deviation is 10 times the square root of 2 pi times e to the minus 1 half times x minus our mean. Our mean is negative right now. Our mean is minus 5, so it's x plus 5 over the standard deviation squared, which is the variance. So that's 100 squared dx. This is what this number is right here, this 7%, or actually 0.07 is the area right under there."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our mean is negative right now. Our mean is minus 5, so it's x plus 5 over the standard deviation squared, which is the variance. So that's 100 squared dx. This is what this number is right here, this 7%, or actually 0.07 is the area right under there. Now, unfortunately for us in the world, this isn't an easy integral to evaluate analytically, even for those of us who know our calculus. So this tends to be done numerically, and kind of an easy way to do this, well, not an easy way, but a function has been defined called the cumulative distribution function that is a useful tool for figuring out this area. So what the cumulative distribution function is essentially, let me call it, cumulative distribution function, it's a function of x."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is what this number is right here, this 7%, or actually 0.07 is the area right under there. Now, unfortunately for us in the world, this isn't an easy integral to evaluate analytically, even for those of us who know our calculus. So this tends to be done numerically, and kind of an easy way to do this, well, not an easy way, but a function has been defined called the cumulative distribution function that is a useful tool for figuring out this area. So what the cumulative distribution function is essentially, let me call it, cumulative distribution function, it's a function of x. It gives us the area under the curve, under this curve, so let's say that this is x right here, that's our x. It tells you the area under the curve up to x. Or so another way to think about it, it tells you what is the probability that you land at some value less than your x value, so it's the area from minus infinity to x of our probability density function, dx."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what the cumulative distribution function is essentially, let me call it, cumulative distribution function, it's a function of x. It gives us the area under the curve, under this curve, so let's say that this is x right here, that's our x. It tells you the area under the curve up to x. Or so another way to think about it, it tells you what is the probability that you land at some value less than your x value, so it's the area from minus infinity to x of our probability density function, dx. And there's actually an Excel, when you actually use the Excel normal distribution function, you say norm distribution, you have to give it your x value, you give it the mean, you give it the standard deviation, and then you say whether you want the cumulative distribution, in which case you say true, or you want just this normal distribution, which you say false. So if you wanted to graph this right here, you would say false, in caps. If you wanted to graph the cumulative distribution function, which I do down here, let me move this down a little bit, let me get out of the pen tool, so the cumulative distribution function is right over here, then you say true when you make that Excel call."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or so another way to think about it, it tells you what is the probability that you land at some value less than your x value, so it's the area from minus infinity to x of our probability density function, dx. And there's actually an Excel, when you actually use the Excel normal distribution function, you say norm distribution, you have to give it your x value, you give it the mean, you give it the standard deviation, and then you say whether you want the cumulative distribution, in which case you say true, or you want just this normal distribution, which you say false. So if you wanted to graph this right here, you would say false, in caps. If you wanted to graph the cumulative distribution function, which I do down here, let me move this down a little bit, let me get out of the pen tool, so the cumulative distribution function is right over here, then you say true when you make that Excel call. So this is a cumulative distribution function for the same, for this, this is a normal distribution, here's a cumulative distribution. And just so you get the intuition, if you want to know what is the probability that I get a value less than 20, so I can get any value less than 20 given this distribution, the cumulative distribution right here, let me make it so you can see the, if you go to 20, you just go right to that point there and you say wow, the probability of getting 20 or less, it's pretty high, it's approaching 100%, that makes sense because most of the area under this curve is less than 20. Or if you said what's the probability of getting less than minus 5, well minus 5 was the mean, so half of your result should be above that and half should be below."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you wanted to graph the cumulative distribution function, which I do down here, let me move this down a little bit, let me get out of the pen tool, so the cumulative distribution function is right over here, then you say true when you make that Excel call. So this is a cumulative distribution function for the same, for this, this is a normal distribution, here's a cumulative distribution. And just so you get the intuition, if you want to know what is the probability that I get a value less than 20, so I can get any value less than 20 given this distribution, the cumulative distribution right here, let me make it so you can see the, if you go to 20, you just go right to that point there and you say wow, the probability of getting 20 or less, it's pretty high, it's approaching 100%, that makes sense because most of the area under this curve is less than 20. Or if you said what's the probability of getting less than minus 5, well minus 5 was the mean, so half of your result should be above that and half should be below. And if you go to this point right here, you can see that this right here is 50%. So the probability of getting less than minus 5 is exactly 50%. So what you do is, if I wanted to know the probability of getting between negative 1 and 1, what I do is, let me get back to my pen tool, what I do is I figure out what is the probability of getting minus 1 or lower, right?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if you said what's the probability of getting less than minus 5, well minus 5 was the mean, so half of your result should be above that and half should be below. And if you go to this point right here, you can see that this right here is 50%. So the probability of getting less than minus 5 is exactly 50%. So what you do is, if I wanted to know the probability of getting between negative 1 and 1, what I do is, let me get back to my pen tool, what I do is I figure out what is the probability of getting minus 1 or lower, right? So I figure out this whole area and then I figure out the probability of getting 1 or lower, which is this whole area, well let me do it in a different color, 1 or lower is everything there. And I subtract the yellow area from the magenta area and I'll just get what's ever left over here, right? So what I do is I take, and that's exactly what I did in the spreadsheet."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what you do is, if I wanted to know the probability of getting between negative 1 and 1, what I do is, let me get back to my pen tool, what I do is I figure out what is the probability of getting minus 1 or lower, right? So I figure out this whole area and then I figure out the probability of getting 1 or lower, which is this whole area, well let me do it in a different color, 1 or lower is everything there. And I subtract the yellow area from the magenta area and I'll just get what's ever left over here, right? So what I do is I take, and that's exactly what I did in the spreadsheet. Let me scroll down. This might be taxing my computer by taking the screen capture with it. So what I did is I evaluated the cumulative distribution function at 1, which would be right there."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what I do is I take, and that's exactly what I did in the spreadsheet. Let me scroll down. This might be taxing my computer by taking the screen capture with it. So what I did is I evaluated the cumulative distribution function at 1, which would be right there. And I evaluate the cumulative distribution function at minus 1, which is right there. And the difference between these two, I subtract this number from this number and that tells me essentially the probability that I'm between those two numbers, or another way to think about it, the area right here. And I really encourage you to play with this and explore the Excel formulas and everything."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what I did is I evaluated the cumulative distribution function at 1, which would be right there. And I evaluate the cumulative distribution function at minus 1, which is right there. And the difference between these two, I subtract this number from this number and that tells me essentially the probability that I'm between those two numbers, or another way to think about it, the area right here. And I really encourage you to play with this and explore the Excel formulas and everything. This area right here, between minus 1 and 1. Now one thing that shows up a lot is what's the probability that you land within the standard deviation of, and just so you know this graph, the central line right here, this is the mean, and then these two lines I drew right here, these are one standard deviation below and one standard deviation above the mean. And some people think, what's the probability that I land within one standard deviation of the mean?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I really encourage you to play with this and explore the Excel formulas and everything. This area right here, between minus 1 and 1. Now one thing that shows up a lot is what's the probability that you land within the standard deviation of, and just so you know this graph, the central line right here, this is the mean, and then these two lines I drew right here, these are one standard deviation below and one standard deviation above the mean. And some people think, what's the probability that I land within one standard deviation of the mean? Well, that's easy to do. What I can do is I'll just click on this and I could call this, what's the probability that I land between, let's see, one standard deviation, the mean is minus 5, one standard deviation below the mean is minus 15. And one standard deviation above the mean is 10 plus minus 5 is 5."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And some people think, what's the probability that I land within one standard deviation of the mean? Well, that's easy to do. What I can do is I'll just click on this and I could call this, what's the probability that I land between, let's see, one standard deviation, the mean is minus 5, one standard deviation below the mean is minus 15. And one standard deviation above the mean is 10 plus minus 5 is 5. So that's between 5 and 15. So 68.3%, and that's actually always the case, that you have a 68.3% probability of landing within one standard deviation of the mean, assuming you have a normal distribution. So once again, that number comes from, that represents the area under the curve here."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And one standard deviation above the mean is 10 plus minus 5 is 5. So that's between 5 and 15. So 68.3%, and that's actually always the case, that you have a 68.3% probability of landing within one standard deviation of the mean, assuming you have a normal distribution. So once again, that number comes from, that represents the area under the curve here. This area under the curve. And the way you get it is with the cumulative distribution function. Let me go down here."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once again, that number comes from, that represents the area under the curve here. This area under the curve. And the way you get it is with the cumulative distribution function. Let me go down here. Every time I move this, I have to get rid of the pen tool. So you go from, you evaluate it at plus 5, which is right here, this was one standard deviation above the mean, which, that's a number right around there. Looks like it's like, I don't know, 80 something percent, maybe 90% roughly."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me go down here. Every time I move this, I have to get rid of the pen tool. So you go from, you evaluate it at plus 5, which is right here, this was one standard deviation above the mean, which, that's a number right around there. Looks like it's like, I don't know, 80 something percent, maybe 90% roughly. And then you evaluate it at one standard deviation below the mean, which is minus 15. And this one looks like, I don't know, roughly 15% or so. 15%, 16%, maybe 17%, let's say 18%."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Looks like it's like, I don't know, 80 something percent, maybe 90% roughly. And then you evaluate it at one standard deviation below the mean, which is minus 15. And this one looks like, I don't know, roughly 15% or so. 15%, 16%, maybe 17%, let's say 18%. But the big picture is when you subtract this value from this value, you get the probability that you land between those two. And that's because this value tells the probability that you're less than. So when you go to the cumulative distribution function, you get that right there."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "15%, 16%, maybe 17%, let's say 18%. But the big picture is when you subtract this value from this value, you get the probability that you land between those two. And that's because this value tells the probability that you're less than. So when you go to the cumulative distribution function, you get that right there. That tells the probability that you are, let me get, I keep scrolling back and forth. Let me, that tells you that you're the, so when you go to 5, and you just go right over here, this essentially tells you this area under the curve. The probability that you're less than or equal to 5, everything up there."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when you go to the cumulative distribution function, you get that right there. That tells the probability that you are, let me get, I keep scrolling back and forth. Let me, that tells you that you're the, so when you go to 5, and you just go right over here, this essentially tells you this area under the curve. The probability that you're less than or equal to 5, everything up there. And then when you evaluate it at minus 15 down here, it tells you the probability that you're down back here. So when you subtract this from the larger thing, you're just left with what's under the curve right there. And just to understand this spreadsheet a little bit better, just because I really want you to play with it and move the, see what happens when, if I make this distribution, the mean was minus 5, now let me make it 5."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability that you're less than or equal to 5, everything up there. And then when you evaluate it at minus 15 down here, it tells you the probability that you're down back here. So when you subtract this from the larger thing, you're just left with what's under the curve right there. And just to understand this spreadsheet a little bit better, just because I really want you to play with it and move the, see what happens when, if I make this distribution, the mean was minus 5, now let me make it 5. It just shifted to the right. It just moved over to the right by 5, right? Whoops."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just to understand this spreadsheet a little bit better, just because I really want you to play with it and move the, see what happens when, if I make this distribution, the mean was minus 5, now let me make it 5. It just shifted to the right. It just moved over to the right by 5, right? Whoops. I'll use the pen tool. It just moved over to the right by 5. If I were to try to make the standard deviation smaller, we'll see that the whole thing just gets a little bit tighter."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Whoops. I'll use the pen tool. It just moved over to the right by 5. If I were to try to make the standard deviation smaller, we'll see that the whole thing just gets a little bit tighter. Let's make it 6. And all of a sudden, this looks a little bit tighter curve, we make it 2, it becomes even tighter. And just so you know how I calculated everything, and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function, and think a lot about how it relates to the binomial distribution, and I covered that in the last video."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I were to try to make the standard deviation smaller, we'll see that the whole thing just gets a little bit tighter. Let's make it 6. And all of a sudden, this looks a little bit tighter curve, we make it 2, it becomes even tighter. And just so you know how I calculated everything, and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function, and think a lot about how it relates to the binomial distribution, and I covered that in the last video. To plot this, I just took each of these points. I went to plot the points between minus 20 and 20, and I just incremented by 1. I just decided to increment by 1."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just so you know how I calculated everything, and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function, and think a lot about how it relates to the binomial distribution, and I covered that in the last video. To plot this, I just took each of these points. I went to plot the points between minus 20 and 20, and I just incremented by 1. I just decided to increment by 1. So this isn't a continuous curve. It's actually just plotting a point at each point and connecting it with a line. Then I did the distance between each of those points and the mean."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I just decided to increment by 1. So this isn't a continuous curve. It's actually just plotting a point at each point and connecting it with a line. Then I did the distance between each of those points and the mean. So I just took, let's say, 0 minus 5. This is this distance. So this just tells you the point minus 20 is 25 less than the mean, right?"}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then I did the distance between each of those points and the mean. So I just took, let's say, 0 minus 5. This is this distance. So this just tells you the point minus 20 is 25 less than the mean, right? That's all I did there. Then I divided that by the standard deviation, and this is the z-score, the standard z-score. So this tells me how many standard deviations is minus 20 away from the mean."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this just tells you the point minus 20 is 25 less than the mean, right? That's all I did there. Then I divided that by the standard deviation, and this is the z-score, the standard z-score. So this tells me how many standard deviations is minus 20 away from the mean. It's 12 and 1 half standard deviations below the mean. And then I used that, and I just plugged it into essentially this formula to figure out the height of the function. So let's say at minus 20, the height is very low."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this tells me how many standard deviations is minus 20 away from the mean. It's 12 and 1 half standard deviations below the mean. And then I used that, and I just plugged it into essentially this formula to figure out the height of the function. So let's say at minus 20, the height is very low. At minus 5, well, let's say at minus 2, the height's a little bit better. The height's going to be someplace, it's going to be like right there. And so that gives me that value."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say at minus 20, the height is very low. At minus 5, well, let's say at minus 2, the height's a little bit better. The height's going to be someplace, it's going to be like right there. And so that gives me that value. But then to actually figure out the probability of that, what I do is I calculate the cumulative distribution function between, well, this is the value, the probability that you're less than that. So the area under the curve below that, which is very, very small. It's not 0."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so that gives me that value. But then to actually figure out the probability of that, what I do is I calculate the cumulative distribution function between, well, this is the value, the probability that you're less than that. So the area under the curve below that, which is very, very small. It's not 0. I know it looks like 0 here, but that's only because I round it. It's going to be 0, 0, 0, 1. It's going to be a really, really small number."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's not 0. I know it looks like 0 here, but that's only because I round it. It's going to be 0, 0, 0, 1. It's going to be a really, really small number. There's some probability that we even get like minus 1,000. And another intuitive thing that you really should have a sense for is the integral over this, or the entire area of the curve, has to be 1. Because that takes into account all possible circumstances."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be a really, really small number. There's some probability that we even get like minus 1,000. And another intuitive thing that you really should have a sense for is the integral over this, or the entire area of the curve, has to be 1. Because that takes into account all possible circumstances. And that should happen if we put a suitably small number here and a suitably large number here. There you go. We get 100%."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because that takes into account all possible circumstances. And that should happen if we put a suitably small number here and a suitably large number here. There you go. We get 100%. Although this isn't 100%. We would have to go from minus infinity to plus infinity to really get 100%. It's just rounding to 100%."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "We get 100%. Although this isn't 100%. We would have to go from minus infinity to plus infinity to really get 100%. It's just rounding to 100%. It's probably 99.999999% or something like that. And so to actually calculate this, what I do is I take the cumulative distribution function of this point, and I subtract from that the cumulative distribution function of that point. And that's where I got this 100% from."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's just rounding to 100%. It's probably 99.999999% or something like that. And so to actually calculate this, what I do is I take the cumulative distribution function of this point, and I subtract from that the cumulative distribution function of that point. And that's where I got this 100% from. Anyway, hopefully that'll give you a good feel for the normal distribution. And I really encourage you to play with the spreadsheet and to even make a spreadsheet like this yourself. And in future exercises, we'll actually use this type of spreadsheet as an input into other models."}, {"video_title": "Introduction to the normal distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's where I got this 100% from. Anyway, hopefully that'll give you a good feel for the normal distribution. And I really encourage you to play with the spreadsheet and to even make a spreadsheet like this yourself. And in future exercises, we'll actually use this type of spreadsheet as an input into other models. So if we're doing a financial model, and if we say our revenue has a normal distribution around some expected value, what is the distribution of our net income? Or we could think of 100 other different types of examples. Anyway, see you in the next video."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "Assume that the conditions for inference were met. What is the approximate p-value for Katerina's test? So like always, pause this video and see if you can figure it out. Well, I just always like to remind ourselves what's going on here. So there's some population here. She has a null hypothesis that the mean is equal to zero, but the alternative is that it's not equal to zero. She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "Well, I just always like to remind ourselves what's going on here. So there's some population here. She has a null hypothesis that the mean is equal to zero, but the alternative is that it's not equal to zero. She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. From that, since we care about, the population parameter we care about is the population mean, she would calculate the sample mean in order to estimate that, and the sample standard deviation. And then from that, we can calculate this T-value. The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. From that, since we care about, the population parameter we care about is the population mean, she would calculate the sample mean in order to estimate that, and the sample standard deviation. And then from that, we can calculate this T-value. The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed based on the null hypothesis, sampling distribution standard deviation, but here we have to estimate it. And so it's going to be our sample standard deviation divided by the square root of N. Now in this example, they calculated all of this for us. They said, hey, this is going to be equal to 2.75."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed based on the null hypothesis, sampling distribution standard deviation, but here we have to estimate it. And so it's going to be our sample standard deviation divided by the square root of N. Now in this example, they calculated all of this for us. They said, hey, this is going to be equal to 2.75. And so we can just use that to figure out our P-value. But let's just think about what that is asking us to do. So the null hypothesis is that the mean is zero."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "They said, hey, this is going to be equal to 2.75. And so we can just use that to figure out our P-value. But let's just think about what that is asking us to do. So the null hypothesis is that the mean is zero. The alternative is that it is not equal to zero. So this is a situation where, if we're looking at the T-distribution right over here, it's my quick drawing of a T-distribution, if this is the mean of our T-distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean, because we care about things that are different from the mean, not just things that are greater than the mean or less than the mean. So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean?"}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So the null hypothesis is that the mean is zero. The alternative is that it is not equal to zero. So this is a situation where, if we're looking at the T-distribution right over here, it's my quick drawing of a T-distribution, if this is the mean of our T-distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean, because we care about things that are different from the mean, not just things that are greater than the mean or less than the mean. So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? And similarly, what's the probability of getting a T-value that is 2.75 or less below, or 2.75 or more below the mean? So this is negative 2.75 right over there. So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? And similarly, what's the probability of getting a T-value that is 2.75 or less below, or 2.75 or more below the mean? So this is negative 2.75 right over there. So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. First of all, you have your degrees of freedom. That's just going to be your sample size minus one. So in this example, our sample size is six, so six minus one is five."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. First of all, you have your degrees of freedom. That's just going to be your sample size minus one. So in this example, our sample size is six, so six minus one is five. And so we are going to be, we are going to be in this row right over here. And then what you wanna do is you wanna look up your T-value. This is T distribution critical value."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So in this example, our sample size is six, so six minus one is five. And so we are going to be, we are going to be in this row right over here. And then what you wanna do is you wanna look up your T-value. This is T distribution critical value. So we wanna look up 2.75 on this row. And we see 2.75, it's a little bit less than that, but that's the closest value. It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "This is T distribution critical value. So we wanna look up 2.75 on this row. And we see 2.75, it's a little bit less than that, but that's the closest value. It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. And so our tail probability, and remember, this is only giving us this probability right over here. Our tail probability is going to be between 0.025 and 0.02, and it's going to be closer than to this one. It's gonna be approximately this."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. And so our tail probability, and remember, this is only giving us this probability right over here. Our tail probability is going to be between 0.025 and 0.02, and it's going to be closer than to this one. It's gonna be approximately this. It'll actually be a little bit greater, because we're gonna go a little bit in that direction, because we are less than 2.757. And so we could say this is approximately 0.02. Well, if that's 0.02 approximately, the T distribution's symmetric, this is going to be approximately 0.02."}, {"video_title": "Using a table to estimate P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "It's gonna be approximately this. It'll actually be a little bit greater, because we're gonna go a little bit in that direction, because we are less than 2.757. And so we could say this is approximately 0.02. Well, if that's 0.02 approximately, the T distribution's symmetric, this is going to be approximately 0.02. And so our P-value, which is going to be the probability of getting a T-value that is at least 2.75 above the mean and, or 2.75 below the mean, the P-value, P-value, is going to be approximately the sum of these areas, which is 0.04. And then, of course, Katarina would wanna compare that to her significance level that she set ahead of time. And if this is lower than that, then she would reject the null hypothesis, and that would suggest the alternative."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Sports utility vehicles, also known as SUVs, make up 12% of the vehicles she registers. Let V be the number of vehicles Amelia registers in a day until she first registers an SUV. Assume that the type of each vehicle is independent. Find the probability that Amelia registers more than four vehicles before she registers an SUV. Let's first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Find the probability that Amelia registers more than four vehicles before she registers an SUV. Let's first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV but the second person is, then V would be equal to two, so forth and so on. So this right over here is a classic geometric random variable right over here. I'll say geometric random variable."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV but the second person is, then V would be equal to two, so forth and so on. So this right over here is a classic geometric random variable right over here. I'll say geometric random variable. We have a very clear success metric for each trial. Do we have an SUV or not? Each trial is independent."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "I'll say geometric random variable. We have a very clear success metric for each trial. Do we have an SUV or not? Each trial is independent. They tell us that. They are independent. The probability of success in each trial is constant."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Each trial is independent. They tell us that. They are independent. The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. The reason why this is not a binomial random variable is that we do not have a finite number of trials. Here, we're going to keep performing trials."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. The reason why this is not a binomial random variable is that we do not have a finite number of trials. Here, we're going to keep performing trials. We're going to keep serving people in the line until we get an SUV. What we have over here, when they say find the probability that Amelia registers more than four vehicles before she registers an SUV, this is the probability that V is greater than four. I encourage you, like always, pause this video and see if you can work through it."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Here, we're going to keep performing trials. We're going to keep serving people in the line until we get an SUV. What we have over here, when they say find the probability that Amelia registers more than four vehicles before she registers an SUV, this is the probability that V is greater than four. I encourage you, like always, pause this video and see if you can work through it. We're going to assume that she's just not going to leave her desk or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. We will just keep looking at people, I guess we could say, over multiple days, forever."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "I encourage you, like always, pause this video and see if you can work through it. We're going to assume that she's just not going to leave her desk or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. We will just keep looking at people, I guess we could say, over multiple days, forever. It will work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. Try to figure this out. I'm assuming you've had a go."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "We will just keep looking at people, I guess we could say, over multiple days, forever. It will work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. Try to figure this out. I'm assuming you've had a go. Some of you might say, well, isn't this going to be equal to the probability that V is equal to five plus the probability that V is equal to six plus the probability that V is equal to seven, and it just goes on and on and on forever. This is actually true. You say, well, how do I calculate this?"}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "I'm assuming you've had a go. Some of you might say, well, isn't this going to be equal to the probability that V is equal to five plus the probability that V is equal to six plus the probability that V is equal to seven, and it just goes on and on and on forever. This is actually true. You say, well, how do I calculate this? It's just summing up an infinite number of things. Now, the key realization here is that one way to think about the probability that V is greater than four is this is the same thing as the probability that V is not less than or equal to four. These two things are equivalent."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "You say, well, how do I calculate this? It's just summing up an infinite number of things. Now, the key realization here is that one way to think about the probability that V is greater than four is this is the same thing as the probability that V is not less than or equal to four. These two things are equivalent. What's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "These two things are equivalent. What's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out. What's the probability that V is not less than or equal to four? That's the same thing as the probability of first four customers or first four, I guess, people, first four, I'll say, customers, or I'll say first four cars, the customers' cars, not SUVs. This one is feeling pretty straightforward."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, pause the video and see if you can figure it out. What's the probability that V is not less than or equal to four? That's the same thing as the probability of first four customers or first four, I guess, people, first four, I'll say, customers, or I'll say first four cars, the customers' cars, not SUVs. This one is feeling pretty straightforward. What's the probability that for each customer she goes to that they're not an SUV? That's one minus 12% or 88% or 0.88. If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "This one is feeling pretty straightforward. What's the probability that for each customer she goes to that they're not an SUV? That's one minus 12% or 88% or 0.88. If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. That's all we have to calculate. Let's get our calculator out. I'm going to get 0.88, and I'm going to raise it to the fourth power."}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. That's all we have to calculate. Let's get our calculator out. I'm going to get 0.88, and I'm going to raise it to the fourth power. I'm just going to round it to the nearest. Let's see. Do they tell me to round it?"}, {"video_title": "Cumulative geometric probability (greater than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "I'm going to get 0.88, and I'm going to raise it to the fourth power. I'm just going to round it to the nearest. Let's see. Do they tell me to round it? Okay, I'll just round it to the nearest, I guess, well, hundredth. I'll just write it as 0.5997 is equal to or approximately equal to 0.5997. If you wanted to write this as a percentage, it would be approximately 59.97%."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just review factorial a little bit. So if I were to say n factorial, that of course is going to be n times n minus, sorry, times n minus one times n minus two, n minus two, and I would just keep going down until I go to times one. So I would keep decrementing n until I get to one, and then I would multiply all of those things together. So for example, in all of this is review, if I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, well, I just keep decrementing until I get to one, but I don't even have to decrement here. I'm already at one, so I just multiply it one."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, in all of this is review, if I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, well, I just keep decrementing until I get to one, but I don't even have to decrement here. I'm already at one, so I just multiply it one. Now what about zero factorial? This is interesting. Zero factorial."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm already at one, so I just multiply it one. Now what about zero factorial? This is interesting. Zero factorial. So one logical thing to say, well, hey, maybe zero factorial is zero. I'm just starting with itself. It's already below one."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Zero factorial. So one logical thing to say, well, hey, maybe zero factorial is zero. I'm just starting with itself. It's already below one. Maybe it is zero. Now what we will see is that this is not the case, that mathematicians have decided. And remember, this is what's interesting."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's already below one. Maybe it is zero. Now what we will see is that this is not the case, that mathematicians have decided. And remember, this is what's interesting. The factorial operation, this is something that humans have invented, that they think is just an interesting thing. It's a useful notation. So they can define what it does."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And remember, this is what's interesting. The factorial operation, this is something that humans have invented, that they think is just an interesting thing. It's a useful notation. So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as, and there's a little bit of a drum roll here, they believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as, and there's a little bit of a drum roll here, they believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations, which is, frankly, where factorial shows up the most. I would say that most of the cases that I've ever seen, factorial in anything, has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations, which is, frankly, where factorial shows up the most. I would say that most of the cases that I've ever seen, factorial in anything, has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that, hey, you know, if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be, it's going to be n factorial over n minus, over n minus k factorial. Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that, hey, you know, if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be, it's going to be n factorial over n minus, over n minus k factorial. Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. The first place has, you know, let's just do this. So this is the first place, this is the second place, this is the third place, all the way you get to the nth place. Well, there would be n possibilities for who's in the first position or which object is in the first position."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. The first place has, you know, let's just do this. So this is the first place, this is the second place, this is the third place, all the way you get to the nth place. Well, there would be n possibilities for who's in the first position or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position because you've already put one into that position. Now for each of these n times n minus one possibilities where you've placed two things, there would be n minus two possibilities of what goes in the third position and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there would be n possibilities for who's in the first position or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position because you've already put one into that position. Now for each of these n times n minus one possibilities where you've placed two things, there would be n minus two possibilities of what goes in the third position and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here. This is equal to, this is equal to n factorial. But if we directly applied this formula, if we applied this formula, this would need to be n factorial over n minus n factorial. And then you might see why this is interesting because this is going to be n factorial over zero factorial."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this thing right over here is exactly what we wrote over here. This is equal to, this is equal to n factorial. But if we directly applied this formula, if we applied this formula, this would need to be n factorial over n minus n factorial. And then you might see why this is interesting because this is going to be n factorial over zero factorial. So in order for this formula to apply for even in the case where k is equal to n, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, hey, this thing we've constructed called factorial, you know, we've said, hey, you put an exclamation mark behind something. In all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them."}, {"video_title": "Zero factorial or 0!   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you might see why this is interesting because this is going to be n factorial over zero factorial. So in order for this formula to apply for even in the case where k is equal to n, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, hey, this thing we've constructed called factorial, you know, we've said, hey, you put an exclamation mark behind something. In all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them. For zero, we're just gonna define this. We're just gonna define, make a mathematical definition. We're just gonna say zero factorial is equal to one."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "And what I want to do is think about the outliers. And to help us with that, let's actually visualize this, the distribution of actual numbers. So let us do that. So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15. So 15, 15. We have one 16. So that's our 16 there."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "15, 15. So 15, 15. We have one 16. So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what?"}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better."}, {"video_title": "Judging outliers in a dataset   Summarizing quantitative data   AP Statistics   Khan Academy.mp3", "Sentence": "And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better. So we are going to start at 6 and go all the way to 19. And then to say that we have these outliers, we would put this, we have outliers over there. So once again, this is a box and whiskers plot of the same data set without outliers."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So a set is really just a collection of distinct objects. So for example, I could have a set. Let's call this set x. And I'll deal with numbers right now, but a set could contain anything. It could contain colors. It could contain people. It could contain other sets."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll deal with numbers right now, but a set could contain anything. It could contain colors. It could contain people. It could contain other sets. It could contain cars. It could contain farm animals. But the numbers will be easy to deal with just because, well, they're numbers."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "It could contain other sets. It could contain cars. It could contain farm animals. But the numbers will be easy to deal with just because, well, they're numbers. So let's say I have a set x. And it has the distinct objects in it, the number 3, the number 12, the number 5, and the number 13. That right there is a set."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the numbers will be easy to deal with just because, well, they're numbers. So let's say I have a set x. And it has the distinct objects in it, the number 3, the number 12, the number 5, and the number 13. That right there is a set. I could have another set. Let's call that set y. I didn't have to call it y. I could have called it a. I could have called it sal. I could have called it a bunch of different things, but I'll just call it y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "That right there is a set. I could have another set. Let's call that set y. I didn't have to call it y. I could have called it a. I could have called it sal. I could have called it a bunch of different things, but I'll just call it y. And let's say that set y, it's a collection of the distinct objects, the number 14, the number 15, the number 6, and the number 3. So fair enough. Those are just two set definitions."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could have called it a bunch of different things, but I'll just call it y. And let's say that set y, it's a collection of the distinct objects, the number 14, the number 15, the number 6, and the number 3. So fair enough. Those are just two set definitions. The way that we typically do it in mathematics is we put these little curly brackets around the objects that are separated by commas. Now let's do some basic operations on sets. And the first operation that I will do is called intersection."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Those are just two set definitions. The way that we typically do it in mathematics is we put these little curly brackets around the objects that are separated by commas. Now let's do some basic operations on sets. And the first operation that I will do is called intersection. And so we would say x intersect the intersection of x and y. And the way that I think about this, this is going to yield another set that contains the elements that are in both x and y. So I often view this intersection symbol right here as and."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the first operation that I will do is called intersection. And so we would say x intersect the intersection of x and y. And the way that I think about this, this is going to yield another set that contains the elements that are in both x and y. So I often view this intersection symbol right here as and. So all of the things that are in x and in y. So what are those things going to be? Well, let's look at both sets x and y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I often view this intersection symbol right here as and. So all of the things that are in x and in y. So what are those things going to be? Well, let's look at both sets x and y. So the number 3 is in set x. Is it in set y as well? Well, sure."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's look at both sets x and y. So the number 3 is in set x. Is it in set y as well? Well, sure. It's in both. So it will be in the intersection of x and y. Now the number 12, that's in set x, but it isn't at y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, sure. It's in both. So it will be in the intersection of x and y. Now the number 12, that's in set x, but it isn't at y. So we're not going to include that. The number 5, it's in x, but it's not in y. And then we have the number 13 is in x, but it's not in y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now the number 12, that's in set x, but it isn't at y. So we're not going to include that. The number 5, it's in x, but it's not in y. And then we have the number 13 is in x, but it's not in y. And so over here, the intersection of x and y is the set that only has one object in it. It only has the number 3. So we are done."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have the number 13 is in x, but it's not in y. And so over here, the intersection of x and y is the set that only has one object in it. It only has the number 3. So we are done. The intersection of x and y is 3. Now another common operation on sets is union. So you could have the union of x and y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we are done. The intersection of x and y is 3. Now another common operation on sets is union. So you could have the union of x and y. And the union I often view, or people often view, as or. So we're thinking about all of the elements that are in x or y. So in some ways, you can kind of imagine that we're bringing these two sets together."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could have the union of x and y. And the union I often view, or people often view, as or. So we're thinking about all of the elements that are in x or y. So in some ways, you can kind of imagine that we're bringing these two sets together. So this is going to be. And the key here is that a set is a collection of distinct objects. And the way we're conceptualizing things right here, this is the number 3."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in some ways, you can kind of imagine that we're bringing these two sets together. So this is going to be. And the key here is that a set is a collection of distinct objects. And the way we're conceptualizing things right here, this is the number 3. This isn't like somebody's score on a test or the number of apples they have. So there you could have multiple people with the same number of apples. Here we're talking about the object, the number 3."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the way we're conceptualizing things right here, this is the number 3. This isn't like somebody's score on a test or the number of apples they have. So there you could have multiple people with the same number of apples. Here we're talking about the object, the number 3. So we can only have a 3 once. But a 3 is in set is in x or y. So I'll put a 3 there."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here we're talking about the object, the number 3. So we can only have a 3 once. But a 3 is in set is in x or y. So I'll put a 3 there. A 12 is in x or y. A 5 is in x or y. The 13 is in x or y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll put a 3 there. A 12 is in x or y. A 5 is in x or y. The 13 is in x or y. And just to simplify things, we really don't care about order if we're just talking about a set. I've just put all the things that are in set x here. And now let's see what we have to add from set y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "The 13 is in x or y. And just to simplify things, we really don't care about order if we're just talking about a set. I've just put all the things that are in set x here. And now let's see what we have to add from set y. So we haven't put a 14 yet. So let's put a 14. We haven't put a 15 yet."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now let's see what we have to add from set y. So we haven't put a 14 yet. So let's put a 14. We haven't put a 15 yet. We haven't put the 6 yet. And we already have a 3 in our set. So there you go."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "We haven't put a 15 yet. We haven't put the 6 yet. And we already have a 3 in our set. So there you go. You have the union of x and y. And one way to visualize sets and visualize intersections and unions and more complicated things is using a Venn diagram. So let's say this whole box is a set of all numbers."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there you go. You have the union of x and y. And one way to visualize sets and visualize intersections and unions and more complicated things is using a Venn diagram. So let's say this whole box is a set of all numbers. So that's all the numbers right over there. We have set x. I'll just draw a circle right over here. And I could even draw the elements of set x."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say this whole box is a set of all numbers. So that's all the numbers right over there. We have set x. I'll just draw a circle right over here. And I could even draw the elements of set x. So you have 3 and 5 and 12, 3, 5, 12, and 13. And then we can draw a set y. And notice, I drew a little overlapping here because they overlap at 3."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I could even draw the elements of set x. So you have 3 and 5 and 12, 3, 5, 12, and 13. And then we can draw a set y. And notice, I drew a little overlapping here because they overlap at 3. 3 is an element in both set x and set y. But set y also has the numbers 14, 15, and 6. And so when we're talking about x intersect y, we're talking about where the two sets overlap."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And notice, I drew a little overlapping here because they overlap at 3. 3 is an element in both set x and set y. But set y also has the numbers 14, 15, and 6. And so when we're talking about x intersect y, we're talking about where the two sets overlap. So we're talking about this region right over here. And the only place that they overlap the way I've drawn it is at the number 3. So this is x intersect y."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so when we're talking about x intersect y, we're talking about where the two sets overlap. So we're talking about this region right over here. And the only place that they overlap the way I've drawn it is at the number 3. So this is x intersect y. And then x union y is the combination of these two sets. So x union y is literally everything right here that we are combining. Let's do one more example just so that we make sure we understand intersection and union."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is x intersect y. And then x union y is the combination of these two sets. So x union y is literally everything right here that we are combining. Let's do one more example just so that we make sure we understand intersection and union. So let's say that I have set A. And set A has the numbers 11, 4, 12, and 7 in it. And I have set B."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do one more example just so that we make sure we understand intersection and union. So let's say that I have set A. And set A has the numbers 11, 4, 12, and 7 in it. And I have set B. And it has the numbers 13, 4, 12, 10, and 3 in it. So first of all, let's think about what A intersect B is going to be equal to. Well, it's the things that are in both sets."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I have set B. And it has the numbers 13, 4, 12, 10, and 3 in it. So first of all, let's think about what A intersect B is going to be equal to. Well, it's the things that are in both sets. So I have 11 here. I don't have an 11 there. So that doesn't make the intersection."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's the things that are in both sets. So I have 11 here. I don't have an 11 there. So that doesn't make the intersection. I have a 4 here. I also have a 4 here. So 4 is in A and B."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that doesn't make the intersection. I have a 4 here. I also have a 4 here. So 4 is in A and B. It's in A and B. So I'll put a 4 here. The number 12, it's in A and B."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 4 is in A and B. It's in A and B. So I'll put a 4 here. The number 12, it's in A and B. So I'll put a 12 here. Number 7's only in A. And then number 13, 10, and 3 is only in B."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "The number 12, it's in A and B. So I'll put a 12 here. Number 7's only in A. And then number 13, 10, and 3 is only in B. So we're done. 4 and 12, the set of 4 and 12 is the intersection of sets A and B. And if we want to, we could even label this as a new set."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then number 13, 10, and 3 is only in B. So we're done. 4 and 12, the set of 4 and 12 is the intersection of sets A and B. And if we want to, we could even label this as a new set. We could say set C is the intersection of A and B. And it's this set right over here. Now let's think about union."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we want to, we could even label this as a new set. We could say set C is the intersection of A and B. And it's this set right over here. Now let's think about union. Let's think about A. I want to do that in orange. Let's think about A union B. What are all the elements that are in A or B?"}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's think about union. Let's think about A. I want to do that in orange. Let's think about A union B. What are all the elements that are in A or B? Well, we can just literally put all the elements in A. 11, 4, 12, 7. And then put the things in B that aren't already in A."}, {"video_title": "Intersection and union of sets   Probability and Statistics   Khan Academy.mp3", "Sentence": "What are all the elements that are in A or B? Well, we can just literally put all the elements in A. 11, 4, 12, 7. And then put the things in B that aren't already in A. So let's see, 13, we already put the 4 and the 12, a 10, and a 3. And I could write this in any order I want. We don't care about order if we're thinking about a set."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Each person snapped their fingers with their dominant hand for 10 seconds and their non-dominant hand for 10 seconds, where if you're right-handed, right hand would be your dominant hand. If you were left-handed, left hand would be your dominant hand. Each participant flipped a coin to determine which hand they would use first, because if you always used your dominant hand first, maybe you're tired by the time you're doing your non-dominant hand or there's something else. So here, it's random which one you use first. Here are the data for how many snaps they performed with each hand, the difference for each participant, and summary statistics. And this is actually real data from the Khan Academy content team. And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So here, it's random which one you use first. Here are the data for how many snaps they performed with each hand, the difference for each participant, and summary statistics. And this is actually real data from the Khan Academy content team. And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. And he was even able to do 35 on his non-dominant hand. And so the difference here, the dominant hand minus the non-dominant was nine. And then they tabulated this data for all five members."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. And he was even able to do 35 on his non-dominant hand. And so the difference here, the dominant hand minus the non-dominant was nine. And then they tabulated this data for all five members. Now, they also calculated summary statistics for them. But this is the really interesting thing right over here. This is the difference between the dominant and the non-dominant hand."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "And then they tabulated this data for all five members. Now, they also calculated summary statistics for them. But this is the really interesting thing right over here. This is the difference between the dominant and the non-dominant hand. And so what they did here, the mean difference, what they did is they took this row right over here, and they calculated the mean, which they got to be 6.8. And then they calculated the standard deviation of these differences right over here, which they got to be approximately 1.64. And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "This is the difference between the dominant and the non-dominant hand. And so what they did here, the mean difference, what they did is they took this row right over here, and they calculated the mean, which they got to be 6.8. And then they calculated the standard deviation of these differences right over here, which they got to be approximately 1.64. And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. So pause this video, see if you can make some headway here. See if you can think about how to approach this. So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. So pause this video, see if you can make some headway here. See if you can think about how to approach this. So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. We're constructing a 95% confidence interval for a mean difference. Now, you might say, wait, wait, wait. You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. We're constructing a 95% confidence interval for a mean difference. Now, you might say, wait, wait, wait. You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. But one way to think about it, this is matched pairs design. So in a matched pairs design, what you do is, for each participant, for each member in your sample, you will make them do the control and the treatment. So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. But one way to think about it, this is matched pairs design. So in a matched pairs design, what you do is, for each participant, for each member in your sample, you will make them do the control and the treatment. So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. And in matched pairs design, you're really concerned about the difference. And so what you can really view this as is you just have one sample size of five for which you are calculating the difference for each member of that sample and the standard deviation across that entire sample. Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. And in matched pairs design, you're really concerned about the difference. And so what you can really view this as is you just have one sample size of five for which you are calculating the difference for each member of that sample and the standard deviation across that entire sample. Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. The first condition we think about is whether our sample is random. Now, if we were trying to make some type of judgment about all human beings and their snapping ability, this would not be a random sample. These people all work at Khan Academy."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. The first condition we think about is whether our sample is random. Now, if we were trying to make some type of judgment about all human beings and their snapping ability, this would not be a random sample. These people all work at Khan Academy. Maybe somehow in our interview process, we select for people who snap particularly well. But whatever inferences we make, we can say, hey, this is roughly true about this group of friends. Now, the next condition we wanna think about is the normal condition."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "These people all work at Khan Academy. Maybe somehow in our interview process, we select for people who snap particularly well. But whatever inferences we make, we can say, hey, this is roughly true about this group of friends. Now, the next condition we wanna think about is the normal condition. Now, there's a couple of ways to think about it. If we had sample size of 30 or larger, the central limit theorem says, okay, the distribution, the sampling distribution would be roughly normal, the sampling distribution of the sample means. But obviously, our sample size is much smaller than that."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Now, the next condition we wanna think about is the normal condition. Now, there's a couple of ways to think about it. If we had sample size of 30 or larger, the central limit theorem says, okay, the distribution, the sampling distribution would be roughly normal, the sampling distribution of the sample means. But obviously, our sample size is much smaller than that. One way to think about it, we could just plot our data points and see whether they seem to be skewed in any way. And if we just do a little dot plot right over here, we could say, let's say make this zero, one, two, three, four, five, six, seven, eight, and nine. So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "But obviously, our sample size is much smaller than that. One way to think about it, we could just plot our data points and see whether they seem to be skewed in any way. And if we just do a little dot plot right over here, we could say, let's say make this zero, one, two, three, four, five, six, seven, eight, and nine. So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. And so this doesn't look massively skewed in any way. Our mean difference was right over here, it was about 6.8. It looks roughly symmetric."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. And so this doesn't look massively skewed in any way. Our mean difference was right over here, it was about 6.8. It looks roughly symmetric. So we can feel okay about this normal distribution. This isn't the best study that one could conduct. This is obviously a small sample size."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "It looks roughly symmetric. So we can feel okay about this normal distribution. This isn't the best study that one could conduct. This is obviously a small sample size. It's not random of the entire population. But maybe we could go with it. Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "This is obviously a small sample size. It's not random of the entire population. But maybe we could go with it. Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. But all of these things, once again, this isn't a super robust study, but this is a fun thing for friends to do if they have nothing else to do. All right. Now the third one is independence."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. But all of these things, once again, this isn't a super robust study, but this is a fun thing for friends to do if they have nothing else to do. All right. Now the third one is independence. And this one actually we can feel pretty good about because Jeff's difference right over here really shouldn't impact David's difference or David's difference really shouldn't impact Kim's difference, especially if they're not observing each other. And let's just say for the sake of argument that they did it all independently in a closed room with a independent observer so they weren't trying to get competitive or something like that. But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Now the third one is independence. And this one actually we can feel pretty good about because Jeff's difference right over here really shouldn't impact David's difference or David's difference really shouldn't impact Kim's difference, especially if they're not observing each other. And let's just say for the sake of argument that they did it all independently in a closed room with a independent observer so they weren't trying to get competitive or something like that. But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. So how do we do that? Well, we've done this so many times. Our confidence interval would be our sample mean."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. So how do we do that? Well, we've done this so many times. Our confidence interval would be our sample mean. So it would be the mean of our difference, the mean of our difference plus or minus. Now we don't know the population standard deviation, so we're going to use our sample standard deviation. And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Our confidence interval would be our sample mean. So it would be the mean of our difference, the mean of our difference plus or minus. Now we don't know the population standard deviation, so we're going to use our sample standard deviation. And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. Now we know most of this data here, and let me just write it down over here. We know the mean, the sample mean right over here is 6.8, so it's going to be 6.8 plus or minus. And now what will be our critical value here?"}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. Now we know most of this data here, and let me just write it down over here. We know the mean, the sample mean right over here is 6.8, so it's going to be 6.8 plus or minus. And now what will be our critical value here? Well, we want to have a 95% confidence interval. And what's our degrees of freedom? Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "And now what will be our critical value here? Well, we want to have a 95% confidence interval. And what's our degrees of freedom? Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. And so we're ready to use a t-table. So this is a truncated t-table that I could fit on my screen here. And so there's a couple of ways to think about it."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. And so we're ready to use a t-table. So this is a truncated t-table that I could fit on my screen here. And so there's a couple of ways to think about it. Here they actually give us the confidence level, and the reason why that corresponds to a tail probability of.025 is if you take the middle 95% of a distribution, you're going to have 2.5% on either end. That's going to be your tail probability. So that's all that's going on over there."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "And so there's a couple of ways to think about it. Here they actually give us the confidence level, and the reason why that corresponds to a tail probability of.025 is if you take the middle 95% of a distribution, you're going to have 2.5% on either end. That's going to be your tail probability. So that's all that's going on over there. So we're going to be in this column right over here. And which degree of freedom do we use, or degrees of freedom? Well, it's gonna be four degrees of freedom."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So that's all that's going on over there. So we're going to be in this column right over here. And which degree of freedom do we use, or degrees of freedom? Well, it's gonna be four degrees of freedom. Our sample size is five, five minus one is four. So this is going to be our critical value, 2.776. So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, it's gonna be four degrees of freedom. Our sample size is five, five minus one is four. So this is going to be our critical value, 2.776. So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. And then we're going to divide that by the square root of our sample size. So the square root of our sample size, we already wrote a five in there. Sometimes I just write an N there."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. And then we're going to divide that by the square root of our sample size. So the square root of our sample size, we already wrote a five in there. Sometimes I just write an N there. And so what is this going to be equal to? First, let's calculate just the margin of error right over here. So this is going to be 2.776 times 1.64 divided by the square root of five."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "Sometimes I just write an N there. And so what is this going to be equal to? First, let's calculate just the margin of error right over here. So this is going to be 2.776 times 1.64 divided by the square root of five. And we get a margin of error of approximately 2.036. So this is going to be 6.8 plus or minus 2.036. It's approximately equal to that, where this is our margin of error."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to be 2.776 times 1.64 divided by the square root of five. And we get a margin of error of approximately 2.036. So this is going to be 6.8 plus or minus 2.036. It's approximately equal to that, where this is our margin of error. And if we actually wanted to write out the interval, we could just take 6.8 minus this and 6.8 plus that. So let's do that again with the calculator. So 6.8 minus 2.036 is equal to 4.764."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "It's approximately equal to that, where this is our margin of error. And if we actually wanted to write out the interval, we could just take 6.8 minus this and 6.8 plus that. So let's do that again with the calculator. So 6.8 minus 2.036 is equal to 4.764. So our confidence interval starts at 4.764 approximately. And it goes to, let's see, I could actually do this one in my head. If I add 2.036 to 6.8, that is going to be 8.836."}, {"video_title": "Confidence interval for a mean with paired data   AP Statistics   Khan Academy.mp3", "Sentence": "So 6.8 minus 2.036 is equal to 4.764. So our confidence interval starts at 4.764 approximately. And it goes to, let's see, I could actually do this one in my head. If I add 2.036 to 6.8, that is going to be 8.836. Now, how would we interpret this confidence interval right over here? One way to interpret it is to say that we are 95% confident that this interval captures the true mean difference in snaps for these friends. We could also say that there appears to be a difference in the mean number of snaps, since zero is not captured in this interval."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "So let's just get to the punch line. Let's solve for the optimal m and b. And just based on what we did in the last videos, there's two ways to do that. We actually now know two points that lie on that line, so we can literally find the slope of that line and then the y-intercept, the b there. Or we could just say it's a solution to this system of equations, and they're actually mathematically equivalent. So let's solve for m first. And if we want to solve for m, we want to cancel out the b."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We actually now know two points that lie on that line, so we can literally find the slope of that line and then the y-intercept, the b there. Or we could just say it's a solution to this system of equations, and they're actually mathematically equivalent. So let's solve for m first. And if we want to solve for m, we want to cancel out the b. So let me rewrite this top equation just the way it's written over here. So we have m times the mean of the x squareds plus b times the mean of... Actually, we could even do it better than that. One step better than that is to actually, based on the work we did in the last video, we can just subtract this bottom equation from this top equation."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And if we want to solve for m, we want to cancel out the b. So let me rewrite this top equation just the way it's written over here. So we have m times the mean of the x squareds plus b times the mean of... Actually, we could even do it better than that. One step better than that is to actually, based on the work we did in the last video, we can just subtract this bottom equation from this top equation. So let me subtract it, or let's add the negative. So if I make this negative, this is negative, this is negative, what do we get? We get m times the mean of the x's minus the mean of the x squareds over the mean of x, over the mean of the x's."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "One step better than that is to actually, based on the work we did in the last video, we can just subtract this bottom equation from this top equation. So let me subtract it, or let's add the negative. So if I make this negative, this is negative, this is negative, what do we get? We get m times the mean of the x's minus the mean of the x squareds over the mean of x, over the mean of the x's. The plus b and the negative b cancel out, is equal to the mean of the y's minus the mean of the x y's over the mean of the x's. And then we can divide both sides of the equation by this. And so we get m is equal to the mean of the y's minus the mean of the x y's over the mean of the x's over this."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We get m times the mean of the x's minus the mean of the x squareds over the mean of x, over the mean of the x's. The plus b and the negative b cancel out, is equal to the mean of the y's minus the mean of the x y's over the mean of the x's. And then we can divide both sides of the equation by this. And so we get m is equal to the mean of the y's minus the mean of the x y's over the mean of the x's over this. The mean of the x's minus the mean of the x squareds over the mean of the x's. Now notice, this is the exact same thing that you would get if you found the slope between these two points over here. Change in y, so the difference between that y and that y is that right over there, over the change in x's."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "And so we get m is equal to the mean of the y's minus the mean of the x y's over the mean of the x's over this. The mean of the x's minus the mean of the x squareds over the mean of the x's. Now notice, this is the exact same thing that you would get if you found the slope between these two points over here. Change in y, so the difference between that y and that y is that right over there, over the change in x's. The change in that x, that x minus that x is exactly this over here. Now to simplify it, we can multiply both the numerator and the denominator by the mean in x. The mean of the x's."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "Change in y, so the difference between that y and that y is that right over there, over the change in x's. The change in that x, that x minus that x is exactly this over here. Now to simplify it, we can multiply both the numerator and the denominator by the mean in x. The mean of the x's. And I do that just so we don't have this in the denominator both places. And so if we multiply the numerator by the mean of the x's, we get the mean of the x's times the mean of the y's minus the mean of the x y's. All of that over, mean of the x's times the mean of the x's is just going to be the mean of the x's squared minus over here you have the mean of the x squared."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "The mean of the x's. And I do that just so we don't have this in the denominator both places. And so if we multiply the numerator by the mean of the x's, we get the mean of the x's times the mean of the y's minus the mean of the x y's. All of that over, mean of the x's times the mean of the x's is just going to be the mean of the x's squared minus over here you have the mean of the x squared. And that's what we get for m. And then if we want to solve for b, we literally can just substitute back into either equation, but this equation right here is simpler. And so if we wanted to solve for b there, we can solve for b in terms of m. We just subtract m times the mean of x's from both sides. We get b is equal to the mean of the y's minus m times the mean of the x's."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "All of that over, mean of the x's times the mean of the x's is just going to be the mean of the x's squared minus over here you have the mean of the x squared. And that's what we get for m. And then if we want to solve for b, we literally can just substitute back into either equation, but this equation right here is simpler. And so if we wanted to solve for b there, we can solve for b in terms of m. We just subtract m times the mean of x's from both sides. We get b is equal to the mean of the y's minus m times the mean of the x's. So what you do is you take your data points, you find the mean of the x's, the mean of the y's, the mean of the x y's, the mean of the x's squared. You find your m. Once you find your m, then you can substitute back in here and you find your b. And then you have your actual optimal line."}, {"video_title": "Proof (part 4) minimizing squared error to regression line   Khan Academy.mp3", "Sentence": "We get b is equal to the mean of the y's minus m times the mean of the x's. So what you do is you take your data points, you find the mean of the x's, the mean of the y's, the mean of the x y's, the mean of the x's squared. You find your m. Once you find your m, then you can substitute back in here and you find your b. And then you have your actual optimal line. And we're done. So these are the two big formula takeaways for our optimal line. What I'm going to do in the next video, and this is where you can kind of, if anyone was skipping up to this point, the next video is where they should reengage."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do another problem from the normal distribution section of ck12.org's AP Statistics book. And I'm using theirs because it's open source. It's actually quite a good book. The problems are, I think, good practice for us. So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "The problems are, I think, good practice for us. So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. 9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Not 9.5. 9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the standard deviation is equal to 1.1 grams. Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's my normal distribution. I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our mean right there. That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is our mean plus one standard deviation. This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what the empirical rule tells us. Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me draw that out. Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. I think you get the idea. I'm not a computer. 9.5 is the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "This height should be the same as that height there. I think you get the idea. I'm not a computer. 9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean. So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "One standard deviation above the mean. So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "They told us the standard deviation is 1.1. That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "One standard deviation below the mean. One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Doing it on the other side. One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms?"}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule?"}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "I said mass because kilograms is actually a unit of mass. But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do part B. Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is two standard deviations below. This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that is this area way out there. I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. This little small area. So what is that probability? So let's turn back to our empirical rule."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's this long tail out here. This little small area. So what is that probability? So let's turn back to our empirical rule. Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's turn back to our empirical rule. Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know the area between minus three standard deviations and plus three standard deviations. We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know this area right here. Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. So what do we have left over for the two tails? Remember, there are two tails. This is one of them."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "I mean, almost all of them. So what do we have left over for the two tails? Remember, there are two tails. This is one of them. And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is one of them. And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal."}, {"video_title": "ck12.org normal distribution problems Empirical rule   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%. And this right here is going to be 0.15%. So the probability of having a one-year-old baby girl in the US that is more than 12.8 kilograms, if you assume a perfect normal distribution, is the area under this curve, the area that is more than three standard deviations above the mean."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we have a type of statistical study described here. I encourage you to pause this video, read it, and see if you can figure out, is this a sample study? Is it an observational study? Is it an experiment? And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Is it an experiment? And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study?"}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study? Observational, or is this an experiment?"}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study? Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here?"}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference."}, {"video_title": "Worked example identifying observational study   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference. If the farmer is petting the cows and treating them humanely and doing other things, it might not matter whether the farmer names them or not. Likewise, if you take some less nice farmers who hit their cows and they have really inhumane conditions, it might not make a difference whether they name the cows or not. And so it's very important that you, from the observational studies, you might, if they're well-constructed, you might be able to make a, you might be able to say there's a correlation."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "If you're not, I encourage you to review the videos on that. And we've already done some hypothesis testing with the chi-squared statistic. And we've even done some hypothesis testing based on two-way tables. And now we're going to extend that by thinking about a chi-squared test for association between two variables. So let's say that we suspect that someone's foot length is related to their hand length, that these things are not independent. Well, what we can do is set up a hypothesis test. And remember, the null hypothesis in a hypothesis test is to always assume no news."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And now we're going to extend that by thinking about a chi-squared test for association between two variables. So let's say that we suspect that someone's foot length is related to their hand length, that these things are not independent. Well, what we can do is set up a hypothesis test. And remember, the null hypothesis in a hypothesis test is to always assume no news. So what we could say is here is that there's no association, no association between foot and hand length. Another way to think about it is that they are independent. And oftentimes what we're doing is called a chi-squared test for independence."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And remember, the null hypothesis in a hypothesis test is to always assume no news. So what we could say is here is that there's no association, no association between foot and hand length. Another way to think about it is that they are independent. And oftentimes what we're doing is called a chi-squared test for independence. And then our alternative hypothesis would be our suspicion there is an association. There is an association. So foot and hand length are not independent."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And oftentimes what we're doing is called a chi-squared test for independence. And then our alternative hypothesis would be our suspicion there is an association. There is an association. So foot and hand length are not independent. So what we can then do is go to a population and we can randomly sample it. And so let's say we randomly sample 100 folks. And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "So foot and hand length are not independent. So what we can then do is go to a population and we can randomly sample it. And so let's say we randomly sample 100 folks. And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. And we also do that for the feet. And we tabulate all of the data. And this is the data that we actually get."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. And we also do that for the feet. And we tabulate all of the data. And this is the data that we actually get. Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity. In a chi-squared test for homogeneity, we sample from two different populations, or we look at two different groups, and we see whether the distribution of a certain variable amongst those two different groups is the same. Here we are just sampling from one group, but we're thinking about two different variables for that one group."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And this is the data that we actually get. Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity. In a chi-squared test for homogeneity, we sample from two different populations, or we look at two different groups, and we see whether the distribution of a certain variable amongst those two different groups is the same. Here we are just sampling from one group, but we're thinking about two different variables for that one group. We're thinking about feet length and we're thinking about hand length. And so you can see here that 11 folks had both their right hand longer and their right foot longer. Three folks had their right hand longer but their left foot was longer."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Here we are just sampling from one group, but we're thinking about two different variables for that one group. We're thinking about feet length and we're thinking about hand length. And so you can see here that 11 folks had both their right hand longer and their right foot longer. Three folks had their right hand longer but their left foot was longer. And then eight folks had their right hand longer but both feet were the same. Likewise, we had nine people where their left foot and hand was longer, but you had two people where the left hand was longer but the right foot was longer. And we could go through all of these."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Three folks had their right hand longer but their left foot was longer. And then eight folks had their right hand longer but both feet were the same. Likewise, we had nine people where their left foot and hand was longer, but you had two people where the left hand was longer but the right foot was longer. And we could go through all of these. But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length. So to help us do that, I'm gonna make a total of our columns here and also a total of our rows here. Let me draw a line here so we know what is going on."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And we could go through all of these. But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length. So to help us do that, I'm gonna make a total of our columns here and also a total of our rows here. Let me draw a line here so we know what is going on. And so what are the total number of people who had a longer right hand? Well, it's gonna be 11 plus three plus eight, which is 22. The total number of people who had a longer left hand is two plus nine plus 14, which is 25."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Let me draw a line here so we know what is going on. And so what are the total number of people who had a longer right hand? Well, it's gonna be 11 plus three plus eight, which is 22. The total number of people who had a longer left hand is two plus nine plus 14, which is 25. And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53. And then if I were to total this column, 22 plus 25 is 47 plus 53, we get 100 right over here. And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "The total number of people who had a longer left hand is two plus nine plus 14, which is 25. And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53. And then if I were to total this column, 22 plus 25 is 47 plus 53, we get 100 right over here. And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. Longer left foot, three plus nine plus 13, that's also 25. And then we could either add these up and we would get 50 or we could say, hey, 25 plus 25 plus what is 100? Well, that is going to be equal to 50."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. Longer left foot, three plus nine plus 13, that's also 25. And then we could either add these up and we would get 50 or we could say, hey, 25 plus 25 plus what is 100? Well, that is going to be equal to 50. Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables. Well, if they are independent, which we are assuming, then our best estimate is that 22% have a longer right hand and our best estimate is that 25% have a longer right foot. And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Well, that is going to be equal to 50. Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables. Well, if they are independent, which we are assuming, then our best estimate is that 22% have a longer right hand and our best estimate is that 25% have a longer right foot. And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. I'm just multiplying the probabilities, which you would do if these were independent variables. And so 0.22 times 0.25, let's see, 1 1\u20444 of 22 is 5 1\u20442, so this is going to be equal to 5.5. Now, what number would you expect to have a longer right hand but a longer left foot?"}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. I'm just multiplying the probabilities, which you would do if these were independent variables. And so 0.22 times 0.25, let's see, 1 1\u20444 of 22 is 5 1\u20442, so this is going to be equal to 5.5. Now, what number would you expect to have a longer right hand but a longer left foot? So that would be 0.22 times 0.25 times 100. Well, we already calculated what that would be. That would be 5.5."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Now, what number would you expect to have a longer right hand but a longer left foot? So that would be 0.22 times 0.25 times 100. Well, we already calculated what that would be. That would be 5.5. And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1\u20442 of 22, which is equal to 11. And we can keep going. This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "That would be 5.5. And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1\u20442 of 22, which is equal to 11. And we can keep going. This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. This value right over here would be 0.25 times 0.25 times 100, which is again 6.25. And then this value right over here, couple of ways we can get it. We can multiply 0.25 times 50 times 100, which would get us to 12.5."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. This value right over here would be 0.25 times 0.25 times 100, which is again 6.25. And then this value right over here, couple of ways we can get it. We can multiply 0.25 times 50 times 100, which would get us to 12.5. Or we could have said this plus this plus this has to equal 25, so this would be 12.5. And now this expected value we can figure out because 5.5 plus 6.25 plus this is going to equal 25. So let's see, 5.5 plus 6.25 is 11.75."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "We can multiply 0.25 times 50 times 100, which would get us to 12.5. Or we could have said this plus this plus this has to equal 25, so this would be 12.5. And now this expected value we can figure out because 5.5 plus 6.25 plus this is going to equal 25. So let's see, 5.5 plus 6.25 is 11.75. 11.75 plus 13.25 is equal to 25. Same thing over here. This would be 13.25, because this is 11.75 plus 13.25 is equal to 25."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "So let's see, 5.5 plus 6.25 is 11.75. 11.75 plus 13.25 is equal to 25. Same thing over here. This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. If we add these two together, we get 26.5. 26.5 plus what is equal to 53? Would be equal to another 26.5."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. If we add these two together, we get 26.5. 26.5 plus what is equal to 53? Would be equal to another 26.5. Now once you figure out all of your expected values, that's a good time to test your conditions. The first condition is that you took a random sample, so let's assume we had done that. The second condition is that your expected value for any of the data points has to be at least equal to five."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Would be equal to another 26.5. Now once you figure out all of your expected values, that's a good time to test your conditions. The first condition is that you took a random sample, so let's assume we had done that. The second condition is that your expected value for any of the data points has to be at least equal to five. And we can see that all of our expected values are at least equal to five. The actual data points we got do not have to be equal to five. So it's okay that we got a two here because the expected value here is five or larger."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "The second condition is that your expected value for any of the data points has to be at least equal to five. And we can see that all of our expected values are at least equal to five. The actual data points we got do not have to be equal to five. So it's okay that we got a two here because the expected value here is five or larger. And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population. So let's assume that that happened as well. So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "So it's okay that we got a two here because the expected value here is five or larger. And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population. So let's assume that that happened as well. So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected. So I did that one. Now I'll do this one."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected. So I did that one. Now I'll do this one. So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11. Then I'll do this one. Two minus 6.25 squared over 6.25."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Now I'll do this one. So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11. Then I'll do this one. Two minus 6.25 squared over 6.25. And I'll keep doing it. I'm gonna do it for all nine of these data points. And I actually calculated this ahead of time to save some time."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Two minus 6.25 squared over 6.25. And I'll keep doing it. I'm gonna do it for all nine of these data points. And I actually calculated this ahead of time to save some time. And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942. Now before we calculate the p-value, we're gonna have to think about what are our degrees of freedom. Now we have a three by three table here."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And I actually calculated this ahead of time to save some time. And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942. Now before we calculate the p-value, we're gonna have to think about what are our degrees of freedom. Now we have a three by three table here. So one way to think about it, it's the number of rows minus one times the number of columns minus one. This is two times two, which is equal to four. Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Now we have a three by three table here. So one way to think about it, it's the number of rows minus one times the number of columns minus one. This is two times two, which is equal to four. Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. And so now we are ready to calculate a p-value. And you could do that using a calculator and you could do that using a chi-squared table. But let's say we did it using a calculator."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. And so now we are ready to calculate a p-value. And you could do that using a calculator and you could do that using a chi-squared table. But let's say we did it using a calculator. And we get a p-value of 0.018. And just to remind ourselves what this is, this is the probability of getting a chi-squared statistic at least this large or larger. And so next, we do what we always do with hypothesis testing."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "But let's say we did it using a calculator. And we get a p-value of 0.018. And just to remind ourselves what this is, this is the probability of getting a chi-squared statistic at least this large or larger. And so next, we do what we always do with hypothesis testing. We compare this to our significance level. And we actually should have set our significance level from the beginning. So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "And so next, we do what we always do with hypothesis testing. We compare this to our significance level. And we actually should have set our significance level from the beginning. So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. You really should do this before you calculate all of this. But then you compare your p-value to your significance level. And we see that this p-value is a good bit less than our significance level."}, {"video_title": "Chi-square test for association (independence)   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. You really should do this before you calculate all of this. But then you compare your p-value to your significance level. And we see that this p-value is a good bit less than our significance level. And so one way to think about it is we got all these expected values, assuming that the null hypothesis was true. But the probability of getting a result this extreme or more extreme is less than 2%, which is lower than our significance level. And so this will lead us to reject our null hypothesis."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "The specific test they use has a false positive rate of 2% and a false negative rate of 1%. Suppose that 5% of all their applicants are actually using illegal drugs and we randomly select an applicant. Given the applicant tests positive, what is the probability that they are actually on drugs? So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs?"}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs."}, {"video_title": "Conditional probability tree diagram example   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs. And so 2% of this larger group of the ones that don't take the drugs, well, this is actually a fairly large number relative to the percentage that do take the drugs and test positive. So I will leave you there. This is fascinating, not just for this particular case, but you will see analysis like this all the time when we're looking at whether a certain medication is effective or a certain procedure is effective."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in a lot of what we're doing in this inferential statistics, we're trying to figure out what is the probability of getting a certain sample mean. So what we've been doing, especially when we have a large sample size, so let me just draw a sampling distribution here, so let's say we have a sampling distribution of the sample mean right here. It has some assumed mean value and some standard deviation. And what we want to do is any result that we get, let's say we get some sample mean out here, we want to figure out the probability of getting a result at least as extreme as this, so you can either figure out the probability of getting a result below this and subtract that from 1, or just figure out this area right over there. And to do that, we've been figuring out how many standard deviations above the mean we actually are. And the way we figure that out is we take our sample mean, we subtract from that our mean itself, what we assume the mean should be, or maybe we don't know what this is, and then we divide that by the standard deviation of the sampling distribution. This is how many standard deviations we are above the mean, that is that distance right over there."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what we want to do is any result that we get, let's say we get some sample mean out here, we want to figure out the probability of getting a result at least as extreme as this, so you can either figure out the probability of getting a result below this and subtract that from 1, or just figure out this area right over there. And to do that, we've been figuring out how many standard deviations above the mean we actually are. And the way we figure that out is we take our sample mean, we subtract from that our mean itself, what we assume the mean should be, or maybe we don't know what this is, and then we divide that by the standard deviation of the sampling distribution. This is how many standard deviations we are above the mean, that is that distance right over there. Now, we usually don't know what this is either. We normally don't know what that is either. And the central limit theorem told us that, assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as the standard deviation of our population divided by the square root of our sample size."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is how many standard deviations we are above the mean, that is that distance right over there. Now, we usually don't know what this is either. We normally don't know what that is either. And the central limit theorem told us that, assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. So this thing right over here can be rewritten as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here, divided by our population mean divided by the square root of our sample size. And this is essentially our best sense of how many standard deviations away from the actual mean we are. And this thing right here, we've learned it before, is a z-score."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the central limit theorem told us that, assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. So this thing right over here can be rewritten as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here, divided by our population mean divided by the square root of our sample size. And this is essentially our best sense of how many standard deviations away from the actual mean we are. And this thing right here, we've learned it before, is a z-score. Or when we're dealing with an actual statistic, when it's derived from the sample mean statistic, we call this a z statistic. And then we could look it up in a z-table or in a normal distribution table to say, what's the probability of getting a value of this z or greater? So that would give us that probability."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this thing right here, we've learned it before, is a z-score. Or when we're dealing with an actual statistic, when it's derived from the sample mean statistic, we call this a z statistic. And then we could look it up in a z-table or in a normal distribution table to say, what's the probability of getting a value of this z or greater? So that would give us that probability. So what's the probability of getting that extreme of a result? Now, normally when we've done this in the last few videos, we also do not know what the standard deviation of the population is. So in order to approximate that, we say that the z-score is approximately, or the z-statistic is approximately going to be, so let me just write the numerator over again."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that would give us that probability. So what's the probability of getting that extreme of a result? Now, normally when we've done this in the last few videos, we also do not know what the standard deviation of the population is. So in order to approximate that, we say that the z-score is approximately, or the z-statistic is approximately going to be, so let me just write the numerator over again. Over, we estimate this using our sample standard deviation. We estimate it using our sample standard deviation. And this is OK if our sample size is greater than 30."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in order to approximate that, we say that the z-score is approximately, or the z-statistic is approximately going to be, so let me just write the numerator over again. Over, we estimate this using our sample standard deviation. We estimate it using our sample standard deviation. And this is OK if our sample size is greater than 30. Or another way to think about it is this will be normally distributed if our sample size is greater than 30. Even this approximation will be approximately normally distributed. Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is OK if our sample size is greater than 30. Or another way to think about it is this will be normally distributed if our sample size is greater than 30. Even this approximation will be approximately normally distributed. Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. So let me rewrite the expression over here. Sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size. We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. So let me rewrite the expression over here. Sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size. We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. If it's not, if this is small, then this is going to have a t distribution. And then you're going to do the exact same thing you did here, but now you would assume that the bell is no longer a normal distribution. So in this example, it was normal."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. If it's not, if this is small, then this is going to have a t distribution. And then you're going to do the exact same thing you did here, but now you would assume that the bell is no longer a normal distribution. So in this example, it was normal. All of the z's are normally distributed. Over here in a t distribution, and this will actually be a normalized t distribution right here, because we subtracted out the mean. So in a normalized t distribution, you're going to have a mean of 0."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in this example, it was normal. All of the z's are normally distributed. Over here in a t distribution, and this will actually be a normalized t distribution right here, because we subtracted out the mean. So in a normalized t distribution, you're going to have a mean of 0. And what you're going to do is you want to figure out the probability of getting a t value at least this extreme. So this is your t value you would get, and then you essentially figure out the area under the curve right over there. And so a very easy rule of thumb is calculate this quantity either way."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in a normalized t distribution, you're going to have a mean of 0. And what you're going to do is you want to figure out the probability of getting a t value at least this extreme. So this is your t value you would get, and then you essentially figure out the area under the curve right over there. And so a very easy rule of thumb is calculate this quantity either way. Calculate this quantity either way. If you have more than 30 samples, if your sample size is more than 30, your sample standard deviation is going to be a good approximator for your population standard deviation, and so this whole thing is going to be approximately normally distributed. And so you can use a z table to figure out the probability of getting a result at least that extreme."}, {"video_title": "Z-statistics vs. T-statistics   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so a very easy rule of thumb is calculate this quantity either way. Calculate this quantity either way. If you have more than 30 samples, if your sample size is more than 30, your sample standard deviation is going to be a good approximator for your population standard deviation, and so this whole thing is going to be approximately normally distributed. And so you can use a z table to figure out the probability of getting a result at least that extreme. If your sample size is small, then this statistic, this quantity, this is going to have a t distribution. And then you're going to have to use a t table to figure out the probability of getting a t value at least this extreme. And we're going to see this in an example a couple of videos from now."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "One day he decided to gather data about the distance in miles that people commuted to get to his restaurant. People reported the following distances traveled. So here are all the distances traveled. He wants to create a graph that helps him understand the spread of distances, this is a key word, the spread of distances, and the median distance, and the median distance that people traveled, or that people travel. What kind of graph should he create? So the answer of what kind of graph he should create, that might be a little bit more straightforward than the actual creation of the graph, which we will also do. But he's trying to visualize the spread of information, and at the same time, he wants the median."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "He wants to create a graph that helps him understand the spread of distances, this is a key word, the spread of distances, and the median distance, and the median distance that people traveled, or that people travel. What kind of graph should he create? So the answer of what kind of graph he should create, that might be a little bit more straightforward than the actual creation of the graph, which we will also do. But he's trying to visualize the spread of information, and at the same time, he wants the median. So what graph captures both of that information? Well, a box and whisker plot. So let's actually try to draw a box and whisker plot."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "But he's trying to visualize the spread of information, and at the same time, he wants the median. So what graph captures both of that information? Well, a box and whisker plot. So let's actually try to draw a box and whisker plot. And to do that, we need to come up with the median, and we'll also see the median of the two halves of the data as well. And whenever we're trying to take the median of something, it's really helpful to order our data. So let's start off by attempting to order our data."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's actually try to draw a box and whisker plot. And to do that, we need to come up with the median, and we'll also see the median of the two halves of the data as well. And whenever we're trying to take the median of something, it's really helpful to order our data. So let's start off by attempting to order our data. So what is the smallest number here? Well, let's see, there's one two, so let me mark it off. And then we have another two, another two, so we got all the twos."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's start off by attempting to order our data. So what is the smallest number here? Well, let's see, there's one two, so let me mark it off. And then we have another two, another two, so we got all the twos. Then we have this three. Then we have this three. I think we got all the threes."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have another two, another two, so we got all the twos. Then we have this three. Then we have this three. I think we got all the threes. Then we have that four. Then we have this four. Do we have any fives?"}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "I think we got all the threes. Then we have that four. Then we have this four. Do we have any fives? No. Do we have any sixes? Yep, we have that six."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Do we have any fives? No. Do we have any sixes? Yep, we have that six. And that looks like the only six. Any sevens? Yep, we have this seven right over here."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yep, we have that six. And that looks like the only six. Any sevens? Yep, we have this seven right over here. And I just realized that I missed this one, so let me put the one at the beginning of our set. So I got that one right over there. Actually, there's two ones."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yep, we have this seven right over here. And I just realized that I missed this one, so let me put the one at the beginning of our set. So I got that one right over there. Actually, there's two ones. I missed both of them. So both of those ones are right over there. So I have ones, twos, threes, fours, no fives."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, there's two ones. I missed both of them. So both of those ones are right over there. So I have ones, twos, threes, fours, no fives. This is one six. There was one seven. There's one eight right over here."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I have ones, twos, threes, fours, no fives. This is one six. There was one seven. There's one eight right over here. And then let's see, any nines? No nines. Any tens?"}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's one eight right over here. And then let's see, any nines? No nines. Any tens? Yep, there's a 10. Any 11s? We have an 11 right over there."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Any tens? Yep, there's a 10. Any 11s? We have an 11 right over there. Any 12s? Nope. 13, 14, then we have a 15, then we have a 20, and then a 22."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have an 11 right over there. Any 12s? Nope. 13, 14, then we have a 15, then we have a 20, and then a 22. So we've ordered all our data. Now it should be relatively straightforward to find the middle of our data, the median. So how many data points do we have?"}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "13, 14, then we have a 15, then we have a 20, and then a 22. So we've ordered all our data. Now it should be relatively straightforward to find the middle of our data, the median. So how many data points do we have? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. So the middle number is going to be a number that has eight numbers larger than it and eight numbers smaller than it. So let's think about it."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many data points do we have? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. So the middle number is going to be a number that has eight numbers larger than it and eight numbers smaller than it. So let's think about it. 1, 2, 3, 4, 5, 6, 7, 8. So the number 6 here is larger than 8 of the values. And if I did the calculations right, it should be smaller than 8 of the values."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about it. 1, 2, 3, 4, 5, 6, 7, 8. So the number 6 here is larger than 8 of the values. And if I did the calculations right, it should be smaller than 8 of the values. 1, 2, 3, 4, 5, 6, 7, 8. So it is indeed the median. Now when we take a box and whisker, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if I did the calculations right, it should be smaller than 8 of the values. 1, 2, 3, 4, 5, 6, 7, 8. So it is indeed the median. Now when we take a box and whisker, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median. And it's essentially dividing our data into two sets. Now let's take the median of each of those sets. And the convention is to take our median out and have the sets that are left over."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now when we take a box and whisker, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median. And it's essentially dividing our data into two sets. Now let's take the median of each of those sets. And the convention is to take our median out and have the sets that are left over. Sometimes people leave it in. But the standard convention, take this median out, and now look separately at this set and look separately at this set. So if we look at this first, the bottom half of our numbers, essentially, what's the median of these numbers?"}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the convention is to take our median out and have the sets that are left over. Sometimes people leave it in. But the standard convention, take this median out, and now look separately at this set and look separately at this set. So if we look at this first, the bottom half of our numbers, essentially, what's the median of these numbers? Well, we have 1, 2, 3, 4, 5, 6, 7, 8 data points. So we're actually going to have two middle numbers. So the two middle numbers are this 2 and this 3."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if we look at this first, the bottom half of our numbers, essentially, what's the median of these numbers? Well, we have 1, 2, 3, 4, 5, 6, 7, 8 data points. So we're actually going to have two middle numbers. So the two middle numbers are this 2 and this 3. Three numbers less than these two, three numbers greater than it. And so when we're looking for a median, you have two middle numbers. We take the mean of these two numbers."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the two middle numbers are this 2 and this 3. Three numbers less than these two, three numbers greater than it. And so when we're looking for a median, you have two middle numbers. We take the mean of these two numbers. So halfway in between 2 and 3 is 2.5. Or you could say 2 plus 3 is 5 divided by 2 is 2.5. So here we have a median of this bottom half of 2.5."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "We take the mean of these two numbers. So halfway in between 2 and 3 is 2.5. Or you could say 2 plus 3 is 5 divided by 2 is 2.5. So here we have a median of this bottom half of 2.5. And then the middle of the top half, once again, we have 8 data points. So our middle two numbers are going to be this 11 and this 14. And so if we want to take the mean of these two numbers, 11 plus 14 is 25."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here we have a median of this bottom half of 2.5. And then the middle of the top half, once again, we have 8 data points. So our middle two numbers are going to be this 11 and this 14. And so if we want to take the mean of these two numbers, 11 plus 14 is 25. Halfway in between the two is 12.5. So 12.5 is exactly halfway between 11 and 14. And now we've figured out all of the information we need to actually plot or actually create or actually draw our box and whisker plot."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if we want to take the mean of these two numbers, 11 plus 14 is 25. Halfway in between the two is 12.5. So 12.5 is exactly halfway between 11 and 14. And now we've figured out all of the information we need to actually plot or actually create or actually draw our box and whisker plot. So let me draw a number line. So my best attempt at a number line. So that's my number line."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now we've figured out all of the information we need to actually plot or actually create or actually draw our box and whisker plot. So let me draw a number line. So my best attempt at a number line. So that's my number line. And let's say that this right over here is 0. I need to make sure I get all the way up to 22 or beyond 22. So let's say that's 0."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's my number line. And let's say that this right over here is 0. I need to make sure I get all the way up to 22 or beyond 22. So let's say that's 0. Let's say this is 5. This is 10. That could be 15."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that's 0. Let's say this is 5. This is 10. That could be 15. And that could be 20. This could be 25. We could keep going."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "That could be 15. And that could be 20. This could be 25. We could keep going. 30, maybe 35. So the first thing we might want to think about, there's several ways to draw it. We want to think about the box part of the box and whisker."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could keep going. 30, maybe 35. So the first thing we might want to think about, there's several ways to draw it. We want to think about the box part of the box and whisker. It essentially represents the middle half of our data. So it's essentially trying to represent this data right over here. So the data between the medians of the two halves."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to think about the box part of the box and whisker. It essentially represents the middle half of our data. So it's essentially trying to represent this data right over here. So the data between the medians of the two halves. So this is a part that we would attempt to represent with the box. So we would start right over here at this lower, this 2.5. This is essentially separating the first quartile from the second quartile, the first quarter of our numbers from the second quarter of our numbers."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the data between the medians of the two halves. So this is a part that we would attempt to represent with the box. So we would start right over here at this lower, this 2.5. This is essentially separating the first quartile from the second quartile, the first quarter of our numbers from the second quarter of our numbers. So let's put it right over here. This is 2.5. 2.5 is halfway between 0 and 5."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is essentially separating the first quartile from the second quartile, the first quarter of our numbers from the second quarter of our numbers. So let's put it right over here. This is 2.5. 2.5 is halfway between 0 and 5. So that's 2.5. And then up here we have 12.5. And 12.5 is right over, let's see, this is 10."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "2.5 is halfway between 0 and 5. So that's 2.5. And then up here we have 12.5. And 12.5 is right over, let's see, this is 10. So this right over here would be this halfway between 10 and 15 is 12.5. So let me do this. So this is 12.5 right over here."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And 12.5 is right over, let's see, this is 10. So this right over here would be this halfway between 10 and 15 is 12.5. So let me do this. So this is 12.5 right over here. 12.5, so that separates the third quartile from the fourth quartile. And then our box is everything in between. So this is literally the middle half of our numbers."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is 12.5 right over here. 12.5, so that separates the third quartile from the fourth quartile. And then our box is everything in between. So this is literally the middle half of our numbers. And we'd want to show where the actual median is. So that was actually one of the things that we wanted to be able to think about in our original, when the owner of the restaurant wanted to think about how far people are traveling from. So the median is 6."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is literally the middle half of our numbers. And we'd want to show where the actual median is. So that was actually one of the things that we wanted to be able to think about in our original, when the owner of the restaurant wanted to think about how far people are traveling from. So the median is 6. So we can plot it right over here. So this right over here looks, this is about 6. So that is, let me do that same pink color."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the median is 6. So we can plot it right over here. So this right over here looks, this is about 6. So that is, let me do that same pink color. So this right over here is 6. And then the whiskers of the box and whisker plot essentially show us the range of our data. And so let me do that."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that is, let me do that same pink color. So this right over here is 6. And then the whiskers of the box and whisker plot essentially show us the range of our data. And so let me do that. I could do this in a different color that I haven't used yet. I'll do this in orange. So essentially if we want to see, look, the numbers go all the way up to 22."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so let me do that. I could do this in a different color that I haven't used yet. I'll do this in orange. So essentially if we want to see, look, the numbers go all the way up to 22. So they go all the way up to, so let's say that this is 22 right over here. Our numbers go all the way up to 22. Our numbers go all the way up to 22."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "So essentially if we want to see, look, the numbers go all the way up to 22. So they go all the way up to, so let's say that this is 22 right over here. Our numbers go all the way up to 22. Our numbers go all the way up to 22. And they go as low as 1. So they go, 1 is right about here. They go as low, let me label that, so that's 1."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our numbers go all the way up to 22. And they go as low as 1. So they go, 1 is right about here. They go as low, let me label that, so that's 1. And they go as low as 1. So there you have it. We have our box and whisker plot."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "They go as low, let me label that, so that's 1. And they go as low as 1. So there you have it. We have our box and whisker plot. And you can see, if you have a plot like this, just visually you can immediately see, OK, what is the median? It's the middle of the box, essentially. It shows you the middle half."}, {"video_title": "Constructing a box and whisker plot   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have our box and whisker plot. And you can see, if you have a plot like this, just visually you can immediately see, OK, what is the median? It's the middle of the box, essentially. It shows you the middle half. So it shows you how far they're spread, or kind of where the meat of the spread is. And then it shows, well, beyond that, we have the range that goes well beyond that, or how far the total spread of our data is. So this gives a pretty good sense of both the median and the spread of our data."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "If we only have a few experiments, it's very possible that our experimental probability could be different than our theoretical probability or even very different. But as we have many, many more experiments, thousands, millions, billions of experiments, the probability that the experimental and the theoretical probabilities are very different goes down dramatically. But let's get an intuitive sense for it. This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happens if we do way more tosses? So now I'm gonna do another, well, let's just do another 200 tosses and see what happens."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happens if we do way more tosses? So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect."}, {"video_title": "Experimental versus theoretical probability simulation   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect. But as we get to many, many more flips, now we're at 1,210, we're getting pretty close to 50% of them turning out heads, but we could keep tossing it more and more and more. And what we'll see is as we get larger and larger and larger, it is likely that we're gonna get closer and closer and closer to 50%. It's not to say that it's impossible that we diverge again, but the likelihood of diverging gets lower and lower and lower the more tosses, the more experiments you make."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 30 people in a room. They're randomly selected. 30 people. And the question is, what is the probability that at least two people have the same birthday? And this is kind of a fun question because, I don't know, that's the size of a lot of classrooms. What's the probability that at least someone in the classroom shares a birthday with someone else in the classroom? And that's actually another good way to phrase it well."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the question is, what is the probability that at least two people have the same birthday? And this is kind of a fun question because, I don't know, that's the size of a lot of classrooms. What's the probability that at least someone in the classroom shares a birthday with someone else in the classroom? And that's actually another good way to phrase it well. This is the same thing as saying, what is the probability that someone shares with at least someone else? They could share it with two other people or four other people on the birthday. Someone else."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's actually another good way to phrase it well. This is the same thing as saying, what is the probability that someone shares with at least someone else? They could share it with two other people or four other people on the birthday. Someone else. And at first this problem seems really hard because, wow, there's a lot of circumstances that make this true. I could have exactly two people have the same birthday. I could have exactly three people have the same birthday."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Someone else. And at first this problem seems really hard because, wow, there's a lot of circumstances that make this true. I could have exactly two people have the same birthday. I could have exactly three people have the same birthday. I could have exactly 29 people have the same birthday. All of these make this true. So do I add the probability of each of those circumstances and then add them up and then that becomes really hard and then I would have to say, OK, whose birthdays am I comparing and I would have to do combinations."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could have exactly three people have the same birthday. I could have exactly 29 people have the same birthday. All of these make this true. So do I add the probability of each of those circumstances and then add them up and then that becomes really hard and then I would have to say, OK, whose birthdays am I comparing and I would have to do combinations. It becomes a really difficult problem. Unless you make kind of one very simplifying, I would say, take on the problem. This is the opposite of, let me draw the probability space."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So do I add the probability of each of those circumstances and then add them up and then that becomes really hard and then I would have to say, OK, whose birthdays am I comparing and I would have to do combinations. It becomes a really difficult problem. Unless you make kind of one very simplifying, I would say, take on the problem. This is the opposite of, let me draw the probability space. Let's say that this is all of the outcomes. Let me draw it with a thicker line. So let's say that's all of the outcomes in my probability space."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the opposite of, let me draw the probability space. Let's say that this is all of the outcomes. Let me draw it with a thicker line. So let's say that's all of the outcomes in my probability space. So that's 100% of the outcomes. And we want to know, let me draw it in a color that won't be offensive to you. That doesn't look that great, but anyway."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that's all of the outcomes in my probability space. So that's 100% of the outcomes. And we want to know, let me draw it in a color that won't be offensive to you. That doesn't look that great, but anyway. Let's say that this is the probability, this area right here, I don't know how big it really is, we'll figure it out, let's say that this is the probability that someone shares a birthday with at least someone else. What's this area over here? What's this green area?"}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "That doesn't look that great, but anyway. Let's say that this is the probability, this area right here, I don't know how big it really is, we'll figure it out, let's say that this is the probability that someone shares a birthday with at least someone else. What's this area over here? What's this green area? Well, that means if these are all the cases where someone shares a birthday with someone else, these are all the areas where no one shares a birthday with anyone. Or you could say all 30 people have different birthdays. So we're trying to figure out, this is what we're trying to figure out, the probability, I'll just call it the probability that someone shares."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's this green area? Well, that means if these are all the cases where someone shares a birthday with someone else, these are all the areas where no one shares a birthday with anyone. Or you could say all 30 people have different birthdays. So we're trying to figure out, this is what we're trying to figure out, the probability, I'll just call it the probability that someone shares. I'll call it the probability of sharing, probability of s. If this whole area is area 1 or area 100%, this green area right here, this is going to be 1 minus p of s. Or if we said that this is the probability, or another way we could say it, actually this is the best way to think about it. If this is different, so this is the probability of different birthdays. This is the probability that all 30 people have 30 different birthdays, right?"}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're trying to figure out, this is what we're trying to figure out, the probability, I'll just call it the probability that someone shares. I'll call it the probability of sharing, probability of s. If this whole area is area 1 or area 100%, this green area right here, this is going to be 1 minus p of s. Or if we said that this is the probability, or another way we could say it, actually this is the best way to think about it. If this is different, so this is the probability of different birthdays. This is the probability that all 30 people have 30 different birthdays, right? No one shares with anyone. The probability that someone shares with someone else plus the probability that no one shares with anyone, they all have distinct birthdays, that's got to be equal to 1. Because we're either going to be in this situation or we're going to be in that situation."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the probability that all 30 people have 30 different birthdays, right? No one shares with anyone. The probability that someone shares with someone else plus the probability that no one shares with anyone, they all have distinct birthdays, that's got to be equal to 1. Because we're either going to be in this situation or we're going to be in that situation. Or you could say they're equal to 100%, either way. 100% and 1 are the same number. It's equal to 100%."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because we're either going to be in this situation or we're going to be in that situation. Or you could say they're equal to 100%, either way. 100% and 1 are the same number. It's equal to 100%. So if we figure out the probability that everyone has the same birthday, we could subtract it from 100. So let's see, if we could just rewrite this. The probability that someone shares a birthday with someone else, that's equal to 100% minus the probability that everyone has distinct separate birthdays."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's equal to 100%. So if we figure out the probability that everyone has the same birthday, we could subtract it from 100. So let's see, if we could just rewrite this. The probability that someone shares a birthday with someone else, that's equal to 100% minus the probability that everyone has distinct separate birthdays. And the reason why I'm doing that is because, as I started off in the video, this is kind of hard to figure out. I could figure out the probability that two people have the same birthday, five people, and it becomes very confusing. But here, if I want to just figure out the probability that everyone has a distinct birthday, it's actually a much easier probability to solve for."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability that someone shares a birthday with someone else, that's equal to 100% minus the probability that everyone has distinct separate birthdays. And the reason why I'm doing that is because, as I started off in the video, this is kind of hard to figure out. I could figure out the probability that two people have the same birthday, five people, and it becomes very confusing. But here, if I want to just figure out the probability that everyone has a distinct birthday, it's actually a much easier probability to solve for. So what's the probability that everyone has a distinct birthday? So let's think about it. Person 1, just for simplicity, let's imagine the case that we only have two people in the room."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "But here, if I want to just figure out the probability that everyone has a distinct birthday, it's actually a much easier probability to solve for. So what's the probability that everyone has a distinct birthday? So let's think about it. Person 1, just for simplicity, let's imagine the case that we only have two people in the room. What's the probability that they have different birthdays? So if I have two, let's see, person 1, their birthday could be 365 days out of 365 days in the year, right? Whatever their birthday is."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Person 1, just for simplicity, let's imagine the case that we only have two people in the room. What's the probability that they have different birthdays? So if I have two, let's see, person 1, their birthday could be 365 days out of 365 days in the year, right? Whatever their birthday is. And then person 2, if we wanted to ensure that they don't have the same birthday, how many days could person 2 be born on? Well, it could be born on any day that person 1 was not born on. So there's 364 possibilities out of 365."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Whatever their birthday is. And then person 2, if we wanted to ensure that they don't have the same birthday, how many days could person 2 be born on? Well, it could be born on any day that person 1 was not born on. So there's 364 possibilities out of 365. So if you had two people, the probability that no one is born on the same birthday, this is just 1, is just going to be equal to 364 over 365, right? Now, what happens if we had three people? So first of all, the first person could be born on any day, then the second person could be born on 364 possible days out of 365, and then the third person, what's the probability that this third person isn't born on either of these people's birthdays?"}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's 364 possibilities out of 365. So if you had two people, the probability that no one is born on the same birthday, this is just 1, is just going to be equal to 364 over 365, right? Now, what happens if we had three people? So first of all, the first person could be born on any day, then the second person could be born on 364 possible days out of 365, and then the third person, what's the probability that this third person isn't born on either of these people's birthdays? So two days are taken up, so the probability is 363 over 365. So this is equal to, you multiply them out, you get 365 times 360. Actually, I should rewrite this one."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So first of all, the first person could be born on any day, then the second person could be born on 364 possible days out of 365, and then the third person, what's the probability that this third person isn't born on either of these people's birthdays? So two days are taken up, so the probability is 363 over 365. So this is equal to, you multiply them out, you get 365 times 360. Actually, I should rewrite this one. Instead of saying this is 1, let me write this as, the numerator is 365 times 364 over 365 squared. Because I want you to see the pattern, right? Here the probability is 365 times 364 times 363 over 365 to the third power."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, I should rewrite this one. Instead of saying this is 1, let me write this as, the numerator is 365 times 364 over 365 squared. Because I want you to see the pattern, right? Here the probability is 365 times 364 times 363 over 365 to the third power. And so in general, if you just kept doing this to 30, if I just kept this process for 30 people, so 30 people, the probability that no one shares the same birthday would be equal to 365 times 364 times 363. We'll have 30 terms up here, right? All the way down to what?"}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here the probability is 365 times 364 times 363 over 365 to the third power. And so in general, if you just kept doing this to 30, if I just kept this process for 30 people, so 30 people, the probability that no one shares the same birthday would be equal to 365 times 364 times 363. We'll have 30 terms up here, right? All the way down to what? 330, all the way down to 336, right? That'll actually be 30 terms divided by 365 to the 30th power, and you could just type this into your calculator right now, it'll take you a little time to type in 30 numbers. And you'll get the probability that no one shares the same birthday with anyone else."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "All the way down to what? 330, all the way down to 336, right? That'll actually be 30 terms divided by 365 to the 30th power, and you could just type this into your calculator right now, it'll take you a little time to type in 30 numbers. And you'll get the probability that no one shares the same birthday with anyone else. But before we do that, let me just show you something that might make it a little bit easier. Is there any way that I can mathematically express this with factorials, or that I could mathematically express this with factorials? Well, let's think about it."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you'll get the probability that no one shares the same birthday with anyone else. But before we do that, let me just show you something that might make it a little bit easier. Is there any way that I can mathematically express this with factorials, or that I could mathematically express this with factorials? Well, let's think about it. 365 factorial is what? 365 factorial is equal to 365 times 364 times 363 times all the way down to 1, right? You just keep multiplying, it's a huge number."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's think about it. 365 factorial is what? 365 factorial is equal to 365 times 364 times 363 times all the way down to 1, right? You just keep multiplying, it's a huge number. Now, if I just want the 365 times the 364 in this case, I have to get rid of all of these numbers back here. So one thing I could do is I could divide this thing by all of these numbers. So 363 times 362 all the way down to 1."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "You just keep multiplying, it's a huge number. Now, if I just want the 365 times the 364 in this case, I have to get rid of all of these numbers back here. So one thing I could do is I could divide this thing by all of these numbers. So 363 times 362 all the way down to 1. So that's the same thing as dividing by 363 factorial. 365 factorial divided by 363 factorial is essentially this, because all of these terms cancel out. So this is equal to 365 factorial over 363 factorial over 365 squared."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 363 times 362 all the way down to 1. So that's the same thing as dividing by 363 factorial. 365 factorial divided by 363 factorial is essentially this, because all of these terms cancel out. So this is equal to 365 factorial over 363 factorial over 365 squared. And of course, for this case, it's almost silly to worry about the factorials, but it becomes useful once we have something larger than two terms up here. So by the same logic, this right here is going to be equal to 365 factorial over 362 factorial over 365 squared. And actually, just another interesting point."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to 365 factorial over 363 factorial over 365 squared. And of course, for this case, it's almost silly to worry about the factorials, but it becomes useful once we have something larger than two terms up here. So by the same logic, this right here is going to be equal to 365 factorial over 362 factorial over 365 squared. And actually, just another interesting point. How did we get this 365? Sorry, how did we get this 363 factorial? Well, 365 minus 2 is 363, right?"}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, just another interesting point. How did we get this 365? Sorry, how did we get this 363 factorial? Well, 365 minus 2 is 363, right? And that makes sense, because we only wanted two terms up here. So we wanted to divide by a factorial that's 2 less. And so we'd only get the highest two terms left."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, 365 minus 2 is 363, right? And that makes sense, because we only wanted two terms up here. So we wanted to divide by a factorial that's 2 less. And so we'd only get the highest two terms left. So this is also equal to, you could write this as 365 factorial divided by 365 minus 2 factorial. 365 minus 2 is 363 factorial. And then you just end up with those two terms, and that's that there."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so we'd only get the highest two terms left. So this is also equal to, you could write this as 365 factorial divided by 365 minus 2 factorial. 365 minus 2 is 363 factorial. And then you just end up with those two terms, and that's that there. And then likewise, this right here, this numerator, you could rewrite as 365 factorial divided by 365 minus 3. And we had 3 people. Factorial."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you just end up with those two terms, and that's that there. And then likewise, this right here, this numerator, you could rewrite as 365 factorial divided by 365 minus 3. And we had 3 people. Factorial. And that should hopefully make sense, right? This is the same thing as 365 factorial. Well, 365 divided by 3 is 362 factorial."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Factorial. And that should hopefully make sense, right? This is the same thing as 365 factorial. Well, 365 divided by 3 is 362 factorial. And so that's equal to 365 times 364 times 363, all the way down, divided by 362 times all the way down. And that'll cancel out with everything else, and you'd be just left with that. And that's that right there."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, 365 divided by 3 is 362 factorial. And so that's equal to 365 times 364 times 363, all the way down, divided by 362 times all the way down. And that'll cancel out with everything else, and you'd be just left with that. And that's that right there. So by that same logic, this top part here can be written as 365 factorial over what? 365 minus 30 factorial. And I did all of that just so I could show you kind of the pattern."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's that right there. So by that same logic, this top part here can be written as 365 factorial over what? 365 minus 30 factorial. And I did all of that just so I could show you kind of the pattern. And because this is, frankly, easier to type into a calculator if you know where the factorial button is. So let's figure out what this entire probability is. So turning on the calculator, we want, so let's do the numerator, 365 factorial divided by, what was 365 minus 30, that's 335, right?"}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I did all of that just so I could show you kind of the pattern. And because this is, frankly, easier to type into a calculator if you know where the factorial button is. So let's figure out what this entire probability is. So turning on the calculator, we want, so let's do the numerator, 365 factorial divided by, what was 365 minus 30, that's 335, right? Divided by 335 factorial. And that's that whole numerator. And now we want to divide the numerator, divided by 365 to the 30th power."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "So turning on the calculator, we want, so let's do the numerator, 365 factorial divided by, what was 365 minus 30, that's 335, right? Divided by 335 factorial. And that's that whole numerator. And now we want to divide the numerator, divided by 365 to the 30th power. Let the calculator think, and we get 0.2936. Equals 0.2936, it keeps, actually, 37 if you round it. Which is equal to 29.37%."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now we want to divide the numerator, divided by 365 to the 30th power. Let the calculator think, and we get 0.2936. Equals 0.2936, it keeps, actually, 37 if you round it. Which is equal to 29.37%. Now, just so you remember what we were doing all along, this was the probability that no one shares a birthday with anyone. This was the probability of everyone having distinct different birthdays from everyone else. And we've said, well, the probability that someone shares a birthday with someone else, or maybe more than one person, is equal to all of the possibilities, kind of the 100%, the probability space, minus the probability that no one shares a birthday with anybody."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which is equal to 29.37%. Now, just so you remember what we were doing all along, this was the probability that no one shares a birthday with anyone. This was the probability of everyone having distinct different birthdays from everyone else. And we've said, well, the probability that someone shares a birthday with someone else, or maybe more than one person, is equal to all of the possibilities, kind of the 100%, the probability space, minus the probability that no one shares a birthday with anybody. So that's equal to 100% minus 29.37%. Or another way you could write it is, that's 1 minus 0.2937. Which is equal to, so if I want to subtract that from 1, 1 minus, that just means the answer, that means 1 minus 0.29."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we've said, well, the probability that someone shares a birthday with someone else, or maybe more than one person, is equal to all of the possibilities, kind of the 100%, the probability space, minus the probability that no one shares a birthday with anybody. So that's equal to 100% minus 29.37%. Or another way you could write it is, that's 1 minus 0.2937. Which is equal to, so if I want to subtract that from 1, 1 minus, that just means the answer, that means 1 minus 0.29. You get 0.7063. So the probability that someone shares a birthday with someone else is 0.7063, it keeps going, which is approximately equal to 70.6%. Which is kind of a neat result."}, {"video_title": "Birthday probability problem   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which is equal to, so if I want to subtract that from 1, 1 minus, that just means the answer, that means 1 minus 0.29. You get 0.7063. So the probability that someone shares a birthday with someone else is 0.7063, it keeps going, which is approximately equal to 70.6%. Which is kind of a neat result. Because if you have 30 people in a room, you might say, oh wow, what are the odds that someone has the same birthday as someone else? It's actually pretty high. Most, 70% of the time, if you have a group of 30 people, at least one person shares a birthday with at least one other person in the room."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Based on the data below, which student score improved the most between the midterm and final exams? So they give us this data in terms of a bar graph for each student. We actually have two bars that show the midterm in blue and then the final exam in red. And they tell us that here, midterm in blue and final exam. Sometimes this is called a two-column bar graph because for each student here you have two columns of data. So if you were to actually look at the data itself, you have the midterm data and then you have the final exam data. Now, they're asking us which student score improved the most between the midterms and the final exams."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And they tell us that here, midterm in blue and final exam. Sometimes this is called a two-column bar graph because for each student here you have two columns of data. So if you were to actually look at the data itself, you have the midterm data and then you have the final exam data. Now, they're asking us which student score improved the most between the midterms and the final exams. So if we look at Jasmine right over here, might as well start with her since she's the most to the left. It looks like she definitely did improve from the midterm to the final. It looks like in the midterm, if I had to guess, this looks like about a, I don't know, it looks like she got maybe about a 72 or 73 on the midterm."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Now, they're asking us which student score improved the most between the midterms and the final exams. So if we look at Jasmine right over here, might as well start with her since she's the most to the left. It looks like she definitely did improve from the midterm to the final. It looks like in the midterm, if I had to guess, this looks like about a, I don't know, it looks like she got maybe about a 72 or 73 on the midterm. I'm just guessing because I don't know the exact number. And it looks like on the final she got, I don't know, that looks like maybe a 77 or 78 approximately. So she improved a little bit, about five points from the midterm to the final."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "It looks like in the midterm, if I had to guess, this looks like about a, I don't know, it looks like she got maybe about a 72 or 73 on the midterm. I'm just guessing because I don't know the exact number. And it looks like on the final she got, I don't know, that looks like maybe a 77 or 78 approximately. So she improved a little bit, about five points from the midterm to the final. The way that they've given us this information, we don't know the exact numbers because it's not super precise in terms of marking off the bars. But hopefully it will become obvious when we look through everyone's scores. Let's look at Jeff."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So she improved a little bit, about five points from the midterm to the final. The way that they've given us this information, we don't know the exact numbers because it's not super precise in terms of marking off the bars. But hopefully it will become obvious when we look through everyone's scores. Let's look at Jeff. So Jeff actually did worse from the midterm to the final exam. He got, looks like over an 85 on the midterm and then on the final he got about an 84 or 85. So he actually went down."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let's look at Jeff. So Jeff actually did worse from the midterm to the final exam. He got, looks like over an 85 on the midterm and then on the final he got about an 84 or 85. So he actually went down. So it's definitely not Jeff. He definitely did not improve the most. He actually went down."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So he actually went down. So it's definitely not Jeff. He definitely did not improve the most. He actually went down. Next, if you think about Nevin, it looks like Nevin actually improved about the same amount as Jasmine. On both scores, I don't know if Nevin is a boy or a girl's name, this person on both tests, this person did better than Jasmine. But it looks like the actual improvement is about the same."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "He actually went down. Next, if you think about Nevin, it looks like Nevin actually improved about the same amount as Jasmine. On both scores, I don't know if Nevin is a boy or a girl's name, this person on both tests, this person did better than Jasmine. But it looks like the actual improvement is about the same. It looks like they went from about an 83 to, I don't know, about an 88. I'm just estimating it, just trying to look at this axis right over there and estimating what those scores are. So Nevin and Jasmine right now, based on the three we've seen, are tied for the lead."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "But it looks like the actual improvement is about the same. It looks like they went from about an 83 to, I don't know, about an 88. I'm just estimating it, just trying to look at this axis right over there and estimating what those scores are. So Nevin and Jasmine right now, based on the three we've seen, are tied for the lead. Now let's look at Alejandra. Now Alejandra, this is, okay, so this jumps out. She definitely improved dramatically from the midterm to the final exam."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So Nevin and Jasmine right now, based on the three we've seen, are tied for the lead. Now let's look at Alejandra. Now Alejandra, this is, okay, so this jumps out. She definitely improved dramatically from the midterm to the final exam. It looks like on the midterm she maybe got an 81 or an 82. So maybe this was an 82 she got on the midterm and then on the final it looks like she got about a 95. A 95 on the final."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "She definitely improved dramatically from the midterm to the final exam. It looks like on the midterm she maybe got an 81 or an 82. So maybe this was an 82 she got on the midterm and then on the final it looks like she got about a 95. A 95 on the final. So it was a dramatic improvement. So right now Alejandra is the leading contender for most improved from the midterm to the final. And finally Marta right over here, it looks like she actually got worse from the midterm to the final."}, {"video_title": "Reading bar graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "A 95 on the final. So it was a dramatic improvement. So right now Alejandra is the leading contender for most improved from the midterm to the final. And finally Marta right over here, it looks like she actually got worse from the midterm to the final. She scored in the mid-90s in the midterm and then low 90s in the final. So she's definitely not the most improved. So the winner here is Alejandra."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Amelia only likes to use the drive-through for restaurants where the average wait time is in the bottom 10% for that town. What is the maximum average wait time for restaurants where Amelia likes to use the drive-through? Round to the nearest whole second. Like always, if you feel like you can tackle this, pause this video and try to do so. I'm assuming you paused it. Now let's work through this together. So let's think about what's going on."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Like always, if you feel like you can tackle this, pause this video and try to do so. I'm assuming you paused it. Now let's work through this together. So let's think about what's going on. They're telling us that the distribution of average wait times is approximately normal. So let's get a visualization of a normal distribution. And they tell us several things about this normal distribution."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So let's think about what's going on. They're telling us that the distribution of average wait times is approximately normal. So let's get a visualization of a normal distribution. And they tell us several things about this normal distribution. They tell us that the mean is 185 seconds. So that's 185 there. The standard deviation is 11 seconds."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And they tell us several things about this normal distribution. They tell us that the mean is 185 seconds. So that's 185 there. The standard deviation is 11 seconds. So for example, this is going to be 11 more than the mean. So this would be 196 seconds. This would be another 11."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "The standard deviation is 11 seconds. So for example, this is going to be 11 more than the mean. So this would be 196 seconds. This would be another 11. Each of these dotted lines are one standard deviation more. So this would be 207. This would be 11 seconds less than the mean."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "This would be another 11. Each of these dotted lines are one standard deviation more. So this would be 207. This would be 11 seconds less than the mean. So this would be 174, and so on and so forth. And we wanna find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants?"}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "This would be 11 seconds less than the mean. So this would be 174, and so on and so forth. And we wanna find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants? That's where the average wait time is in the bottom 10% for that town. So how do we think about it? Well, there's going to be some value."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, what are those restaurants? That's where the average wait time is in the bottom 10% for that town. So how do we think about it? Well, there's going to be some value. Let me mark it off right over here in this red color. So we're gonna have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, there's going to be some value. Let me mark it off right over here in this red color. So we're gonna have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. And so this area right over here is going to be 10% of the total, or it's going to be 0.10. So the way we can tackle this is we can get up a z-table and figure out what z-score gives us a proportion of only 0.10 being less than that z-score. And then using that z-score, we can figure out this value, the actual wait time."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. And so this area right over here is going to be 10% of the total, or it's going to be 0.10. So the way we can tackle this is we can get up a z-table and figure out what z-score gives us a proportion of only 0.10 being less than that z-score. And then using that z-score, we can figure out this value, the actual wait time. So let's get our z-table out. And since we know that this is below the mean, the mean would be the 50th percentile, we know we're gonna have a negative z-score. So I'm gonna take out the part of the table that has the negative z-scores on it."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And then using that z-score, we can figure out this value, the actual wait time. So let's get our z-table out. And since we know that this is below the mean, the mean would be the 50th percentile, we know we're gonna have a negative z-score. So I'm gonna take out the part of the table that has the negative z-scores on it. And remember, we're looking for 10%, but we don't wanna go beyond 10%. We wanna be sure that that value is within the 10th percentile, that any higher will be out of the 10th percentile. So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So I'm gonna take out the part of the table that has the negative z-scores on it. And remember, we're looking for 10%, but we don't wanna go beyond 10%. We wanna be sure that that value is within the 10th percentile, that any higher will be out of the 10th percentile. So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet. So let's scroll down a little bit. And let's remember as we do so that this is zero in the hundredths place, one, two, three, four, five, six, seven, eight, nine. So let's remember those columns."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet. So let's scroll down a little bit. And let's remember as we do so that this is zero in the hundredths place, one, two, three, four, five, six, seven, eight, nine. So let's remember those columns. So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28. And that's a little bit crossing the 10th percentile."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So let's remember those columns. So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28. And that's a little bit crossing the 10th percentile. But if we get a little bit more negative than that, we are in the 10th percentile. So this is negative 1.29. And this does seem to be the highest z-score for which we are within the 10th percentile."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And that's a little bit crossing the 10th percentile. But if we get a little bit more negative than that, we are in the 10th percentile. So this is negative 1.29. And this does seem to be the highest z-score for which we are within the 10th percentile. So negative 1.29 is our z-score. So this is z equals negative 1.29. And if we wanna figure out the actual value for that, we would start with the mean, which is 185."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And this does seem to be the highest z-score for which we are within the 10th percentile. So negative 1.29 is our z-score. So this is z equals negative 1.29. And if we wanna figure out the actual value for that, we would start with the mean, which is 185. And then we would say, well, we wanna go 1.29 standard deviations below the mean. The negative says we're going below the mean. So we could say minus 1.29 times the standard deviation."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And if we wanna figure out the actual value for that, we would start with the mean, which is 185. And then we would say, well, we wanna go 1.29 standard deviations below the mean. The negative says we're going below the mean. So we could say minus 1.29 times the standard deviation. And they tell us up here the standard deviation is 11 seconds. So it's going to be 1.29 times 11. And this is going to be equal to 1.29 times 11 is equal to 14.19."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So we could say minus 1.29 times the standard deviation. And they tell us up here the standard deviation is 11 seconds. So it's going to be 1.29 times 11. And this is going to be equal to 1.29 times 11 is equal to 14.19. And then I'll make that negative and then add that to 185. Plus 185 is equal to 170.81. 170.81."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And this is going to be equal to 1.29 times 11 is equal to 14.19. And then I'll make that negative and then add that to 185. Plus 185 is equal to 170.81. 170.81. Now they say round to the nearest whole second. There's a couple of ways to think about it. If you really wanna ensure that you're not gonna cross the 10th percentile, you might wanna round to the nearest second that is below this threshold."}, {"video_title": "Threshold for low percentile   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "170.81. Now they say round to the nearest whole second. There's a couple of ways to think about it. If you really wanna ensure that you're not gonna cross the 10th percentile, you might wanna round to the nearest second that is below this threshold. So you might say that this is approximately 170 seconds. If you were to just round normally, this would go to 171. But just by doing that, you might have crossed the threshold."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "We're told Della has over 500 songs on her mobile phone, and she wants to estimate what proportion of the songs are by a female artist. She takes a simple random sample, that's what SRS stands for, of 50 songs on her phone, and finds that 20 of the songs sampled are by a female artist. Based on this sample, which of the following is a 99% confidence interval for the proportion of songs on her phone that are by a female artist? So like always, pause this video and see if you can figure it out on your own. Della has a library of 500 songs right over here, and she's trying to figure out the proportion that are sung by a female artist. She doesn't have the time to go through all 500 songs to figure out the true population proportion, P, so instead she takes a sample of 50 songs, N is equal to 50, and from that she calculates a sample proportion, which we could denote with P hat, and she finds that 20 out of the 50 are sung by a female. 20 out of the 50, which is the same thing, is 0.4, and then she wants to construct a 99% confidence interval."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "So like always, pause this video and see if you can figure it out on your own. Della has a library of 500 songs right over here, and she's trying to figure out the proportion that are sung by a female artist. She doesn't have the time to go through all 500 songs to figure out the true population proportion, P, so instead she takes a sample of 50 songs, N is equal to 50, and from that she calculates a sample proportion, which we could denote with P hat, and she finds that 20 out of the 50 are sung by a female. 20 out of the 50, which is the same thing, is 0.4, and then she wants to construct a 99% confidence interval. So before we even go about constructing the confidence interval, you wanna check to make sure that we're making some valid assumptions, we're using a valid technique. So before we actually calculate the confidence interval, let's just make sure that our sampling distribution is not distorted in some way, and so that we can, with confidence, make a confidence interval. So the first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "20 out of the 50, which is the same thing, is 0.4, and then she wants to construct a 99% confidence interval. So before we even go about constructing the confidence interval, you wanna check to make sure that we're making some valid assumptions, we're using a valid technique. So before we actually calculate the confidence interval, let's just make sure that our sampling distribution is not distorted in some way, and so that we can, with confidence, make a confidence interval. So the first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it. The next condition is to assume that your sampling distribution of the sample proportions is approximately normal, and there, you wanna be confident, or you wanna see that in your sample, you have at least 10 successes and at least 10 failures. Well, here, we have 20 successes, which means, well, 50 minus 20, we have 30 failures. So both of those are more than 10, and so meets that condition."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "So the first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it. The next condition is to assume that your sampling distribution of the sample proportions is approximately normal, and there, you wanna be confident, or you wanna see that in your sample, you have at least 10 successes and at least 10 failures. Well, here, we have 20 successes, which means, well, 50 minus 20, we have 30 failures. So both of those are more than 10, and so meets that condition. And then the last condition is, sometimes it'll called the independence test or the independence rule or the 10% rule. If you were doing this sample with replacement, so if she were to look at one song, test whether it's a female or not, and then put it back in her pile and then look at another song, then each of those observations would truly be independent. But we don't know that."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "So both of those are more than 10, and so meets that condition. And then the last condition is, sometimes it'll called the independence test or the independence rule or the 10% rule. If you were doing this sample with replacement, so if she were to look at one song, test whether it's a female or not, and then put it back in her pile and then look at another song, then each of those observations would truly be independent. But we don't know that. In fact, we'll assume that she didn't do it with the replacement, and so if you don't do it with the replacement, you can assume rough independence for each observation of a song if this is no more than 10% of the population. And so it looks like it is exactly 10% of the population, so Della just squeezes through on our independence test right over there. So with that out of the way, let's just think about what the confidence interval is going to be."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "But we don't know that. In fact, we'll assume that she didn't do it with the replacement, and so if you don't do it with the replacement, you can assume rough independence for each observation of a song if this is no more than 10% of the population. And so it looks like it is exactly 10% of the population, so Della just squeezes through on our independence test right over there. So with that out of the way, let's just think about what the confidence interval is going to be. Well, it's going to be her sample proportion, plus or minus, there's going to be some critical value, and this critical value is going to be dictated by our confidence level we wanna have, and then that critical value times the standard deviation of the sampling distribution of the sample proportions, which we don't know. And so instead of having that, we use the standard error of the sample proportion, and in this case, it would be p hat times one minus p hat, all of that over n, our sample size, all of that over 50. So what's this going to be?"}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "So with that out of the way, let's just think about what the confidence interval is going to be. Well, it's going to be her sample proportion, plus or minus, there's going to be some critical value, and this critical value is going to be dictated by our confidence level we wanna have, and then that critical value times the standard deviation of the sampling distribution of the sample proportions, which we don't know. And so instead of having that, we use the standard error of the sample proportion, and in this case, it would be p hat times one minus p hat, all of that over n, our sample size, all of that over 50. So what's this going to be? We're gonna get p hat, our sample proportion here, is 0.4 plus or minus, I'll save the z star here, our critical value, for a little bit. We're gonna use a z table for that. And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6, all of that over 50."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "So what's this going to be? We're gonna get p hat, our sample proportion here, is 0.4 plus or minus, I'll save the z star here, our critical value, for a little bit. We're gonna use a z table for that. And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6, all of that over 50. So we can already look at some choices that look interesting here. This choice and this choice both look interesting, and the main thing we have to reason through is which one has a correct critical value. Do we wanna go 1.96 standard errors above and below our sample proportion, or do we wanna go 2.576 standard errors above and below our sample proportion?"}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6, all of that over 50. So we can already look at some choices that look interesting here. This choice and this choice both look interesting, and the main thing we have to reason through is which one has a correct critical value. Do we wanna go 1.96 standard errors above and below our sample proportion, or do we wanna go 2.576 standard errors above and below our sample proportion? And the key is the 99% confidence level. Now, if we have a 99% confidence level, one way to think about it is, so let me just do my best shot at drawing a normal distribution here. And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99% under the curve right over here, that area."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "Do we wanna go 1.96 standard errors above and below our sample proportion, or do we wanna go 2.576 standard errors above and below our sample proportion? And the key is the 99% confidence level. Now, if we have a 99% confidence level, one way to think about it is, so let me just do my best shot at drawing a normal distribution here. And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99% under the curve right over here, that area. And so if this is 99%, then this right over here is going to be 0.5%, and this right over here is 0.5%. We want the z value that's going to leave 0.5% above it. And so that's actually going to be 99.5% is what we wanna look up on the table."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99% under the curve right over here, that area. And so if this is 99%, then this right over here is going to be 0.5%, and this right over here is 0.5%. We want the z value that's going to leave 0.5% above it. And so that's actually going to be 99.5% is what we wanna look up on the table. And that's because many z tables, including the one that you might see on something like an AP Stats exam, they will have the area up to and including, up to and including a certain value. And so they're not going to leave this free right over here. So let's just look up 99.5% on our z table."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "And so that's actually going to be 99.5% is what we wanna look up on the table. And that's because many z tables, including the one that you might see on something like an AP Stats exam, they will have the area up to and including, up to and including a certain value. And so they're not going to leave this free right over here. So let's just look up 99.5% on our z table. All right, so let me move this down so you can see it. All right, that's our z table. Let's see, we're at 99 point, okay, it's gonna be right in this area right over here."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just look up 99.5% on our z table. All right, so let me move this down so you can see it. All right, that's our z table. Let's see, we're at 99 point, okay, it's gonna be right in this area right over here. And so that is 2.5, looks like 2.57 or 2.58 around that. And so this right over here is about 2.57. It's between 2.57 and 2.58, which gives us enough information to answer this question."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "Let's see, we're at 99 point, okay, it's gonna be right in this area right over here. And so that is 2.5, looks like 2.57 or 2.58 around that. And so this right over here is about 2.57. It's between 2.57 and 2.58, which gives us enough information to answer this question. It's definitely not going to be this one right over here. We have 2.576, which is indeed between 2.57 and 2.58. So let's remind ourselves."}, {"video_title": "Example constructing and interpreting a confidence interval for p   AP Statistics   Khan Academy.mp3", "Sentence": "It's between 2.57 and 2.58, which gives us enough information to answer this question. It's definitely not going to be this one right over here. We have 2.576, which is indeed between 2.57 and 2.58. So let's remind ourselves. We've been able to construct our confidence interval right over here. But what does that actually mean? That means that if we were to repeatedly take samples of size 50 and repeatedly use this technique to construct confidence intervals, that roughly 99% of those intervals constructed this way are going to contain our true population parameter."}, {"video_title": "Mean of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So the expected value of X, which I could also denote as the mean of our random variable X, let's say I expect to see three dogs a day, and similarly for the cats, the expected value of Y is equal to, I could also denote that as the mean of Y, is going to be equal to, and this is just for the sake of argument, let's say I expect to see four cats a day, and in previous videos, we defined how do you take the mean of a random variable, or the expected value of a random variable. What we're going to think about now is, what would be the expected value of X plus Y? Or another way of saying that, the mean of the sum of these two random variables. Well, it turns out, and I'm not proving it just yet, that the mean of the sum of random variables is equal to the sum of the means. So this is going to be equal to the mean of random variable X plus the mean of random variable Y. And so in this particular case, if I were to say, well what's the expected number of dogs and cats that I would see in a given day? Well, I would add these two means."}, {"video_title": "Mean of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, it turns out, and I'm not proving it just yet, that the mean of the sum of random variables is equal to the sum of the means. So this is going to be equal to the mean of random variable X plus the mean of random variable Y. And so in this particular case, if I were to say, well what's the expected number of dogs and cats that I would see in a given day? Well, I would add these two means. It would be three plus four, it would be equal to seven. So in this particular case, it is equal to three plus four, which is equal to seven. And similarly, if I were to ask you the difference, if I were to say, well what's the, how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X?"}, {"video_title": "Mean of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, I would add these two means. It would be three plus four, it would be equal to seven. So in this particular case, it is equal to three plus four, which is equal to seven. And similarly, if I were to ask you the difference, if I were to say, well what's the, how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X? What would that be? Well, intuitively, you might say, well, hey, if we can add random, if the expected value of the sum is the sum of the expected values, then the expected value or the mean of the difference will be the differences of the means, and that is absolutely true. So this is the same thing as the mean of Y minus X, which is equal to the mean of Y, is going to be equal to the mean of Y minus the mean of X minus the mean of X, and in this particular case, it would be equal to four minus three, minus three is equal to one."}, {"video_title": "Mean of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And similarly, if I were to ask you the difference, if I were to say, well what's the, how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X? What would that be? Well, intuitively, you might say, well, hey, if we can add random, if the expected value of the sum is the sum of the expected values, then the expected value or the mean of the difference will be the differences of the means, and that is absolutely true. So this is the same thing as the mean of Y minus X, which is equal to the mean of Y, is going to be equal to the mean of Y minus the mean of X minus the mean of X, and in this particular case, it would be equal to four minus three, minus three is equal to one. So another way of thinking about this intuitively is I would expect to see on a given day one more cat than dogs. Now, the example that I've just used, this is discrete random variables, on a given day, I wouldn't see 2.2 dogs or pi dogs. The expected value itself does not have to be a whole number because you could, of course, average it over many days, but this same idea that the mean of a sum is the same thing as the sum of means and that the mean of a difference of random variables is the same as the difference of the means."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just to review a little bit of the last video, we said we're trying to model out the probability distribution of how many cars might pass in an hour. And the first thing we did is we sat at that intersection, and we found a pretty good expected value of our random variable, and this random variable, just to go back to the top, we defined the random variable as the number of cars that pass in an hour at a certain point on a certain road, and we said that we measure it a bunch. We sat out there a bunch of hours, and we got a pretty good estimate of this, and we say it's lambda. And we said, OK, we wanted to model it as a binomial distribution. So if this is a binomial distribution, then this lambda would be equal to the number of trials times the probability of success per trial. And so if we could view a trial as an interval of time, this is the total number of successes in an hour. Success in hour, and so this would be success in an smaller interval, an interval, and this would be the probability of success in that smaller interval."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we said, OK, we wanted to model it as a binomial distribution. So if this is a binomial distribution, then this lambda would be equal to the number of trials times the probability of success per trial. And so if we could view a trial as an interval of time, this is the total number of successes in an hour. Success in hour, and so this would be success in an smaller interval, an interval, and this would be the probability of success in that smaller interval. And in the last video, we tried it out. We said, oh, well, what if we make this interval a minute, and this is the probability of success per minute? We'd have maybe a reasonable description of what we're describing, but what if more than one car passes in a minute?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Success in hour, and so this would be success in an smaller interval, an interval, and this would be the probability of success in that smaller interval. And in the last video, we tried it out. We said, oh, well, what if we make this interval a minute, and this is the probability of success per minute? We'd have maybe a reasonable description of what we're describing, but what if more than one car passes in a minute? And they said, oh, let's make this per second, and this is the probability of success per second. But then we still have the problem. More than one car could pass in a second very easily."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We'd have maybe a reasonable description of what we're describing, but what if more than one car passes in a minute? And they said, oh, let's make this per second, and this is the probability of success per second. But then we still have the problem. More than one car could pass in a second very easily. So what we want to do is we want to take the limit as this approaches infinity, and then see what kind of formula we get from the math gods. So if we describe this as a binomial distribution with the limit as it approaches infinity, we could say that the probability that x is equal to some number, so the probability that our random variable is equal to, I don't know, three cars in a particular hour, exactly three cars in an hour, is equal to, oh, we want to take the limit as it approaches infinity, right? The limit as n approaches infinity of n choose k. We're going to have k moments in time, right?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "More than one car could pass in a second very easily. So what we want to do is we want to take the limit as this approaches infinity, and then see what kind of formula we get from the math gods. So if we describe this as a binomial distribution with the limit as it approaches infinity, we could say that the probability that x is equal to some number, so the probability that our random variable is equal to, I don't know, three cars in a particular hour, exactly three cars in an hour, is equal to, oh, we want to take the limit as it approaches infinity, right? The limit as n approaches infinity of n choose k. We're going to have k moments in time, right? Because as n approaches infinity, these intervals become super, super, duper small, right? So these become moments in time. So we're going to have close to an infinite number of moments, and this is the number of successful moments where cars pass."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "The limit as n approaches infinity of n choose k. We're going to have k moments in time, right? Because as n approaches infinity, these intervals become super, super, duper small, right? So these become moments in time. So we're going to have close to an infinite number of moments, and this is the number of successful moments where cars pass. If we have three moments where there was a success where a car passed, then we had a total of three cars pass, right? Or seven cars. Seven moments where it was true that a car passed, then we would have a total of seven cars pass in the hour."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're going to have close to an infinite number of moments, and this is the number of successful moments where cars pass. If we have three moments where there was a success where a car passed, then we had a total of three cars pass, right? Or seven cars. Seven moments where it was true that a car passed, then we would have a total of seven cars pass in the hour. So just finishing up with our binomial distribution, n moments choose k successes times the probability of success. What's the probability of success? We said if this is, so this would be n, what's p equal to?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Seven moments where it was true that a car passed, then we would have a total of seven cars pass in the hour. So just finishing up with our binomial distribution, n moments choose k successes times the probability of success. What's the probability of success? We said if this is, so this would be n, what's p equal to? p is equal to lambda divided by n, right? n times p is lambda, so let me just write that down. p is equal to lambda divided by n. I just rearranged this up here, right?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We said if this is, so this would be n, what's p equal to? p is equal to lambda divided by n, right? n times p is lambda, so let me just write that down. p is equal to lambda divided by n. I just rearranged this up here, right? So our probability of success is lambda times n. And we're saying, what's the probability that we have k successes? And then what's the probability that we have a failure? Well, it's 1 minus the probability of success."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "p is equal to lambda divided by n. I just rearranged this up here, right? So our probability of success is lambda times n. And we're saying, what's the probability that we have k successes? And then what's the probability that we have a failure? Well, it's 1 minus the probability of success. And how many failures are we going to have? How many moments will not have a car pass? Well, we have a total of n moments, and k of them were successes."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's 1 minus the probability of success. And how many failures are we going to have? How many moments will not have a car pass? Well, we have a total of n moments, and k of them were successes. So we'll have n minus k failures. Let's see what we can do with this. So this is equal to, let me rewrite it all, and I'll change colors."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we have a total of n moments, and k of them were successes. So we'll have n minus k failures. Let's see what we can do with this. So this is equal to, let me rewrite it all, and I'll change colors. The limit as n approaches infinity. Let me write out this binomial coefficient. That's n factorial over n minus k factorial times k factorial, normally I write these the other way around, but it's the same thing, times, let's see, lambda to the k, I'm just using my exponent properties, over n to the k. And then this expression right here, I can actually separate out the exponents."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to, let me rewrite it all, and I'll change colors. The limit as n approaches infinity. Let me write out this binomial coefficient. That's n factorial over n minus k factorial times k factorial, normally I write these the other way around, but it's the same thing, times, let's see, lambda to the k, I'm just using my exponent properties, over n to the k. And then this expression right here, I can actually separate out the exponents. This is the same thing as 1 minus lambda over n to the n times 1 minus lambda over n to the minus k. You have the same base, you could add the exponents and you would get this up here. And let me simplify a little bit more. Let me swap spots with these two."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's n factorial over n minus k factorial times k factorial, normally I write these the other way around, but it's the same thing, times, let's see, lambda to the k, I'm just using my exponent properties, over n to the k. And then this expression right here, I can actually separate out the exponents. This is the same thing as 1 minus lambda over n to the n times 1 minus lambda over n to the minus k. You have the same base, you could add the exponents and you would get this up here. And let me simplify a little bit more. Let me swap spots with these two. You can kind of view them both as being in the denominator. So you can change the order of division or multiplication depending how you view it. So this is equal to the limit, let me switch colors, the limit as n approaches infinity, I don't like that color."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me swap spots with these two. You can kind of view them both as being in the denominator. So you can change the order of division or multiplication depending how you view it. So this is equal to the limit, let me switch colors, the limit as n approaches infinity, I don't like that color. Actually, let me just rewrite what we did in the last video. What is this thing right here? And we showed it at the end of the last video."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to the limit, let me switch colors, the limit as n approaches infinity, I don't like that color. Actually, let me just rewrite what we did in the last video. What is this thing right here? And we showed it at the end of the last video. n factorial divided by n minus k factorial. It was n times n minus 1 times n minus 2, all the way to n minus k plus 1. If this was 7 over 7 minus 2 factorial, we would have 7 times 6, right?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we showed it at the end of the last video. n factorial divided by n minus k factorial. It was n times n minus 1 times n minus 2, all the way to n minus k plus 1. If this was 7 over 7 minus 2 factorial, we would have 7 times 6, right? And 6 is 1 more than 7 minus 2. So that's where we got that. And we did that in the last video, if you're getting confused."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "If this was 7 over 7 minus 2 factorial, we would have 7 times 6, right? And 6 is 1 more than 7 minus 2. So that's where we got that. And we did that in the last video, if you're getting confused. And we also said that there's going to be exactly k terms here. So if you counted these, there's 1, 2, 3, all the way there's going to be k terms here. And so we took care of that, we just rewrote that."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we did that in the last video, if you're getting confused. And we also said that there's going to be exactly k terms here. So if you counted these, there's 1, 2, 3, all the way there's going to be k terms here. And so we took care of that, we just rewrote that. And I said I would switch these two things around. So that's divided by n to the k times lambda to the k over k factorial. And then what do we have here?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so we took care of that, we just rewrote that. And I said I would switch these two things around. So that's divided by n to the k times lambda to the k over k factorial. And then what do we have here? I can just rewrite that. This is continuing the same line. 1 minus lambda over n to the n times 1 minus lambda over n to the minus k. Now we can take the limit."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then what do we have here? I can just rewrite that. This is continuing the same line. 1 minus lambda over n to the n times 1 minus lambda over n to the minus k. Now we can take the limit. So what happens when we take the limit? So just so you know, if you take the limit, this is another property, just so you don't get too overwhelmed. Another property of limits, if I take the limit as x approaches anything, a of f of x times g of x, that's equal to the limit as x approaches a of f of x times the limit as x approaches a of g of x."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 minus lambda over n to the n times 1 minus lambda over n to the minus k. Now we can take the limit. So what happens when we take the limit? So just so you know, if you take the limit, this is another property, just so you don't get too overwhelmed. Another property of limits, if I take the limit as x approaches anything, a of f of x times g of x, that's equal to the limit as x approaches a of f of x times the limit as x approaches a of g of x. So we could take each of these limits in the product and then multiply them and then we'll be all set. So let's do that. And I want to leave this stuff up here."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Another property of limits, if I take the limit as x approaches anything, a of f of x times g of x, that's equal to the limit as x approaches a of f of x times the limit as x approaches a of g of x. So we could take each of these limits in the product and then multiply them and then we'll be all set. So let's do that. And I want to leave this stuff up here. So first of all, what's this limit? Let me write this out. And let me pick a good color."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I want to leave this stuff up here. So first of all, what's this limit? Let me write this out. And let me pick a good color. Yellow. So we have the limit as n approaches infinity. So this thing up here, this n times n minus 1 times n minus 2, all the way down to n minus k plus 1, what's it going to look like?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let me pick a good color. Yellow. So we have the limit as n approaches infinity. So this thing up here, this n times n minus 1 times n minus 2, all the way down to n minus k plus 1, what's it going to look like? It's going to be a polynomial, right? We're multiplying a bunch of binomials. We're doing it k times."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this thing up here, this n times n minus 1 times n minus 2, all the way down to n minus k plus 1, what's it going to look like? It's going to be a polynomial, right? We're multiplying a bunch of binomials. We're doing it k times. So the largest degree term is going to be n to the k, right? It's going to be n to the k plus something times n to the k minus 1. It's going to be this big kind of binomial, this big polynomial, kth degree polynomial."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're doing it k times. So the largest degree term is going to be n to the k, right? It's going to be n to the k plus something times n to the k minus 1. It's going to be this big kind of binomial, this big polynomial, kth degree polynomial. And that's really all we need to know for this derivation. So it's going to be n to the k plus blah, blah, blah, blah, blah, blah, blah, a bunch of other stuff. This thing, when you multiply it out."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be this big kind of binomial, this big polynomial, kth degree polynomial. And that's really all we need to know for this derivation. So it's going to be n to the k plus blah, blah, blah, blah, blah, blah, blah, a bunch of other stuff. This thing, when you multiply it out. Over, we have this n to the k, right? So we just, that's this part of it. Times the limit as, well actually, we don't have to worry, this is a constant."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "This thing, when you multiply it out. Over, we have this n to the k, right? So we just, that's this part of it. Times the limit as, well actually, we don't have to worry, this is a constant. So we could actually bring this out front. So we don't even have to write a limit. So times lambda to the k over k factorial, right?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Times the limit as, well actually, we don't have to worry, this is a constant. So we could actually bring this out front. So we don't even have to write a limit. So times lambda to the k over k factorial, right? There's no n here, so this is a constant with respect to n. Times the limit as n approaches infinity of 1 minus lambda over n to the n times 1 minus lambda over n to the minus k. All right. I know you can barely read this. So first of all, what's this limit?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So times lambda to the k over k factorial, right? There's no n here, so this is a constant with respect to n. Times the limit as n approaches infinity of 1 minus lambda over n to the n times 1 minus lambda over n to the minus k. All right. I know you can barely read this. So first of all, what's this limit? The limit as n approaches infinity of some polynomial where it's n to the kth power plus blah, blah, blah, blah, where all of these other terms have a lower degree. This is the highest degree term. So you have n to the k in the numerator and you have n to the k in the denominator."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So first of all, what's this limit? The limit as n approaches infinity of some polynomial where it's n to the kth power plus blah, blah, blah, blah, where all of these other terms have a lower degree. This is the highest degree term. So you have n to the k in the numerator and you have n to the k in the denominator. So the highest degrees are the same, the coefficients are 1. So this limit is 1. Another way you could do it, you could divide the numerator and the denominator by n to the k and you would get, this would be 1 plus 1 over n plus 1 over everything else would have a 1 over n in it."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have n to the k in the numerator and you have n to the k in the denominator. So the highest degrees are the same, the coefficients are 1. So this limit is 1. Another way you could do it, you could divide the numerator and the denominator by n to the k and you would get, this would be 1 plus 1 over n plus 1 over everything else would have a 1 over n in it. And this would just be a 1. And if you take the limit as you approach infinity, then all of these other terms would be 0 and you'd get left with 1 over 1. But either way, you have the same degree in the top and the bottom and their coefficients are the same."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Another way you could do it, you could divide the numerator and the denominator by n to the k and you would get, this would be 1 plus 1 over n plus 1 over everything else would have a 1 over n in it. And this would just be a 1. And if you take the limit as you approach infinity, then all of these other terms would be 0 and you'd get left with 1 over 1. But either way, you have the same degree in the top and the bottom and their coefficients are the same. So the limit as n approaches infinity of this is 1, which is a nice simplification. So you end up with 1 times lambda k over k factorial. Now what's the limit as n approaches infinity of this thing right here?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "But either way, you have the same degree in the top and the bottom and their coefficients are the same. So the limit as n approaches infinity of this is 1, which is a nice simplification. So you end up with 1 times lambda k over k factorial. Now what's the limit as n approaches infinity of this thing right here? 1 minus lambda over n to the n. Well, in the last video we showed that it would be, I'll write it right here, that the limit as n approaches infinity of 1 plus a over n to the n is equal to e to the a, right? That's exactly what we have here, but instead of an a, we have a minus lambda, right? Minus lambda, so this is going to be equal to e to the minus lambda, right?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what's the limit as n approaches infinity of this thing right here? 1 minus lambda over n to the n. Well, in the last video we showed that it would be, I'll write it right here, that the limit as n approaches infinity of 1 plus a over n to the n is equal to e to the a, right? That's exactly what we have here, but instead of an a, we have a minus lambda, right? Minus lambda, so this is going to be equal to e to the minus lambda, right? We have a minus lambda instead of an a. And then finally, what's the limit as n approaches infinity? Let me write it a little bit neater."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Minus lambda, so this is going to be equal to e to the minus lambda, right? We have a minus lambda instead of an a. And then finally, what's the limit as n approaches infinity? Let me write it a little bit neater. I'm just rewriting this term. 1 minus lambda over n to the minus k power. What happens as n approaches infinity?"}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write it a little bit neater. I'm just rewriting this term. 1 minus lambda over n to the minus k power. What happens as n approaches infinity? Well, this term, right, lambda's a constant. As this approaches infinity, this term's going to approach 0, so you have 1 to the minus k. 1 to any power is 1, so that term becomes 1. So we have another 1 there."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "What happens as n approaches infinity? Well, this term, right, lambda's a constant. As this approaches infinity, this term's going to approach 0, so you have 1 to the minus k. 1 to any power is 1, so that term becomes 1. So we have another 1 there. So there you have it. We're done. The probability that our random variable, the number of cars that pass in an hour, is equal to a particular number, you know, it's equal to 7 cars pass in an hour, is equal to the limit as n approaches infinity of n choose k times, well, we said it was lambda over n to the k successes, times 1 minus lambda over n to the n minus k failures."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have another 1 there. So there you have it. We're done. The probability that our random variable, the number of cars that pass in an hour, is equal to a particular number, you know, it's equal to 7 cars pass in an hour, is equal to the limit as n approaches infinity of n choose k times, well, we said it was lambda over n to the k successes, times 1 minus lambda over n to the n minus k failures. And we just showed that this is equal to lambda to the kth power over k factorial times e to the minus lambda. And that's pretty neat, because when you just see it in kind of a vacuum, if you have no context for it, you wouldn't guess that this is in any way related to the binomial theorem. I mean, it's got an e in there."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability that our random variable, the number of cars that pass in an hour, is equal to a particular number, you know, it's equal to 7 cars pass in an hour, is equal to the limit as n approaches infinity of n choose k times, well, we said it was lambda over n to the k successes, times 1 minus lambda over n to the n minus k failures. And we just showed that this is equal to lambda to the kth power over k factorial times e to the minus lambda. And that's pretty neat, because when you just see it in kind of a vacuum, if you have no context for it, you wouldn't guess that this is in any way related to the binomial theorem. I mean, it's got an e in there. It's got a factorial, but you know, a lot of things have factorials in life, so not clear that that would make it a binomial theorem. But this is just the limit as you take smaller and smaller and smaller intervals, and the probability of success in each interval becomes smaller. But as you take the limit, you end up with e. And if you think about it, it makes sense, because one of our derivations of e actually came out of compound interest, and we kind of did something similar there."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "I mean, it's got an e in there. It's got a factorial, but you know, a lot of things have factorials in life, so not clear that that would make it a binomial theorem. But this is just the limit as you take smaller and smaller and smaller intervals, and the probability of success in each interval becomes smaller. But as you take the limit, you end up with e. And if you think about it, it makes sense, because one of our derivations of e actually came out of compound interest, and we kind of did something similar there. We took smaller and smaller intervals of compounding, and over each interval we compounded by a much smaller number, and when you took the limit, you got e again. And that's actually where that whole formula up here came from to begin with. But anyway, just so that you know how to use this thing, so let's say that I were to go out, I'm the traffic engineer, and I figure out that on average, 9 cars pass per hour."}, {"video_title": "Poisson process 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "But as you take the limit, you end up with e. And if you think about it, it makes sense, because one of our derivations of e actually came out of compound interest, and we kind of did something similar there. We took smaller and smaller intervals of compounding, and over each interval we compounded by a much smaller number, and when you took the limit, you got e again. And that's actually where that whole formula up here came from to begin with. But anyway, just so that you know how to use this thing, so let's say that I were to go out, I'm the traffic engineer, and I figure out that on average, 9 cars pass per hour. So I want to know the probability that 2 cars pass in a given hour, exactly 2 cars pass, it's going to be equal to 9 cars per hour to the 2th power, or squared, instead of the 2th power, divided by 2 factorial times e to the minus 9 power. So it's equal to 81 over 2 times e to the minus 9 power. And let's see, maybe I should just get the graphing calculator out there."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "I think now is as good a time as any to play around a little bit with the formula for variance and see where it goes. And I think just by doing this, we'll also get a little bit better intuition of just manipulating sigma notation or even what it means. So we learned several times that the formula for variance, and let's just do variance of a population. It's almost the same thing as variance of a sample. You just divide by n instead of n minus 1. Variance of a population is equal to, when you take each of the data points, x sub i, you subtract from that the mean, you square it, and then you take the average of all of these. So you add the squared distance for each of these points from i equals 1 to i is equal to n, and you divide it by n. So let's see what happens if we can, I don't know, maybe we want to multiply out the squared term and see where it takes us."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's almost the same thing as variance of a sample. You just divide by n instead of n minus 1. Variance of a population is equal to, when you take each of the data points, x sub i, you subtract from that the mean, you square it, and then you take the average of all of these. So you add the squared distance for each of these points from i equals 1 to i is equal to n, and you divide it by n. So let's see what happens if we can, I don't know, maybe we want to multiply out the squared term and see where it takes us. So let's see. And I think it'll take us someplace interesting. So this is the same thing as the sum from i is equal to 1 to n. Let's see, this we just multiply it out."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you add the squared distance for each of these points from i equals 1 to i is equal to n, and you divide it by n. So let's see what happens if we can, I don't know, maybe we want to multiply out the squared term and see where it takes us. So let's see. And I think it'll take us someplace interesting. So this is the same thing as the sum from i is equal to 1 to n. Let's see, this we just multiply it out. This is the same thing as x sub i squared minus, this is your little algebra going on here. So when you square it, I mean, we could multiply it out. We could write it x sub i minus mu times x sub i minus mu, so we have x sub i times x sub i, that's x sub i squared."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the same thing as the sum from i is equal to 1 to n. Let's see, this we just multiply it out. This is the same thing as x sub i squared minus, this is your little algebra going on here. So when you square it, I mean, we could multiply it out. We could write it x sub i minus mu times x sub i minus mu, so we have x sub i times x sub i, that's x sub i squared. Then you have x sub i times mu, times minus mu, and then you have minus u times x sub i. So when you add those two together, you get minus 2 x sub i mu, because you have it twice. x sub i times mu, that's 1 minus x sub i mu."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could write it x sub i minus mu times x sub i minus mu, so we have x sub i times x sub i, that's x sub i squared. Then you have x sub i times mu, times minus mu, and then you have minus u times x sub i. So when you add those two together, you get minus 2 x sub i mu, because you have it twice. x sub i times mu, that's 1 minus x sub i mu. And then you have another one, minus mu x sub i. When you add them together, you get minus 2 x sub i mu. I know it's confusing with me saying sub i and all of that, but it's really no different than when you did a minus b squared, just the variables look a little bit more complicated."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "x sub i times mu, that's 1 minus x sub i mu. And then you have another one, minus mu x sub i. When you add them together, you get minus 2 x sub i mu. I know it's confusing with me saying sub i and all of that, but it's really no different than when you did a minus b squared, just the variables look a little bit more complicated. And then the last term is minus mu times minus mu, which is plus mu squared. Fair enough. Let me switch colors, just to keep it interesting."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "I know it's confusing with me saying sub i and all of that, but it's really no different than when you did a minus b squared, just the variables look a little bit more complicated. And then the last term is minus mu times minus mu, which is plus mu squared. Fair enough. Let me switch colors, just to keep it interesting. Let me cordon that off. OK, so how can we, well, the sum of this is the same thing as the sum of, because you think about it, we're going to take each x sub i, for each of the numbers in our population, we're going to perform this thing, and we're going to sum it up. But if you think about it, this is the same thing as, if you're not familiar with sigmation, this is a good kind of thing to know in general, just a little bit of intuition, that this is the same thing as, I'll do it here to have space, as the sum from i is equal to 1 to n of the first term, x sub i squared minus, and actually we can bring out the constant terms."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me switch colors, just to keep it interesting. Let me cordon that off. OK, so how can we, well, the sum of this is the same thing as the sum of, because you think about it, we're going to take each x sub i, for each of the numbers in our population, we're going to perform this thing, and we're going to sum it up. But if you think about it, this is the same thing as, if you're not familiar with sigmation, this is a good kind of thing to know in general, just a little bit of intuition, that this is the same thing as, I'll do it here to have space, as the sum from i is equal to 1 to n of the first term, x sub i squared minus, and actually we can bring out the constant terms. You just can't take, when you're summing, the only thing that matters is the thing that has the ith term. So in this case, it's x sub i, so x sub 1, x sub 2. So that's the thing that you have to leave on the right hand side of the sigmonotation."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "But if you think about it, this is the same thing as, if you're not familiar with sigmation, this is a good kind of thing to know in general, just a little bit of intuition, that this is the same thing as, I'll do it here to have space, as the sum from i is equal to 1 to n of the first term, x sub i squared minus, and actually we can bring out the constant terms. You just can't take, when you're summing, the only thing that matters is the thing that has the ith term. So in this case, it's x sub i, so x sub 1, x sub 2. So that's the thing that you have to leave on the right hand side of the sigmonotation. And if you've done the calculus playlist already, sigmonotation is really, it's kind of like a discrete integral on some level. Because in an integral, you're summing up a bunch of things, you're multiplying them times dx, which is a really small interval, but here you're just taking a sum. And that's what we showed in the calculus playlist, that an integral actually is kind of this infinite sum of infinitely small things, but I don't want to digress too much, but this was just a long way of saying that the sum from i equals 1 to n of the second term is the same thing as minus 2 times mu of the sum from i is equal to 1 to n of x sub i."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's the thing that you have to leave on the right hand side of the sigmonotation. And if you've done the calculus playlist already, sigmonotation is really, it's kind of like a discrete integral on some level. Because in an integral, you're summing up a bunch of things, you're multiplying them times dx, which is a really small interval, but here you're just taking a sum. And that's what we showed in the calculus playlist, that an integral actually is kind of this infinite sum of infinitely small things, but I don't want to digress too much, but this was just a long way of saying that the sum from i equals 1 to n of the second term is the same thing as minus 2 times mu of the sum from i is equal to 1 to n of x sub i. And then finally, you have plus, well this is just a constant term, right? This is just a constant term, so you can take it out, times mu squared times the sum from i equals 1 to n. And what's going to be here? It's going to be a 1."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's what we showed in the calculus playlist, that an integral actually is kind of this infinite sum of infinitely small things, but I don't want to digress too much, but this was just a long way of saying that the sum from i equals 1 to n of the second term is the same thing as minus 2 times mu of the sum from i is equal to 1 to n of x sub i. And then finally, you have plus, well this is just a constant term, right? This is just a constant term, so you can take it out, times mu squared times the sum from i equals 1 to n. And what's going to be here? It's going to be a 1. We just divided a 1, we just divided this by 1, took it out of the sigma sign, out of the sum, and you're just left with a 1 there. And actually, we could have just left the mu squared there. But either way, let's just keep simplifying it."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be a 1. We just divided a 1, we just divided this by 1, took it out of the sigma sign, out of the sum, and you're just left with a 1 there. And actually, we could have just left the mu squared there. But either way, let's just keep simplifying it. So this we can't really do. Well actually, we could. Well no, we don't know what the x sub i's are, so we just have to leave that the same."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "But either way, let's just keep simplifying it. So this we can't really do. Well actually, we could. Well no, we don't know what the x sub i's are, so we just have to leave that the same. So that's the sum. Oh sorry, this is just the numerator, right? This whole simplification, we're just simplifying the numerator, and later we're just going to divide by n. So that is equal to that divided by n, which is equal to this thing divided by n. I'll divide by n at the end, because it's the numerator that's the confusing part, right?"}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well no, we don't know what the x sub i's are, so we just have to leave that the same. So that's the sum. Oh sorry, this is just the numerator, right? This whole simplification, we're just simplifying the numerator, and later we're just going to divide by n. So that is equal to that divided by n, which is equal to this thing divided by n. I'll divide by n at the end, because it's the numerator that's the confusing part, right? We just want to simplify this term up here. So let's keep doing this. So this is equal to the sum from i equals 1 to n of x sub i squared, and let's see, minus 2 times mu."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "This whole simplification, we're just simplifying the numerator, and later we're just going to divide by n. So that is equal to that divided by n, which is equal to this thing divided by n. I'll divide by n at the end, because it's the numerator that's the confusing part, right? We just want to simplify this term up here. So let's keep doing this. So this is equal to the sum from i equals 1 to n of x sub i squared, and let's see, minus 2 times mu. Sorry, that mu doesn't look good. Edit, undo. Minus 2 times mu times the sum from i is equal to 1 to n of x i, and then what is this?"}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to the sum from i equals 1 to n of x sub i squared, and let's see, minus 2 times mu. Sorry, that mu doesn't look good. Edit, undo. Minus 2 times mu times the sum from i is equal to 1 to n of x i, and then what is this? What is another way to write this, right? Essentially we're going to add 1 to itself n times, right? This is kind of saying just look, whatever you have here, just iterate through it n times."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Minus 2 times mu times the sum from i is equal to 1 to n of x i, and then what is this? What is another way to write this, right? Essentially we're going to add 1 to itself n times, right? This is kind of saying just look, whatever you have here, just iterate through it n times. If you had an x sub i here, you would use the first x term and then the second x term. When you have a 1 here, this is just essentially saying add 1 to itself n times, right? Which is the same thing as n. So this is going to be plus mu squared times n. All right."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is kind of saying just look, whatever you have here, just iterate through it n times. If you had an x sub i here, you would use the first x term and then the second x term. When you have a 1 here, this is just essentially saying add 1 to itself n times, right? Which is the same thing as n. So this is going to be plus mu squared times n. All right. Then let's see if there's anything else we can do here. Remember, this was just the numerator. So this looks fine."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which is the same thing as n. So this is going to be plus mu squared times n. All right. Then let's see if there's anything else we can do here. Remember, this was just the numerator. So this looks fine. We add up each of those terms. We have minus 2 mu, right? From i equals 1 to, oh, well think about this."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this looks fine. We add up each of those terms. We have minus 2 mu, right? From i equals 1 to, oh, well think about this. What is this? What is this thing right here? Well, actually, let's bring back that n. So this simplified to that divided by n, which simplifies to that whole thing, which simplifies to this whole thing divided by n, which simplifies to this whole thing divided by n, which is the same thing as each of the terms divided by n, which is the same thing as that, which is the same thing as that, which is the same thing as that, right?"}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "From i equals 1 to, oh, well think about this. What is this? What is this thing right here? Well, actually, let's bring back that n. So this simplified to that divided by n, which simplifies to that whole thing, which simplifies to this whole thing divided by n, which simplifies to this whole thing divided by n, which is the same thing as each of the terms divided by n, which is the same thing as that, which is the same thing as that, which is the same thing as that, right? And now, how does this simplify is the interesting part. Well, nothing much I can do here. So that just becomes the sum from i is equal to 1 to n times x sub i squared divided by big N. Now, this is interesting."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, actually, let's bring back that n. So this simplified to that divided by n, which simplifies to that whole thing, which simplifies to this whole thing divided by n, which simplifies to this whole thing divided by n, which is the same thing as each of the terms divided by n, which is the same thing as that, which is the same thing as that, which is the same thing as that, right? And now, how does this simplify is the interesting part. Well, nothing much I can do here. So that just becomes the sum from i is equal to 1 to n times x sub i squared divided by big N. Now, this is interesting. What is, if I take each of the terms in my population and I add them up and then I divide it by n, what is that? This thing right here. If I sum up all the terms in my population and divide by the number of terms there are."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that just becomes the sum from i is equal to 1 to n times x sub i squared divided by big N. Now, this is interesting. What is, if I take each of the terms in my population and I add them up and then I divide it by n, what is that? This thing right here. If I sum up all the terms in my population and divide by the number of terms there are. That's the mean, right? That's the mean of my population. So this thing right here is also mu."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I sum up all the terms in my population and divide by the number of terms there are. That's the mean, right? That's the mean of my population. So this thing right here is also mu. So this thing simplifies to what? Minus 2 times what? This whole thing is mu too, so times mu squared."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this thing right here is also mu. So this thing simplifies to what? Minus 2 times what? This whole thing is mu too, so times mu squared. Mu times mu. This is the mean of the population. So that was a nice simplification."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "This whole thing is mu too, so times mu squared. Mu times mu. This is the mean of the population. So that was a nice simplification. And then plus, what do you have here? Let's see. You have mu, you have n over n. Those cancel out."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that was a nice simplification. And then plus, what do you have here? Let's see. You have mu, you have n over n. Those cancel out. So you have plus mu squared. So that was a very nice simplification. And then this simplifies to, can't do much on this side, so the sum from i is equal to 1 to n of x sub i squared over n. And then see, we have minus 2 mu squared plus mu squared."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have mu, you have n over n. Those cancel out. So you have plus mu squared. So that was a very nice simplification. And then this simplifies to, can't do much on this side, so the sum from i is equal to 1 to n of x sub i squared over n. And then see, we have minus 2 mu squared plus mu squared. Well, that's the same thing as minus mu squared. Minus the mean squared. So this, already, we've kind of come up with a neat way of writing the variance."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then this simplifies to, can't do much on this side, so the sum from i is equal to 1 to n of x sub i squared over n. And then see, we have minus 2 mu squared plus mu squared. Well, that's the same thing as minus mu squared. Minus the mean squared. So this, already, we've kind of come up with a neat way of writing the variance. You can essentially take the average of the squares of all of the numbers, in this case a population, and then subtract from that the mean squared of your population. So this could be, depending on how you're calculating things, maybe a slightly faster way of calculating the variance. So just playing with a little algebra we got from this thing, where you have to, each time, take each of your data points, subtract the mean from it, and then square it."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this, already, we've kind of come up with a neat way of writing the variance. You can essentially take the average of the squares of all of the numbers, in this case a population, and then subtract from that the mean squared of your population. So this could be, depending on how you're calculating things, maybe a slightly faster way of calculating the variance. So just playing with a little algebra we got from this thing, where you have to, each time, take each of your data points, subtract the mean from it, and then square it. And then, of course, before you had to do anything, you had to calculate the mean. And you take the square, then you sum it all up, then you take the average, essentially, when you divide it, when you sum and divide it by n. We've simplified it, just using a little bit of algebra, to this formula. And this is, we're getting to something called the raw score method."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just playing with a little algebra we got from this thing, where you have to, each time, take each of your data points, subtract the mean from it, and then square it. And then, of course, before you had to do anything, you had to calculate the mean. And you take the square, then you sum it all up, then you take the average, essentially, when you divide it, when you sum and divide it by n. We've simplified it, just using a little bit of algebra, to this formula. And this is, we're getting to something called the raw score method. What we want to do is write this right here, just in terms of xi's. And then we really are at what you call the raw score method, which is oftentimes a faster way of calculating the variance. So let's see, what is mu equal to?"}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is, we're getting to something called the raw score method. What we want to do is write this right here, just in terms of xi's. And then we really are at what you call the raw score method, which is oftentimes a faster way of calculating the variance. So let's see, what is mu equal to? What is the mean? The mean is just equal to the sum from i is equal to 1 to n of each of the terms. You just take the sum of each of the terms, and you divide by the number of terms there are."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's see, what is mu equal to? What is the mean? The mean is just equal to the sum from i is equal to 1 to n of each of the terms. You just take the sum of each of the terms, and you divide by the number of terms there are. So that is equal to, so if we look at this thing, this thing can be written as, let me draw a line here, this thing can be written as the sum from i is equal to 1 to n of xi squared, all of that over n, minus mu squared. Well, mu is this. So this thing squared."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "You just take the sum of each of the terms, and you divide by the number of terms there are. So that is equal to, so if we look at this thing, this thing can be written as, let me draw a line here, this thing can be written as the sum from i is equal to 1 to n of xi squared, all of that over n, minus mu squared. Well, mu is this. So this thing squared. So this thing squared is what? This is x sub i, take the sum to n, i is equal to 1. You're going to square this thing, and then you're going to divide it by, we squared, right?"}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this thing squared. So this thing squared is what? This is x sub i, take the sum to n, i is equal to 1. You're going to square this thing, and then you're going to divide it by, we squared, right? You divide it by n squared. And this might seem like a more, out of all of them, this is actually seems like the simplest formula for me, where you essentially just take, if you know the mean of your population, you just say, OK, my mean is whatever, and I can just square that, and just put that aside for a second. But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're going to square this thing, and then you're going to divide it by, we squared, right? You divide it by n squared. And this might seem like a more, out of all of them, this is actually seems like the simplest formula for me, where you essentially just take, if you know the mean of your population, you just say, OK, my mean is whatever, and I can just square that, and just put that aside for a second. But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have. I don't know if I wrote, no, I've erased the last set of numbers, but we could show you that you'll get to the same variance. So to me, this is almost the simplest formula. But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have. I don't know if I wrote, no, I've erased the last set of numbers, but we could show you that you'll get to the same variance. So to me, this is almost the simplest formula. But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time. You can just say, OK, for each xi, I just perform this operation, and then I divide by n squared or n accordingly, and I'll also get to the variance. So you don't have to do this calculation before you figure out the whole variance. But anyway, I thought it would be instructive and hopefully give you a little bit more intuition behind the algebra dealing with sigma notation, if we kind of worked out these other ways to write variances."}, {"video_title": "Statistics Alternate variance formulas   Probability and Statistics   Khan Academy.mp3", "Sentence": "But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time. You can just say, OK, for each xi, I just perform this operation, and then I divide by n squared or n accordingly, and I'll also get to the variance. So you don't have to do this calculation before you figure out the whole variance. But anyway, I thought it would be instructive and hopefully give you a little bit more intuition behind the algebra dealing with sigma notation, if we kind of worked out these other ways to write variances. And frankly, some books will just kind of say, oh yeah, you know what, the variance could be written like this, or, and we're talking about the variance of a population, or it could be written like this, or maybe they'll even write it like this. And it's good to know that you can just do a little bit of simple algebraic manipulation and get from one to the other. Anyway, I've run out of time."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "She was curious if this figure was higher in her city, so she tested. Her null hypothesis is that the proportion in her city is the same as all Americans, 26%. Her alternative hypothesis is it's actually greater than 26%, where P represents the proportion of people in her city that can speak more than one language. She found that 40 of 120 people sampled could speak more than one language. So what's going on is, here's the population of her city. She took a sample. Her sample size is 120."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "She found that 40 of 120 people sampled could speak more than one language. So what's going on is, here's the population of her city. She took a sample. Her sample size is 120. And then she calculates her sample proportion, which is 40 out of 120. And this is going to be equal to 1 3rd, which is approximately equal to 0.33. And then she calculates the test statistic for these results was z is approximately equal to 1.83."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "Her sample size is 120. And then she calculates her sample proportion, which is 40 out of 120. And this is going to be equal to 1 3rd, which is approximately equal to 0.33. And then she calculates the test statistic for these results was z is approximately equal to 1.83. We do this in other videos, but just as a reminder of how she gets this, she's really trying to say, well, how many standard deviations above the assumed proportion, remember, when we're doing the significance test, we're assuming that the null hypothesis is true, and then we figure out, well, what's the probability of getting something at least this extreme or more? And then if it's below a threshold, then we would reject the null hypothesis, which would suggest the alternative. But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that?"}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "And then she calculates the test statistic for these results was z is approximately equal to 1.83. We do this in other videos, but just as a reminder of how she gets this, she's really trying to say, well, how many standard deviations above the assumed proportion, remember, when we're doing the significance test, we're assuming that the null hypothesis is true, and then we figure out, well, what's the probability of getting something at least this extreme or more? And then if it's below a threshold, then we would reject the null hypothesis, which would suggest the alternative. But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that? So the z statistic, and we did this in previous videos, you would find the difference between this, what we got for our sample, our sample proportion, and the assumed true proportion, so 0.33 minus 0.26, all of that over the standard deviation of the sampling distribution of the sample proportions. And we've seen that in previous videos. That is just going to be the assumed proportion, so it would be just this."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that? So the z statistic, and we did this in previous videos, you would find the difference between this, what we got for our sample, our sample proportion, and the assumed true proportion, so 0.33 minus 0.26, all of that over the standard deviation of the sampling distribution of the sample proportions. And we've seen that in previous videos. That is just going to be the assumed proportion, so it would be just this. It'd be the assumed population proportion times one minus the assumed population proportion over n. In this particular situation, that would be 0.26 times one minus 0.26, all of that over our n, that's our sample size, 120. And if you calculate this, this should give us approximately 1.83. So they did all of that for us."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "That is just going to be the assumed proportion, so it would be just this. It'd be the assumed population proportion times one minus the assumed population proportion over n. In this particular situation, that would be 0.26 times one minus 0.26, all of that over our n, that's our sample size, 120. And if you calculate this, this should give us approximately 1.83. So they did all of that for us. And they say, assuming that the necessary conditions are met, they're talking about the necessary conditions to assume that the sampling distribution of the sample proportions is roughly normal, and that's the random condition, the normal condition, the independence condition that we have talked about in the past. What is the approximate p-value? Well, this p-value, this is the p-value would be equal to the probability of in a normal distribution, we're assuming that the sampling distribution is normal because we met the necessary conditions."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So they did all of that for us. And they say, assuming that the necessary conditions are met, they're talking about the necessary conditions to assume that the sampling distribution of the sample proportions is roughly normal, and that's the random condition, the normal condition, the independence condition that we have talked about in the past. What is the approximate p-value? Well, this p-value, this is the p-value would be equal to the probability of in a normal distribution, we're assuming that the sampling distribution is normal because we met the necessary conditions. So in a normal distribution, what is the probability of getting a z greater than or equal to 1.83? So to help us visualize this, imagine, let's visualize what the sampling distribution would look like, we're assuming it is roughly normal. The mean of the sampling distribution right over here would be the assumed population proportion."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "Well, this p-value, this is the p-value would be equal to the probability of in a normal distribution, we're assuming that the sampling distribution is normal because we met the necessary conditions. So in a normal distribution, what is the probability of getting a z greater than or equal to 1.83? So to help us visualize this, imagine, let's visualize what the sampling distribution would look like, we're assuming it is roughly normal. The mean of the sampling distribution right over here would be the assumed population proportion. So that would be p-naught, when we put that little zero there, that means the assumed population proportion from the null hypothesis, and that's 0.26. And this result that we got from our sample is 1.83 standard deviations above the mean of the sampling distribution, so 1.83. So that would be 1.83 standard deviations."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "The mean of the sampling distribution right over here would be the assumed population proportion. So that would be p-naught, when we put that little zero there, that means the assumed population proportion from the null hypothesis, and that's 0.26. And this result that we got from our sample is 1.83 standard deviations above the mean of the sampling distribution, so 1.83. So that would be 1.83 standard deviations. And so what we wanna do, this probability is this area under our normal curve right over here. So now let's get our z table. So notice, this z table gives us the area to the left of a certain z value."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So that would be 1.83 standard deviations. And so what we wanna do, this probability is this area under our normal curve right over here. So now let's get our z table. So notice, this z table gives us the area to the left of a certain z value. We wanted it to the right of a certain z value, but a normal distribution is symmetric, so instead of saying anything greater than or equal to 1.83 standard deviations above the mean, we could say anything less than or equal to 1.83 standard deviations below the mean. So this is negative 1.83. And so we could look at that on this z table right over here."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So notice, this z table gives us the area to the left of a certain z value. We wanted it to the right of a certain z value, but a normal distribution is symmetric, so instead of saying anything greater than or equal to 1.83 standard deviations above the mean, we could say anything less than or equal to 1.83 standard deviations below the mean. So this is negative 1.83. And so we could look at that on this z table right over here. Negative one, let me, negative 1.8, negative 1.83 is this right over here. So 0.0336. So there we have it."}, {"video_title": "Calculating a P-value given a z statistic   AP Statistics   Khan Academy.mp3", "Sentence": "And so we could look at that on this z table right over here. Negative one, let me, negative 1.8, negative 1.83 is this right over here. So 0.0336. So there we have it. So this is approximately 0.0336, 0.0336, or a little over 3% or a little less than 4%. And so what Faye would then do is compare that to the significance level that she should have set before conducting this significance test. And so if her significance level was, say, 5%, well then in that situation, since this is lower than that significance level, she would be able to reject the null hypothesis."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's say you're in the babysitting business and you like to keep a log of whom you are babysitting. So in the last month, you babysat six children and you wrote the ages of all six children in your log. But then when you go back to your log, you notice that some blue ink spilled over one of the ages and you forgot how old that child is. And at first you're really worried, your whole system of keeping records seems to, you know, you've lost information. But then you remember that every time you wrote down a new age that month, you recalculated the mean. And so you have the mean here of being four, the mean age is four for the six children. So given that, given that you know the mean and that you know five out of six of the ages, can you figure out what the sixth age is?"}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And at first you're really worried, your whole system of keeping records seems to, you know, you've lost information. But then you remember that every time you wrote down a new age that month, you recalculated the mean. And so you have the mean here of being four, the mean age is four for the six children. So given that, given that you know the mean and that you know five out of six of the ages, can you figure out what the sixth age is? And I encourage you to pause the video and try to figure it out on your own. So assuming you've had a shot at it, so let's just call this missing age, let's call that question mark. So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is?"}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So given that, given that you know the mean and that you know five out of six of the ages, can you figure out what the sixth age is? And I encourage you to pause the video and try to figure it out on your own. So assuming you've had a shot at it, so let's just call this missing age, let's call that question mark. So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? Well, we would take the total, we would take the total of ages, of ages, we would then divide that by the number of children, we would then divide that by the number of ages that we had, and then that would be equal to, that would be equal to the mean. Or another way to think about it, if you multiply both sides times the number of ages, the number of ages on that side and the number of, of ages on that side, then this is going to cancel with that, and we're going to be left with the total, the total is going to be equal to, is going to be equal to the mean times the number of ages. Mean times, and I'll just write times the number, times number of data points, or number of ages."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? Well, we would take the total, we would take the total of ages, of ages, we would then divide that by the number of children, we would then divide that by the number of ages that we had, and then that would be equal to, that would be equal to the mean. Or another way to think about it, if you multiply both sides times the number of ages, the number of ages on that side and the number of, of ages on that side, then this is going to cancel with that, and we're going to be left with the total, the total is going to be equal to, is going to be equal to the mean times the number of ages. Mean times, and I'll just write times the number, times number of data points, or number of ages. So maybe we can use this information, because we're just going to have this missing question mark here, and we know the mean and we know the number of ages, so we just have to solve for the question mark, so let's do that. So let's go back to the beginning here, just so that this makes sense with some numbers. The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two, plus four, plus eight."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Mean times, and I'll just write times the number, times number of data points, or number of ages. So maybe we can use this information, because we're just going to have this missing question mark here, and we know the mean and we know the number of ages, so we just have to solve for the question mark, so let's do that. So let's go back to the beginning here, just so that this makes sense with some numbers. The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two, plus four, plus eight. We're going to divide by the number of ages. We're going to divide it by the number of ages. Well, we have six ages here."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two, plus four, plus eight. We're going to divide by the number of ages. We're going to divide it by the number of ages. Well, we have six ages here. One, two, three, four, five, six. Six ages, and that's going to be equal to the mean. This is going to be equal to the mean."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, we have six ages here. One, two, three, four, five, six. Six ages, and that's going to be equal to the mean. This is going to be equal to the mean. The mean here is four. This is just how you calculate the mean. Let's see if we can simplify this."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is going to be equal to the mean. The mean here is four. This is just how you calculate the mean. Let's see if we can simplify this. Five plus two is seven. Let me do this, that's the wrong color. Five plus two is seven."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's see if we can simplify this. Five plus two is seven. Let me do this, that's the wrong color. Five plus two is seven. Two plus four is six, plus eight is 14. 14, and then seven plus 14 is 21. We're left with 21 plus question mark over six is equal to four."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Five plus two is seven. Two plus four is six, plus eight is 14. 14, and then seven plus 14 is 21. We're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have. We can multiply both sides times six."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have. We can multiply both sides times six. We can multiply both sides, both sides times six. Six on that side, six on this side. Six in the numerator, six in the denominator, those cancel."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We can multiply both sides times six. We can multiply both sides, both sides times six. Six on that side, six on this side. Six in the numerator, six in the denominator, those cancel. All we're left is, on the left-hand side, we're left with 21 plus question mark. All of these other green numbers, those are simplified, five plus two plus two plus four plus eight is 21, and we still have the question mark. We get 21 plus question mark."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Six in the numerator, six in the denominator, those cancel. All we're left is, on the left-hand side, we're left with 21 plus question mark. All of these other green numbers, those are simplified, five plus two plus two plus four plus eight is 21, and we still have the question mark. We get 21 plus question mark. I'm going to do that green color. 21 plus this question mark. The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We get 21 plus question mark. I'm going to do that green color. 21 plus this question mark. The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six. What's four times six? That's 24. What's the question mark?"}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six. What's four times six? That's 24. What's the question mark? 21 plus what is equal to 24? We can, of course, you might just, well, it's going to be three, or if you want to, you could say, well, question mark is going to be, question mark is going to be equal to, is going to be equal to 24 minus 21, which is, of course, three. Which, of course, let me just write this down."}, {"video_title": "How to find a missing value given the mean   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "What's the question mark? 21 plus what is equal to 24? We can, of course, you might just, well, it's going to be three, or if you want to, you could say, well, question mark is going to be, question mark is going to be equal to, is going to be equal to 24 minus 21, which is, of course, three. Which, of course, let me just write this down. The question mark is equal to three. The missing age, you were able to figure it out based on the information you had, because you had the mean, you were able to figure out that behind the splotch, that behind the splotch, you had a three. It's exciting."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "Who knows what that's in some blood pressure units. Construct a 95% confidence interval for the true expected blood pressure increase for all patients in a population. So there's some population distribution here. It's a reasonable assumption to think that it is normal. It's a biological process. So if you gave this drug to every person who has ever lived, that will result in some mean increase in blood pressure. Or who knows, maybe it's actually a decrease."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's a reasonable assumption to think that it is normal. It's a biological process. So if you gave this drug to every person who has ever lived, that will result in some mean increase in blood pressure. Or who knows, maybe it's actually a decrease. And there's also going to be some standard deviation here. It is a normal distribution. And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or who knows, maybe it's actually a decrease. And there's also going to be some standard deviation here. It is a normal distribution. And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. It's going to be the sum of many thousands and millions of random events. And things that are sums of many millions and thousands of random events tend to be normal distribution. So this is a population distribution."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. It's going to be the sum of many thousands and millions of random events. And things that are sums of many millions and thousands of random events tend to be normal distribution. So this is a population distribution. This is the population distribution. And we don't know anything really about it outside of the sample that we have here. Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is a population distribution. This is the population distribution. And we don't know anything really about it outside of the sample that we have here. Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go. So we have our seven data points. And you can add them up and divide by 7 and get your sample mean. So our sample mean here is 2.34."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go. So we have our seven data points. And you can add them up and divide by 7 and get your sample mean. So our sample mean here is 2.34. And then you can also calculate your sample standard deviation. Find the square distance from each of these points to your sample mean, add them up, divide by n minus 1 because it's a sample, then take the square root, and you get your sample standard deviation. And I did this ahead of time just to save time."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our sample mean here is 2.34. And then you can also calculate your sample standard deviation. Find the square distance from each of these points to your sample mean, add them up, divide by n minus 1 because it's a sample, then take the square root, and you get your sample standard deviation. And I did this ahead of time just to save time. Sample standard deviation is 1.04. And we don't know anything about the population distribution. The thing that we've been doing from the get go is estimating that character with our sample standard deviation."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I did this ahead of time just to save time. Sample standard deviation is 1.04. And we don't know anything about the population distribution. The thing that we've been doing from the get go is estimating that character with our sample standard deviation. So we've been estimating the true standard deviation of the population with our sample standard deviation. Now, in this problem, this exact problem, we're going to run into a problem. We're estimating our standard deviation with an n of only 7."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "The thing that we've been doing from the get go is estimating that character with our sample standard deviation. So we've been estimating the true standard deviation of the population with our sample standard deviation. Now, in this problem, this exact problem, we're going to run into a problem. We're estimating our standard deviation with an n of only 7. So this is probably going to be a not so good estimate. Let me just write, because n is small. In general, this is considered a bad estimate if n is less than 30."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're estimating our standard deviation with an n of only 7. So this is probably going to be a not so good estimate. Let me just write, because n is small. In general, this is considered a bad estimate if n is less than 30. Above 30, you're dealing in the realm of pretty good estimates. And so the whole focus of this video is when we think about the sampling distribution, which is what we're going to use to generate our interval, instead of assuming that the sampling distribution is normal, like we did in many other videos using the central limit theorem and all of that, we're going to tweak the sampling distribution. We're not going to assume it's a normal distribution, because this is a bad estimate."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "In general, this is considered a bad estimate if n is less than 30. Above 30, you're dealing in the realm of pretty good estimates. And so the whole focus of this video is when we think about the sampling distribution, which is what we're going to use to generate our interval, instead of assuming that the sampling distribution is normal, like we did in many other videos using the central limit theorem and all of that, we're going to tweak the sampling distribution. We're not going to assume it's a normal distribution, because this is a bad estimate. We're going to assume that it's something called a t distribution. And the t distribution is essentially, the best way to think about it is it's almost engineered. It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not going to assume it's a normal distribution, because this is a bad estimate. We're going to assume that it's something called a t distribution. And the t distribution is essentially, the best way to think about it is it's almost engineered. It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size. And it looks very similar to a normal distribution. It has some mean. So this is your mean of your sampling distribution still."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size. And it looks very similar to a normal distribution. It has some mean. So this is your mean of your sampling distribution still. But it also has fatter tails. And the way I think about why it has fatter tails is when you make an assumption that this is the standard deviation for, well, let me take one more step. So normally what we do is we find the estimate of the true standard deviation."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is your mean of your sampling distribution still. But it also has fatter tails. And the way I think about why it has fatter tails is when you make an assumption that this is the standard deviation for, well, let me take one more step. So normally what we do is we find the estimate of the true standard deviation. And then we say that the standard deviation of the sampling distribution is equal to the true standard deviation of our population divided by the square root of n. In this case, n is equal to 7. And then we say, OK, we never know the true standard, or we seldom know. Sometimes you do know."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So normally what we do is we find the estimate of the true standard deviation. And then we say that the standard deviation of the sampling distribution is equal to the true standard deviation of our population divided by the square root of n. In this case, n is equal to 7. And then we say, OK, we never know the true standard, or we seldom know. Sometimes you do know. We seldom know the true standard deviation. So if we don't know that, the best thing we can put in there is our sample standard deviation. So the best thing we can put in there is our sample standard deviation."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sometimes you do know. We seldom know the true standard deviation. So if we don't know that, the best thing we can put in there is our sample standard deviation. So the best thing we can put in there is our sample standard deviation. And this right here, this is the whole reason why we don't say that this is just a 95 probability interval. This is the whole reason why we call it a confidence interval, because we're making some assumptions here. This thing is going to change from sample to sample."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the best thing we can put in there is our sample standard deviation. And this right here, this is the whole reason why we don't say that this is just a 95 probability interval. This is the whole reason why we call it a confidence interval, because we're making some assumptions here. This thing is going to change from sample to sample. And in particular, this is going to be a particularly bad estimate when we have a small sample size, a size less than 30. So when you are estimating the standard deviation where you don't know it, you're estimating it with your sample standard deviation. And your sample size is small."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "This thing is going to change from sample to sample. And in particular, this is going to be a particularly bad estimate when we have a small sample size, a size less than 30. So when you are estimating the standard deviation where you don't know it, you're estimating it with your sample standard deviation. And your sample size is small. And you're going to use this to estimate the standard deviation of your sampling distribution. You don't assume your sampling distribution is a normal distribution. You assume it has fatter tails."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And your sample size is small. And you're going to use this to estimate the standard deviation of your sampling distribution. You don't assume your sampling distribution is a normal distribution. You assume it has fatter tails. And it has fatter tails, because you're essentially underestimating the standard deviation over here. Anyway, with all of that said, let's just actually go through this problem. So we need to think about a 95% confidence interval around this mean right over here."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "You assume it has fatter tails. And it has fatter tails, because you're essentially underestimating the standard deviation over here. Anyway, with all of that said, let's just actually go through this problem. So we need to think about a 95% confidence interval around this mean right over here. So a 95% confidence interval, if this was a normal distribution, you would just look it up in a z table. But it's not. This is a t distribution."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we need to think about a 95% confidence interval around this mean right over here. So a 95% confidence interval, if this was a normal distribution, you would just look it up in a z table. But it's not. This is a t distribution. This is a t distribution. We're looking for a 95% confidence interval. So some interval around the mean that encapsulates 95% of the area."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is a t distribution. This is a t distribution. We're looking for a 95% confidence interval. So some interval around the mean that encapsulates 95% of the area. For t distribution, you use a t table. And I have a t table ahead of time right over here. And what you want to do is use the two-sided."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So some interval around the mean that encapsulates 95% of the area. For t distribution, you use a t table. And I have a t table ahead of time right over here. And what you want to do is use the two-sided. You want to use a two-sided row for what we're doing right over here. And the best way to think about it is that we're symmetric around the mean. And that's why they call it two-sided."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what you want to do is use the two-sided. You want to use a two-sided row for what we're doing right over here. And the best way to think about it is that we're symmetric around the mean. And that's why they call it two-sided. One-sided if it was kind of a cumulative percentage up to some critical threshold. But in this case, it's two-sided. We're symmetric."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's why they call it two-sided. One-sided if it was kind of a cumulative percentage up to some critical threshold. But in this case, it's two-sided. We're symmetric. Or another way to think about it is we're excluding the two sides. So we want the 95% in the middle. And this is a sampling distribution of the sample mean for n is equal to 7."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're symmetric. Or another way to think about it is we're excluding the two sides. So we want the 95% in the middle. And this is a sampling distribution of the sample mean for n is equal to 7. And I won't go into the details here. But when n is equal to 7, you have 6 degrees of freedom, or n minus 1. And the way that t tables are set up, you go and find the degrees of freedom."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is a sampling distribution of the sample mean for n is equal to 7. And I won't go into the details here. But when n is equal to 7, you have 6 degrees of freedom, or n minus 1. And the way that t tables are set up, you go and find the degrees of freedom. So you don't go to the n. You go to the n minus 1. So you go to the 6 right here. So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the way that t tables are set up, you go and find the degrees of freedom. So you don't go to the n. You go to the n minus 1. So you go to the 6 right here. So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. And this t table assumes that you are approximating that standard deviation using your sample standard deviation. So it's another way to think of it. You have to go 2.447 of these approximated standard deviations."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. And this t table assumes that you are approximating that standard deviation using your sample standard deviation. So it's another way to think of it. You have to go 2.447 of these approximated standard deviations. So let me go right here. So you have to go 2.447. This distance right here is 2.447 times this approximated standard deviation."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have to go 2.447 of these approximated standard deviations. So let me go right here. So you have to go 2.447. This distance right here is 2.447 times this approximated standard deviation. And sometimes, you'll see this in some statistics books, this thing right here, this exact number, is shown like this. They put a little hat on top of the standard deviation to show that it has been approximated using the sample standard deviation. So we'll put a little hat over here, because frankly, this is the only thing that we can calculate."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "This distance right here is 2.447 times this approximated standard deviation. And sometimes, you'll see this in some statistics books, this thing right here, this exact number, is shown like this. They put a little hat on top of the standard deviation to show that it has been approximated using the sample standard deviation. So we'll put a little hat over here, because frankly, this is the only thing that we can calculate. So this is how far you have to go in each direction. And we know what this value is. We know what the sample distribution is."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we'll put a little hat over here, because frankly, this is the only thing that we can calculate. So this is how far you have to go in each direction. And we know what this value is. We know what the sample distribution is. So let's get our calculator out. So we know our sample standard deviation is 1.04. And we want to divide that by the square root of 7."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know what the sample distribution is. So let's get our calculator out. So we know our sample standard deviation is 1.04. And we want to divide that by the square root of 7. So we get 0.39. So this right here is 0.39. And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we want to divide that by the square root of 7. So we get 0.39. So this right here is 0.39. And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. So let's do that. So times 2.447 is equal to 0.96. So this distance right here is 0.96."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. So let's do that. So times 2.447 is equal to 0.96. So this distance right here is 0.96. And then this distance right here is 0.96. So if you take a random sample, and that's exactly what we did when we found these seven samples. When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this distance right here is 0.96. And then this distance right here is 0.96. So if you take a random sample, and that's exactly what we did when we found these seven samples. When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. And so the probability, so we can view it. We can say that there's a 95% chance. And we have to actually caveat everything with a confident, because we're doing all of these estimations here."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. And so the probability, so we can view it. We can say that there's a 95% chance. And we have to actually caveat everything with a confident, because we're doing all of these estimations here. So it's not a true, precise 95% chance. We're just confident that there's a 95% chance that our random sampling mean right here, so that 2.34, which we can kind of use. We just pick that 2.34 from this distribution right here."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we have to actually caveat everything with a confident, because we're doing all of these estimations here. So it's not a true, precise 95% chance. We're just confident that there's a 95% chance that our random sampling mean right here, so that 2.34, which we can kind of use. We just pick that 2.34 from this distribution right here. So there's a 95% chance that 2.34 is within 0.96 of the true sampling distribution mean, which we know is also the same thing as the population mean. So I'll just say of the population mean. Or we can just rearrange the sentence and say that there is a 95% chance that the mean, the true mean, which is the same thing as a sampling distribution mean, is within 0.96 of our sample mean of 2.34."}, {"video_title": "Small sample size confidence intervals   Probability and Statistics   Khan Academy.mp3", "Sentence": "We just pick that 2.34 from this distribution right here. So there's a 95% chance that 2.34 is within 0.96 of the true sampling distribution mean, which we know is also the same thing as the population mean. So I'll just say of the population mean. Or we can just rearrange the sentence and say that there is a 95% chance that the mean, the true mean, which is the same thing as a sampling distribution mean, is within 0.96 of our sample mean of 2.34. So at the low end, so if you go 2.34 minus 0.96, that's the low end of our confidence interval, 1.38. And the high end of our confidence interval, 2.34 plus 0.96 is equal to 3.3. So our 95% confidence interval is from 1.38 to 3.3."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So there's four suits, each of them have nine cards, so that gives us 36 unique cards. A hand is a collection of nine cards, which can be sorted however the player chooses. So they're essentially telling us that order does not matter. What is the probability of getting all four of the ones? So they want to know the probability of getting all four of the ones. So all four ones in my hand of nine. Now this is kind of daunting at first."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "What is the probability of getting all four of the ones? So they want to know the probability of getting all four of the ones. So all four ones in my hand of nine. Now this is kind of daunting at first. You're like, gee, I have nine cards and I'm kicking them out of 36. I have to figure out how do I get all of the ones. But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now this is kind of daunting at first. You're like, gee, I have nine cards and I'm kicking them out of 36. I have to figure out how do I get all of the ones. But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens. So this is what the definition of the probability is. It's going to be the number of ways in which event can happen, and when we talk about the event, we're talking about having all four ones in my hand. That's the event."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens. So this is what the definition of the probability is. It's going to be the number of ways in which event can happen, and when we talk about the event, we're talking about having all four ones in my hand. That's the event. And all of these different ways, that's sometimes called the event space. But we actually want to count how many ways that if I get a hand of nine, picking from 36, that I can get the four ones in it, so this is the number of ways in which my event can happen. And we want to divide that into all of the possibilities."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "That's the event. And all of these different ways, that's sometimes called the event space. But we actually want to count how many ways that if I get a hand of nine, picking from 36, that I can get the four ones in it, so this is the number of ways in which my event can happen. And we want to divide that into all of the possibilities. Or maybe I should write it this way. The total number of hands that I can get. So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And we want to divide that into all of the possibilities. Or maybe I should write it this way. The total number of hands that I can get. So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. Now let's figure out the total number of hands first, because at some level this might be more intuitive, and we've actually done this before. Now, the total number of hands, we're picking nine cards, and we're picking them from a set of 36 unique cards. And we've done this many, many times."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. Now let's figure out the total number of hands first, because at some level this might be more intuitive, and we've actually done this before. Now, the total number of hands, we're picking nine cards, and we're picking them from a set of 36 unique cards. And we've done this many, many times. Let me write this. Total number of hands, or total number of possible hands. That's equal to, you can imagine you have nine cards to pick from."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And we've done this many, many times. Let me write this. Total number of hands, or total number of possible hands. That's equal to, you can imagine you have nine cards to pick from. The first card you pick is going to be one of 36 cards, then the next one's going to be one of 35, then the next one's going to be one of 34, 33, 32, 31. We're going to do this nine times, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So that would be the total number of hands if order mattered."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "That's equal to, you can imagine you have nine cards to pick from. The first card you pick is going to be one of 36 cards, then the next one's going to be one of 35, then the next one's going to be one of 34, 33, 32, 31. We're going to do this nine times, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So that would be the total number of hands if order mattered. But we know, and we've gone over this before, that we don't care about the order. All we care about the actual cards that are in there. So we're over counting here."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So that would be the total number of hands if order mattered. But we know, and we've gone over this before, that we don't care about the order. All we care about the actual cards that are in there. So we're over counting here. We're over counting for all of the different rearrangements that these cards would have. It doesn't matter whether the ace of diamonds is the first card I pick or the last card I pick. The way I've counted them right now, we are counting those as two separate hands."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So we're over counting here. We're over counting for all of the different rearrangements that these cards would have. It doesn't matter whether the ace of diamonds is the first card I pick or the last card I pick. The way I've counted them right now, we are counting those as two separate hands. But they aren't two separate hands, so order doesn't matter. So what we have to do is, we have to divide this by the number of ways you can arrange nine things. So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "The way I've counted them right now, we are counting those as two separate hands. But they aren't two separate hands, so order doesn't matter. So what we have to do is, we have to divide this by the number of ways you can arrange nine things. So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on. It essentially becomes 9 factorial, times 2, times 1. And we've seen this multiple times. This is essentially 36 choose 9."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on. It essentially becomes 9 factorial, times 2, times 1. And we've seen this multiple times. This is essentially 36 choose 9. This expression right here is the same thing, just so you can relate it to the, I guess, combinatorics formulas that you might be familiar with. This is the same thing as 36 factorial over 36 minus 9 factorial, that's what this orange part is over here, divided by 9 factorial, or over 9 factorial. What's green is what's green, and what is orange is what's orange there."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "This is essentially 36 choose 9. This expression right here is the same thing, just so you can relate it to the, I guess, combinatorics formulas that you might be familiar with. This is the same thing as 36 factorial over 36 minus 9 factorial, that's what this orange part is over here, divided by 9 factorial, or over 9 factorial. What's green is what's green, and what is orange is what's orange there. So that's the total number of hands. Now, a little bit more of a nuanced thought process is, how do we figure out the number of ways in which the event can happen, in which we can have all four 1's? So let's figure that out."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "What's green is what's green, and what is orange is what's orange there. So that's the total number of hands. Now, a little bit more of a nuanced thought process is, how do we figure out the number of ways in which the event can happen, in which we can have all four 1's? So let's figure that out. So number of ways, or maybe we should say this, number of hands with four 1's. And just as a little bit of thought experiment, imagine if we were only taking four cards, if a hand only had four cards in it. Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let's figure that out. So number of ways, or maybe we should say this, number of hands with four 1's. And just as a little bit of thought experiment, imagine if we were only taking four cards, if a hand only had four cards in it. Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination. You just have four 1's. That's the only combination with four 1's if we were only picking four cards. But here, we're not only picking four cards."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination. You just have four 1's. That's the only combination with four 1's if we were only picking four cards. But here, we're not only picking four cards. Four of the cards are going to be 1's. So four of the cards are going to be 1's. I mean, 1, 2, 3, 4, but the other five cards are going to be different."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "But here, we're not only picking four cards. Four of the cards are going to be 1's. So four of the cards are going to be 1's. I mean, 1, 2, 3, 4, but the other five cards are going to be different. So 1, 2, 3, 4, 5. So for the other five cards, if you imagine this slot, considering that of the 36, we would have to pick four of them already in order for us to have four 1's. So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "I mean, 1, 2, 3, 4, but the other five cards are going to be different. So 1, 2, 3, 4, 5. So for the other five cards, if you imagine this slot, considering that of the 36, we would have to pick four of them already in order for us to have four 1's. So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand. And then there'd be 31 in that position of the hand. And then there'd be 30, because every time we're picking a card, we're using it up. And now we only have 30 to pick from."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand. And then there'd be 31 in that position of the hand. And then there'd be 30, because every time we're picking a card, we're using it up. And now we only have 30 to pick from. Then we only have 29 to pick from. And then we have 28 to pick from. And just like we did before, we don't care about order."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And now we only have 30 to pick from. Then we only have 29 to pick from. And then we have 28 to pick from. And just like we did before, we don't care about order. We don't care if we pick the five of clubs first or whether we pick the five of clubs last. So we shouldn't double count it. So we have to divide by the different number of ways that five cards can be arranged."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And just like we did before, we don't care about order. We don't care if we pick the five of clubs first or whether we pick the five of clubs last. So we shouldn't double count it. So we have to divide by the different number of ways that five cards can be arranged. So we have to divide this by the different ways that five cards can be arranged. So the first card or the first position could be any one of five cards, then four cards, then three cards, then two cards, then one cards. So the number of hands with four 1's is actually just this number."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So we have to divide by the different number of ways that five cards can be arranged. So we have to divide this by the different ways that five cards can be arranged. So the first card or the first position could be any one of five cards, then four cards, then three cards, then two cards, then one cards. So the number of hands with four 1's is actually just this number. You're actually looking at all of the different ways you can fill up the remaining cards. These four 1's are just going to be four 1's. There's only one way to get that."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So the number of hands with four 1's is actually just this number. You're actually looking at all of the different ways you can fill up the remaining cards. These four 1's are just going to be four 1's. There's only one way to get that. It's the remaining cards that's going to give all of the different combinations of having four 1's. So this will be a count of all of the different combinations. Because all of the different extra stuff that you have will be all of the different hands."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "There's only one way to get that. It's the remaining cards that's going to give all of the different combinations of having four 1's. So this will be a count of all of the different combinations. Because all of the different extra stuff that you have will be all of the different hands. Now, we know the total number of hands with four 1's. It's this number. And now we can divide it by the total number of possible hands."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Because all of the different extra stuff that you have will be all of the different hands. Now, we know the total number of hands with four 1's. It's this number. And now we can divide it by the total number of possible hands. And I didn't multiply them out on purpose so that we can cancel things out. So let's do that. Let's take this and divide by that."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And now we can divide it by the total number of possible hands. And I didn't multiply them out on purpose so that we can cancel things out. So let's do that. Let's take this and divide by that. So let me just copy and paste it. Let me take that. Let me copy it and let me paste it."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let's take this and divide by that. So let me just copy and paste it. Let me take that. Let me copy it and let me paste it. Let's take that and let's divide it by that. But dividing by a fraction is the same thing as multiplying by the reciprocal. So let's just multiply by the reciprocal."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let me copy it and let me paste it. Let's take that and let's divide it by that. But dividing by a fraction is the same thing as multiplying by the reciprocal. So let's just multiply by the reciprocal. So let's multiply. So this is the denominator. Let's make this the numerator."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let's just multiply by the reciprocal. So let's multiply. So this is the denominator. Let's make this the numerator. So let me copy it and then let me paste it. So that's the numerator. And then that's the denominator up there."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let's make this the numerator. So let me copy it and then let me paste it. So that's the numerator. And then that's the denominator up there. Because we're dividing by that expression. So let me put that there. Let me get the Select tool."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then that's the denominator up there. Because we're dividing by that expression. So let me put that there. Let me get the Select tool. And then let me make sure I'm selecting all of the numbers. Let me copy it and then let me paste that. It's a little messy with those lines there, but I think this will suit our purposes."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let me get the Select tool. And then let me make sure I'm selecting all of the numbers. Let me copy it and then let me paste that. It's a little messy with those lines there, but I think this will suit our purposes. So when we're multiplying by this, we're essentially dividing by this expression up here. Now, this we can simplify pretty easily. We have a, well actually I forgot to do, this should be 9 factorial."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "It's a little messy with those lines there, but I think this will suit our purposes. So when we're multiplying by this, we're essentially dividing by this expression up here. Now, this we can simplify pretty easily. We have a, well actually I forgot to do, this should be 9 factorial. This should be 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. So let me put that in both places. Actually, let me just clear that both places."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We have a, well actually I forgot to do, this should be 9 factorial. This should be 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. So let me put that in both places. Actually, let me just clear that both places. I'm sorry if that confused you when I wrote it earlier. This would be 9 factorial 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Let me copy and paste that now."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Actually, let me just clear that both places. I'm sorry if that confused you when I wrote it earlier. This would be 9 factorial 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Let me copy and paste that now. Copy and then you paste it. That's that right there. And then we have this in the numerator."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let me copy and paste that now. Copy and then you paste it. That's that right there. And then we have this in the numerator. We have 5 times 4 times 3 times 1 in the denominator. So this will cancel out with that part right over there. And then we have 32 times 31 times 30 times 29 times 28."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then we have this in the numerator. We have 5 times 4 times 3 times 1 in the denominator. So this will cancel out with that part right over there. And then we have 32 times 31 times 30 times 29 times 28. That is going to cancel with that. That and that cancels out. So what we're left with is just this part over here."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then we have 32 times 31 times 30 times 29 times 28. That is going to cancel with that. That and that cancels out. So what we're left with is just this part over here. Let me rewrite it. So we're left with 9 times 8 times 7 times 6 over, and this will just be an exercise in simplifying this expression, 36 times 35 times 34 times 33. And let's see."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So what we're left with is just this part over here. Let me rewrite it. So we're left with 9 times 8 times 7 times 6 over, and this will just be an exercise in simplifying this expression, 36 times 35 times 34 times 33. And let's see. If we divide the numerator and denominator by 9, that becomes a 1. This becomes a 4. You can divide the numerator and denominator by 4."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And let's see. If we divide the numerator and denominator by 9, that becomes a 1. This becomes a 4. You can divide the numerator and denominator by 4. This becomes a 2. This becomes a 1. You divide numerator and denominator by 7."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "You can divide the numerator and denominator by 4. This becomes a 2. This becomes a 1. You divide numerator and denominator by 7. This becomes a 1. This becomes a 5. You can divide both by 2 again."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "You divide numerator and denominator by 7. This becomes a 1. This becomes a 5. You can divide both by 2 again. And then this becomes a 1. This becomes a 17. And you could divide this and this by 3."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "You can divide both by 2 again. And then this becomes a 1. This becomes a 17. And you could divide this and this by 3. This becomes a 2. And then this becomes an 11. So we're left with the probability of having all 4 1's in my hand of 9 that I'm selecting from 36 unique cards is equal to, and the numerator I'm just left with this 2, times 1 times 1 times 1."}, {"video_title": "Example Combinatorics and probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And you could divide this and this by 3. This becomes a 2. And then this becomes an 11. So we're left with the probability of having all 4 1's in my hand of 9 that I'm selecting from 36 unique cards is equal to, and the numerator I'm just left with this 2, times 1 times 1 times 1. So it's equal to 2 over 5 times 17 times 11. And that is, so drum roll, this was kind of an involved problem, 5 times 17 times 11 is equal to 935. So it's equal to 2 over 935."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "She recorded the height in centimeters of each customer and the frame size in centimeters of the bicycle that customer rented. After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer, and this is the equation. So before I even look at this question, let's just think about what she did. So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted?"}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one."}, {"video_title": "Calculating residual example   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to zoom in right over here, you can't see it that well, but let me draw, so if we zoom in, let's say we were to zoom in the line, and it looks like this, and our data point is right, our data point is right over here. We know we're below the line, and this is going to be a negative residual, and the magnitude of that residual is how far we are below the line. And in this case, it is negative one."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "We are told that a conservation group with a long-term goal of preserving species believes that all at-risk species will disappear when land inhabited by those species is developed. It has an opportunity to purchase land in an area about to be developed. The group has a choice of creating one large nature preserve with an area of 45 square kilometers and containing 70 at-risk species, or five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species unique to that preserve. Which choice would you recommend and why? And there's some interesting data here. Here it looks like some data they have gathered for different islands, and we have their areas, and then this is the number of species at risk in 1990, and then the species extinct by 2000. And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "Which choice would you recommend and why? And there's some interesting data here. Here it looks like some data they have gathered for different islands, and we have their areas, and then this is the number of species at risk in 1990, and then the species extinct by 2000. And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area. It's the natural log of the area."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area. It's the natural log of the area. And why did they do this? Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "It's the natural log of the area. And why did they do this? Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our tools of linear regression to think about what would be the proportion extinct for the 45 square kilometers versus for the five small three-kilometer islands. So pause this video and see if you can figure it out on your own. And they give us the regression data for a line that fits this data."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our tools of linear regression to think about what would be the proportion extinct for the 45 square kilometers versus for the five small three-kilometer islands. So pause this video and see if you can figure it out on your own. And they give us the regression data for a line that fits this data. All right, now let's work through it together and to make some space because all of it is already plotted right over here and we have our regression data. So the regression line, we know it's slope and y-intercept. The y-intercept is right over here, 0.28996."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And they give us the regression data for a line that fits this data. All right, now let's work through it together and to make some space because all of it is already plotted right over here and we have our regression data. So the regression line, we know it's slope and y-intercept. The y-intercept is right over here, 0.28996. So 0.2, this is, let's see, one, two, three, four, five, so 28996. It's almost two nine. So it's gonna be right over here would be the y-intercept."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "The y-intercept is right over here, 0.28996. So 0.2, this is, let's see, one, two, three, four, five, so 28996. It's almost two nine. So it's gonna be right over here would be the y-intercept. And its slope is negative 0.05 approximately. And I could eyeball it. It probably is going to look something like this."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So it's gonna be right over here would be the y-intercept. And its slope is negative 0.05 approximately. And I could eyeball it. It probably is going to look something like this. That's the regression line. Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323. And we have to be careful here."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "It probably is going to look something like this. That's the regression line. Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323. And we have to be careful here. You might be tempted to say times the area, but no, the horizontal axis here is the natural log of the area, times the natural log of the area. And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation. And then how many actual species will get extinct."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And we have to be careful here. You might be tempted to say times the area, but no, the horizontal axis here is the natural log of the area, times the natural log of the area. And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation. And then how many actual species will get extinct. And then the one that maybe has fewer species that get extinct is maybe the best one. Or the one that the more that we can preserve is maybe the best one. And so let's look at the two scenarios."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And then how many actual species will get extinct. And then the one that maybe has fewer species that get extinct is maybe the best one. Or the one that the more that we can preserve is maybe the best one. And so let's look at the two scenarios. So the first scenario is the 45 square kilometer island. And this is just one, so times one. And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression?"}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And so let's look at the two scenarios. So the first scenario is the 45 square kilometer island. And this is just one, so times one. And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? Well it's going to be 0.28996 minus 0.05323 times the natural log of 45. And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species. So times our 70 species."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? Well it's going to be 0.28996 minus 0.05323 times the natural log of 45. And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species. So times our 70 species. And so we can get our calculator out to figure that out. So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression. So this would be equal to, so it looks like almost 9%."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So times our 70 species. And so we can get our calculator out to figure that out. So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression. So this would be equal to, so it looks like almost 9%. And if we wanna figure out the actual number we would expect to go extinct, we would just multiply that times the number of species on that island. So times 70. And we get, so approximately about 6.11."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So this would be equal to, so it looks like almost 9%. And if we wanna figure out the actual number we would expect to go extinct, we would just multiply that times the number of species on that island. So times 70. And we get, so approximately about 6.11. So let me write that down. So this is going to be approximately 6.11. So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And we get, so approximately about 6.11. So let me write that down. So this is going to be approximately 6.11. So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves. So it's going to be three square kilometers times five islands."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves. So it's going to be three square kilometers times five islands. And we're gonna just do the same exercise. Our proportion that goes extinct is going to be 0.28996. That's just the y-intercept for our regression line."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So it's going to be three square kilometers times five islands. And we're gonna just do the same exercise. Our proportion that goes extinct is going to be 0.28996. That's just the y-intercept for our regression line. Minus 0.05323, and I have a negative sign there because we have a negative slope. And this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "That's just the y-intercept for our regression line. Minus 0.05323, and I have a negative sign there because we have a negative slope. And this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers. Three square kilometers. And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species. So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "It's going to be three square kilometers. Three square kilometers. And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species. So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. So times 80. So let's figure out what this is, get the calculator out again. And we are going to get, so this is going to be the proportion."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. So times 80. So let's figure out what this is, get the calculator out again. And we are going to get, so this is going to be the proportion. It's a much higher proportion. And then we'll multiply that times our number of species. So times 80 to figure out how many species will go extinct."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "And we are going to get, so this is going to be the proportion. It's a much higher proportion. And then we'll multiply that times our number of species. So times 80 to figure out how many species will go extinct. And we have here, it's approximately 18.52. So this is approximately 18.52. So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So times 80 to figure out how many species will go extinct. And we have here, it's approximately 18.52. So this is approximately 18.52. So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. And then if we have 19 extinct, how many are we going to save? We're going to have 61 saved. 61 saved."}, {"video_title": "Worked example of linear regression using transformed data   AP Statistics   Khan Academy.mp3", "Sentence": "So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. And then if we have 19 extinct, how many are we going to save? We're going to have 61 saved. 61 saved. And even if you said 18 1\u20442 here and 61.5 here, on either measure, the 45 square, the big island is better. You're going to have fewer species that are extinct and more that are saved. So which choice would you recommend and why?"}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "The school nurse plans to provide additional screening to students whose resting pulse rates are in the top 30% of the students who were tested. What is the minimum resting pulse rate at that school for students who will receive additional screening, round to the nearest whole number? If you feel like you know how to tackle this, I encourage you to pause this video and try to work it out. All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number."}, {"video_title": "Finding z-score for a percentile   AP Statistics   Khan Academy.mp3", "Sentence": "0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number. So we will just round to 85 beats per minute. So that's the threshold. If you have that resting heartbeat, then the school nurse is going to give you some additional screening."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "A hand is chosen, a hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are 36 unique cards, and I won't worry about there's 9 numbers in each suit, and there are 4 suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick 9 of them. So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let's think about it. There are 36 unique cards, and I won't worry about there's 9 numbers in each suit, and there are 4 suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick 9 of them. So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick 9 cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick 9 cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 possible hands. Now, this would be true if order mattered. This would be true if maybe I have card 15 here."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 possible hands. Now, this would be true if order mattered. This would be true if maybe I have card 15 here. Maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have, and that's one hand, and then I have another, so then I have cards 1, 2, 3, 4, 5, 6, 7, 8. I have 8 other cards."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "This would be true if maybe I have card 15 here. Maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have, and that's one hand, and then I have another, so then I have cards 1, 2, 3, 4, 5, 6, 7, 8. I have 8 other cards. Or maybe another hand is I have the 8 cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "I have 8 other cards. Or maybe another hand is I have the 8 cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses. So order doesn't matter. So we're over-counting. We're counting all of the different ways that the same number of cards can be arranged."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "But they're telling us that the cards can be sorted however the player chooses. So order doesn't matter. So we're over-counting. We're counting all of the different ways that the same number of cards can be arranged. So in order to, I guess, not over-count, we have to divide this by the way 9 cards can be rearranged. So how many ways can 9 cards be rearranged? If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We're counting all of the different ways that the same number of cards can be arranged. So in order to, I guess, not over-count, we have to divide this by the way 9 cards can be rearranged. So how many ways can 9 cards be rearranged? If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took 1 to put in the first, so I have 8 left, then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be one card left to put in it. So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took 1 to put in the first, so I have 8 left, then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be one card left to put in it. So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. And you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands, if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. And you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands, if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer. And this will be the correct answer. Now, this is a super, super, duper large number. Let's figure out how large of a number this is."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And this will be an answer. And this will be the correct answer. Now, this is a super, super, duper large number. Let's figure out how large of a number this is. We have 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 divided by 9. Well, I could do it this way. I could put a parentheses."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let's figure out how large of a number this is. We have 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 divided by 9. Well, I could do it this way. I could put a parentheses. Divided by parentheses 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Now, hopefully the calculator can handle this. And it gave us this number."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "I could put a parentheses. Divided by parentheses 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Now, hopefully the calculator can handle this. And it gave us this number. Was it 94,143,280? Let me put this on the side so I can read it. So this number right here gives us 94,143,280."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And it gave us this number. Was it 94,143,280? Let me put this on the side so I can read it. So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9-card hands in this situation. Now, we kind of just worked through it."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9-card hands in this situation. Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is they say, look, we have 36 things."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is they say, look, we have 36 things. And we are going to choose 9 of them. We don't care about order. So sometimes it'll be written as n choose k. Let me write it this way."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And the way that people denote this formula is they say, look, we have 36 things. And we are going to choose 9 of them. We don't care about order. So sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. But we stopped only 9 away from 36. So this is 36 factorial. So this part right here, that part right there, is not just 36 factorial."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "It would just keep going. But we stopped only 9 away from 36. So this is 36 factorial. So this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36 minus 9 factorial. What is 36 minus 9? It's 27."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36 minus 9 factorial. What is 36 minus 9? It's 27. So 27 factorial. So let's think about this. 36 factorial, it'd be 36 times 35."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "It's 27. So 27 factorial. So let's think about this. 36 factorial, it'd be 36 times 35. You keep going. All the way times 28 times 27, all the way down to 1. That is 36 factorial."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "36 factorial, it'd be 36 times 35. You keep going. All the way times 28 times 27, all the way down to 1. That is 36 factorial. Now what is 36 minus 9 factorial? That's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "That is 36 factorial. Now what is 36 minus 9 factorial? That's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26. So that and that would cancel out."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26. So that and that would cancel out. So if you do 36 divided by 36 minus 9 factorial, you just get the first, I guess the largest 9 terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So that and that would cancel out. So if you do 36 divided by 36 minus 9 factorial, you just get the first, I guess the largest 9 terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial. And we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then we divided it by 9 factorial. And we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this. n choose k, and they'll write the formula as equal to n factorial over n minus k factorial. And also in the denominator, k factorial. And this is a general formula that if you have n things and you want to find out all of the possible ways you can pick k things from those n things and you don't care about the order."}, {"video_title": "Example 9 card hands   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And sometimes you'll see this formula written like this. n choose k, and they'll write the formula as equal to n factorial over n minus k factorial. And also in the denominator, k factorial. And this is a general formula that if you have n things and you want to find out all of the possible ways you can pick k things from those n things and you don't care about the order. All you care is about which k things you picked. You don't care about the order in which you picked those k things. So that's what we did here."}, {"video_title": "Example Different ways to pick officers   Precalculus   Khan Academy.mp3", "Sentence": "So to think about the probability of Marsha, so let me write this president, president is equal to Marsha, or vice president is equal to Sabitha, and secretary is equal to Robert. This is going to be, this right here is one possible outcome, one specific outcome, so it's one outcome out of the total number of outcomes over the total number of possibilities. Now what is the total number of possibilities? Well to think about that, let's just think about the three positions. You have president, you have vice president, and you have secretary. Now let's just assume that we're going to fill the slot of president first. We don't have to do president first, but we're just going to pick here."}, {"video_title": "Example Different ways to pick officers   Precalculus   Khan Academy.mp3", "Sentence": "Well to think about that, let's just think about the three positions. You have president, you have vice president, and you have secretary. Now let's just assume that we're going to fill the slot of president first. We don't have to do president first, but we're just going to pick here. So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from. So there are nine possibilities here. Now when we go to selecting our vice president, we would have already assigned one person to the president."}, {"video_title": "Example Different ways to pick officers   Precalculus   Khan Academy.mp3", "Sentence": "We don't have to do president first, but we're just going to pick here. So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from. So there are nine possibilities here. Now when we go to selecting our vice president, we would have already assigned one person to the president. So we only have eight people to pick from. And when we assign our secretary, we would have already assigned our president and vice president, so we're only going to have seven people to pick from. So the total permutations here, or the total number of possibilities, or the total number of ways to pick president, vice president, and secretary from nine people is going to be 9 times 8 times 7, which is, let's see, 9 times 8 is 72, 72 times 7, 2 times 7 is 14, 7 times 7 is 49, plus 1 is 50."}, {"video_title": "Example Different ways to pick officers   Precalculus   Khan Academy.mp3", "Sentence": "Now when we go to selecting our vice president, we would have already assigned one person to the president. So we only have eight people to pick from. And when we assign our secretary, we would have already assigned our president and vice president, so we're only going to have seven people to pick from. So the total permutations here, or the total number of possibilities, or the total number of ways to pick president, vice president, and secretary from nine people is going to be 9 times 8 times 7, which is, let's see, 9 times 8 is 72, 72 times 7, 2 times 7 is 14, 7 times 7 is 49, plus 1 is 50. So there's 504 possibilities. So to answer the question, the probability of Marsha being president, Savita being vice president, Robert being secretary, is 1 over the total number of possibilities, which is 1 over 504. That's the probability."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "Each problem has only one correct answer. What is the probability of randomly guessing the correct answer on both problems? Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities?"}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both?"}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes."}, {"video_title": "Test taking probability and independent events   Precalculus   Khan Academy.mp3", "Sentence": "And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes. Or, since these are independent events, you can multiply. You see that there are 12 outcomes because there's 12 possible outcomes. So there's four possible outcomes for problem number one times the three possible outcomes for problem number two, and that's also where you get a 12."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I ask them, and there's only two options. They can either have an unfavorable rating or they could have a favorable rating. And let's say after I survey every single member of this population, 40% have an unfavorable rating and 60% have a favorable rating. So if I were to draw the probability distribution, it's going to be a discrete one because there's only two values that any person can take on. They could either have an unfavorable view or they could have a favorable view. And 40% have an unfavorable view. And let me color code this a little bit."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I were to draw the probability distribution, it's going to be a discrete one because there's only two values that any person can take on. They could either have an unfavorable view or they could have a favorable view. And 40% have an unfavorable view. And let me color code this a little bit. So this is the 40% right over here, so 0.4. Maybe I'll just write 40% right over there. And then 60% have a favorable view."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let me color code this a little bit. So this is the 40% right over here, so 0.4. Maybe I'll just write 40% right over there. And then 60% have a favorable view. Let me just make sure I'm doing this right. 60% have a favorable view. And notice these two numbers add up to 100% because everyone had to pick between these two options."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then 60% have a favorable view. Let me just make sure I'm doing this right. 60% have a favorable view. And notice these two numbers add up to 100% because everyone had to pick between these two options. Now, if I were to go and ask you to pick a random member of that population and say, what is the expected favorability rating of that member, what would it be? Or another way to think about it is, what is the mean of this distribution? And for a discrete distribution like this, your mean or your expected value is just going to be the probability-weighted sum of the different values that your distribution can take on."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And notice these two numbers add up to 100% because everyone had to pick between these two options. Now, if I were to go and ask you to pick a random member of that population and say, what is the expected favorability rating of that member, what would it be? Or another way to think about it is, what is the mean of this distribution? And for a discrete distribution like this, your mean or your expected value is just going to be the probability-weighted sum of the different values that your distribution can take on. Now, the way I've written it right here, you can't take a probability-weighted sum of u and f. You can't say 40% times u plus 60% times f. You won't get any type of a number. So what we're going to do is define u and f to be some type of value. So let's say that u is 0 and f is 1."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And for a discrete distribution like this, your mean or your expected value is just going to be the probability-weighted sum of the different values that your distribution can take on. Now, the way I've written it right here, you can't take a probability-weighted sum of u and f. You can't say 40% times u plus 60% times f. You won't get any type of a number. So what we're going to do is define u and f to be some type of value. So let's say that u is 0 and f is 1. And now the notion of taking a probability-weighted sum makes some sense. So the mean of this distribution is going to be 0.4. That's this probability right here."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that u is 0 and f is 1. And now the notion of taking a probability-weighted sum makes some sense. So the mean of this distribution is going to be 0.4. That's this probability right here. Times 0 plus 0.6 times 1, which is going to be equal to this is just going to be 0. 0.6 times 1 is 0.6. So clearly, no individual can take on the value of 0.6."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's this probability right here. Times 0 plus 0.6 times 1, which is going to be equal to this is just going to be 0. 0.6 times 1 is 0.6. So clearly, no individual can take on the value of 0.6. No one can tell you I 60% am favorable and 40% am unfavorable. Everyone has to pick either favorable or unfavorable. So you will never actually find someone who has a 0.6 favorability value."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So clearly, no individual can take on the value of 0.6. No one can tell you I 60% am favorable and 40% am unfavorable. Everyone has to pick either favorable or unfavorable. So you will never actually find someone who has a 0.6 favorability value. It will either be a 1 or a 0. So this is an interesting case where the mean or the expected value is not a value that the distribution can actually take on. It's a value someplace over here that obviously cannot happen, but this is the mean."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you will never actually find someone who has a 0.6 favorability value. It will either be a 1 or a 0. So this is an interesting case where the mean or the expected value is not a value that the distribution can actually take on. It's a value someplace over here that obviously cannot happen, but this is the mean. This is the expected value. And the reason why that makes sense is if you surveyed 100 people, you multiply 100 times this number, you would expect 60 people to say yes. Or if you summed them all up, 60 would say yes and then 40 would say 0."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's a value someplace over here that obviously cannot happen, but this is the mean. This is the expected value. And the reason why that makes sense is if you surveyed 100 people, you multiply 100 times this number, you would expect 60 people to say yes. Or if you summed them all up, 60 would say yes and then 40 would say 0. You sum them all up, you would get 60% saying yes. And that's exactly what our population distribution told us. Now what is the variance?"}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if you summed them all up, 60 would say yes and then 40 would say 0. You sum them all up, you would get 60% saying yes. And that's exactly what our population distribution told us. Now what is the variance? What is the variance of this population right over here? So the variance, let me write it over here. Let me pick a new color."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what is the variance? What is the variance of this population right over here? So the variance, let me write it over here. Let me pick a new color. The variance is just, you could view it as the probability weighted sum of the squared distances from the mean or the expected value of the squared distances from the mean. So what's that going to be? Well, there's two different values that anything can take on."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me pick a new color. The variance is just, you could view it as the probability weighted sum of the squared distances from the mean or the expected value of the squared distances from the mean. So what's that going to be? Well, there's two different values that anything can take on. You can either have a 0 or you could either have a 1. The probability that you get a 0 is 0.4. So there's a 0.4 probability that you get a 0."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's two different values that anything can take on. You can either have a 0 or you could either have a 1. The probability that you get a 0 is 0.4. So there's a 0.4 probability that you get a 0. And if you get a 0, what's the distance from 0 to the mean? The distance from 0 to the mean is 0 minus 0.6. Or I could even say 0.6 minus 0."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a 0.4 probability that you get a 0. And if you get a 0, what's the distance from 0 to the mean? The distance from 0 to the mean is 0 minus 0.6. Or I could even say 0.6 minus 0. Same thing because we're going to square it. 0 minus 0.6 squared. Remember, the variance is the probability or the weighted sum of the squared distances."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or I could even say 0.6 minus 0. Same thing because we're going to square it. 0 minus 0.6 squared. Remember, the variance is the probability or the weighted sum of the squared distances. So this is the difference between 0 and the mean. And then plus, there's a 0.6 chance that you get a 1. The difference between 1 and 0.6, 1 and our mean, 0.6, is that, and then we are also going to square this over here."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, the variance is the probability or the weighted sum of the squared distances. So this is the difference between 0 and the mean. And then plus, there's a 0.6 chance that you get a 1. The difference between 1 and 0.6, 1 and our mean, 0.6, is that, and then we are also going to square this over here. Now what is this value going to be? This is going to be 0.4 times 0.6 squared. This is 0.4 times 0."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "The difference between 1 and 0.6, 1 and our mean, 0.6, is that, and then we are also going to square this over here. Now what is this value going to be? This is going to be 0.4 times 0.6 squared. This is 0.4 times 0. Because 0 minus 0.6 is negative 0.6. If you square it, you get positive 0.36. So this value right here, I'm going to color code it, this value right here is times 0.36."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 0.4 times 0. Because 0 minus 0.6 is negative 0.6. If you square it, you get positive 0.36. So this value right here, I'm going to color code it, this value right here is times 0.36. And then this value right here, let me do this in another, so then we're going to have plus 0.6, plus this 0.6 times 1 minus 0.6 squared. Now 1 minus 0.6 is 0.4. 0.4 squared, or 0.4 squared, is 0.16."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this value right here, I'm going to color code it, this value right here is times 0.36. And then this value right here, let me do this in another, so then we're going to have plus 0.6, plus this 0.6 times 1 minus 0.6 squared. Now 1 minus 0.6 is 0.4. 0.4 squared, or 0.4 squared, is 0.16. So let me do this. So this value right here is going to be 0.16. Let me get my calculator out to actually calculate these values."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "0.4 squared, or 0.4 squared, is 0.16. So let me do this. So this value right here is going to be 0.16. Let me get my calculator out to actually calculate these values. Let me get my calculator out. So this is going to be 0.4 times 0.36 plus 0.6 times 0.16, which is equal to 0.24. So our standard deviation of this distribution is 0.24."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me get my calculator out to actually calculate these values. Let me get my calculator out. So this is going to be 0.4 times 0.36 plus 0.6 times 0.16, which is equal to 0.24. So our standard deviation of this distribution is 0.24. Or if you want to think about the variance of this distribution is 0.24, and the standard deviation of this distribution, which is just the square root of this, is going to be the square root of 0.24. And let's calculate what that is. That is going to be, let's take the square root of 0.24, which is equal to 0.48."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our standard deviation of this distribution is 0.24. Or if you want to think about the variance of this distribution is 0.24, and the standard deviation of this distribution, which is just the square root of this, is going to be the square root of 0.24. And let's calculate what that is. That is going to be, let's take the square root of 0.24, which is equal to 0.48. Well, I'll just round it up, 0.49. So this is equal to 0.49. So if you were to look at this distribution, the mean of this distribution is 0.6."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is going to be, let's take the square root of 0.24, which is equal to 0.48. Well, I'll just round it up, 0.49. So this is equal to 0.49. So if you were to look at this distribution, the mean of this distribution is 0.6. So 0.6 is the mean. And the standard deviation is 0.5. So the standard deviation is, so it's actually out here, because if you go add one standard deviation, you're almost getting to 1.1."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you were to look at this distribution, the mean of this distribution is 0.6. So 0.6 is the mean. And the standard deviation is 0.5. So the standard deviation is, so it's actually out here, because if you go add one standard deviation, you're almost getting to 1.1. So this is one standard deviation above. And then one standard deviation below gets you right about here. And that kind of makes sense."}, {"video_title": "Mean and variance of Bernoulli distribution example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the standard deviation is, so it's actually out here, because if you go add one standard deviation, you're almost getting to 1.1. So this is one standard deviation above. And then one standard deviation below gets you right about here. And that kind of makes sense. It's hard to really have a good intuition for a discrete distribution, because you really can't take on those values. But it makes sense that the distribution is skewed to the right over here. Anyway, I did this example with particular numbers, because I wanted to show you why this distribution is useful, and the next video I'll do these with just general numbers, where this is going to be p, where this is the probability of success, and this is the 1 minus p, which is the probability of failure."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "What we're going to do in this video is talk about the idea of a residual plot for a given regression and the data that it's trying to explain. So right over here, we have a fairly simple least squares regression. We're trying to fit four points. And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate."}, {"video_title": "Residual plots   Exploring bivariate numerical data   AP Statistics   Khan Academy.mp3", "Sentence": "When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate. Maybe some type of nonlinear model, some type of nonlinear curve might better fit the data, or the relationship between the y and the x is nonlinear. Another way you could think about it is when you have a lot of residuals that are pretty far away from the x-axis in the residual plot, you would also say this line isn't such a good fit. If you calculate the r value here, it would only be slightly positive, but it would not be close to one."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "The data were skewed to the right with a sample mean of 38.75. She's considering using her data to make a confidence interval to estimate the mean age of faculty members at her university. Which conditions for constructing a t-interval have been met? So pause this video and see if you can answer this on your own. Okay, now let's try to answer this together. So there's 700 faculty members over here. She's trying to estimate the population mean, the mean age."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So pause this video and see if you can answer this on your own. Okay, now let's try to answer this together. So there's 700 faculty members over here. She's trying to estimate the population mean, the mean age. She can't talk to all 700, so she takes a sample, a simple random sample of 20. So the n is equal to 20 here. From this 20, she calculates a sample mean of 38.75."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "She's trying to estimate the population mean, the mean age. She can't talk to all 700, so she takes a sample, a simple random sample of 20. So the n is equal to 20 here. From this 20, she calculates a sample mean of 38.75. Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic. And so that interval would look something like this. It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "From this 20, she calculates a sample mean of 38.75. Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic. And so that interval would look something like this. It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals. The first is, is that our sample is random. Well, they tell us that here, that she took a simple random sample of 20."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals. The first is, is that our sample is random. Well, they tell us that here, that she took a simple random sample of 20. And so we know that we are meeting that constraint, and that's actually choice A. The data is a random sample from the population of interest. So we can circle that in."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Well, they tell us that here, that she took a simple random sample of 20. And so we know that we are meeting that constraint, and that's actually choice A. The data is a random sample from the population of interest. So we can circle that in. So the next condition is the normal condition. Now the normal condition when we're doing a t-interval is a little bit more involved, because we do need to assume that the sampling distribution of the sample means is roughly normal. Now there's a couple of ways that we can get there."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So we can circle that in. So the next condition is the normal condition. Now the normal condition when we're doing a t-interval is a little bit more involved, because we do need to assume that the sampling distribution of the sample means is roughly normal. Now there's a couple of ways that we can get there. Either our sample size is greater than or equal to 30. The central limit theorem tells us that then our sampling distribution, regardless of what the distribution is in the population, that the sampling distribution actually would then be approximately normal. She didn't meet that constraint right over here."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Now there's a couple of ways that we can get there. Either our sample size is greater than or equal to 30. The central limit theorem tells us that then our sampling distribution, regardless of what the distribution is in the population, that the sampling distribution actually would then be approximately normal. She didn't meet that constraint right over here. Here, her sample size is only 20. So, so far, this isn't looking good. Now that's not the only way to meet the normal condition."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "She didn't meet that constraint right over here. Here, her sample size is only 20. So, so far, this isn't looking good. Now that's not the only way to meet the normal condition. Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal. So original, distribution, normal. Or even if it's roughly symmetric around the mean."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Now that's not the only way to meet the normal condition. Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal. So original, distribution, normal. Or even if it's roughly symmetric around the mean. So approximately symmetric. But if you look at this, they tell us that it has a right skew. They say the data were skewed to the right with the sample mean of 38.75."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Or even if it's roughly symmetric around the mean. So approximately symmetric. But if you look at this, they tell us that it has a right skew. They say the data were skewed to the right with the sample mean of 38.75. So that tells us that the data set that we're getting in our sample is not symmetric. And the original distribution is unlikely to be normal. Think about it."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "They say the data were skewed to the right with the sample mean of 38.75. So that tells us that the data set that we're getting in our sample is not symmetric. And the original distribution is unlikely to be normal. Think about it. It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this. And that's actually what's causing that skew to the right. So this one does not meet the normal condition."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Think about it. It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this. And that's actually what's causing that skew to the right. So this one does not meet the normal condition. We can't feel good that our sampling distribution of the sample means is going to be normal. So I'm not gonna fill that one in. Choice C, individual observations can be considered independent."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So this one does not meet the normal condition. We can't feel good that our sampling distribution of the sample means is going to be normal. So I'm not gonna fill that one in. Choice C, individual observations can be considered independent. So there's two ways to meet this constraint. One is, is if we sample with replacement. Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Choice C, individual observations can be considered independent. So there's two ways to meet this constraint. One is, is if we sample with replacement. Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. It does not look like she did that. It doesn't look like she sampled with replacement. And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. It does not look like she did that. It doesn't look like she sampled with replacement. And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. And the 10% of this population is 70. 70 is 10% of 700. And so this is definitely less than or equal to 10%."}, {"video_title": "Conditions for valid t intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. And the 10% of this population is 70. 70 is 10% of 700. And so this is definitely less than or equal to 10%. And so it can be considered independent. And so we can actually meet that constraint as well. So the main issue where our t interval might not be so good is that our sampling distribution, we can't feel so confident that that is going to be normal."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Let me write that down. Theoretical, theoretical probability. Well, maybe the simplest example, or one of the simplest examples, is if you're flipping a coin. And let's say in theory you're flipping a completely fair coin and you're flipping it in a way that is completely fair. Well, there you know you have two outcomes. The coin will either result, either heads will be on top or head tails will be on top. And so theoretically you say, well look, if I want to figure out the probability of getting a heads, in theory I have two equally likely possibilities and heads is one of those two equally likely possibilities."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And let's say in theory you're flipping a completely fair coin and you're flipping it in a way that is completely fair. Well, there you know you have two outcomes. The coin will either result, either heads will be on top or head tails will be on top. And so theoretically you say, well look, if I want to figure out the probability of getting a heads, in theory I have two equally likely possibilities and heads is one of those two equally likely possibilities. So you have a 1 1\u20442 probability. And once again, if in theory the coin is definitely fair, it's a fair coin and it's flipped in a very fair way, then this is true. You have a 1 1\u20442 probability."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And so theoretically you say, well look, if I want to figure out the probability of getting a heads, in theory I have two equally likely possibilities and heads is one of those two equally likely possibilities. So you have a 1 1\u20442 probability. And once again, if in theory the coin is definitely fair, it's a fair coin and it's flipped in a very fair way, then this is true. You have a 1 1\u20442 probability. We could also do that with rolling a die. A fair six-sided die is going to have six possible outcomes. One, two, three, four, five, and six."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You have a 1 1\u20442 probability. We could also do that with rolling a die. A fair six-sided die is going to have six possible outcomes. One, two, three, four, five, and six. And if you said, what is the probability of rolling or getting a result that is greater than or equal to three? Well, we have six equally likely possibilities. You see them there."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "One, two, three, four, five, and six. And if you said, what is the probability of rolling or getting a result that is greater than or equal to three? Well, we have six equally likely possibilities. You see them there. And in theory, if they're all equally likely, four of these possibilities meet our constraint of being greater than or equal to three. So we have four out of the six equal of these possibilities meet our constraints. So we have a 2\u20443, 4\u2076 is the same thing as 2\u20443 probability of it happening."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You see them there. And in theory, if they're all equally likely, four of these possibilities meet our constraint of being greater than or equal to three. So we have four out of the six equal of these possibilities meet our constraints. So we have a 2\u20443, 4\u2076 is the same thing as 2\u20443 probability of it happening. Now these are for simple things like die or flipping a coin. And if you have fancy computers or spreadsheets, you can even say, hey, I'm gonna flip a coin a bunch of times and do all the combinatorics and all of that. But there are things that are even beyond what a computer can find the exact theoretical probability."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So we have a 2\u20443, 4\u2076 is the same thing as 2\u20443 probability of it happening. Now these are for simple things like die or flipping a coin. And if you have fancy computers or spreadsheets, you can even say, hey, I'm gonna flip a coin a bunch of times and do all the combinatorics and all of that. But there are things that are even beyond what a computer can find the exact theoretical probability. Let's say you're playing a game, say football, American football, and you wanted to figure out the probability of scoring a certain number of points. Well, that isn't very simple because that's going to involve what human beings are doing. Minds are very unpredictable, how people will respond to things."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "But there are things that are even beyond what a computer can find the exact theoretical probability. Let's say you're playing a game, say football, American football, and you wanted to figure out the probability of scoring a certain number of points. Well, that isn't very simple because that's going to involve what human beings are doing. Minds are very unpredictable, how people will respond to things. The weather might get involved. There might be, someone might fall sick. The ball might be wet or just how the ball might interact with some player's jersey."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Minds are very unpredictable, how people will respond to things. The weather might get involved. There might be, someone might fall sick. The ball might be wet or just how the ball might interact with some player's jersey. Who knows what actually might result in the score being one point this way or seven points this way or seven points that way. And so for situations like that, it makes more sense to think more in terms of experimental probability. And experimental probability, we're really just trying to get an estimate of something happening based on data and experience that we've had in the past."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "The ball might be wet or just how the ball might interact with some player's jersey. Who knows what actually might result in the score being one point this way or seven points this way or seven points that way. And so for situations like that, it makes more sense to think more in terms of experimental probability. And experimental probability, we're really just trying to get an estimate of something happening based on data and experience that we've had in the past. So for example, let's say that you had, this is data from your football team and it's a couple of games or many games into the season, and you've been tabulating the number of points. You have a histogram of the number of games that scored between zero and nine points. You had two games that scored between zero and nine points, four games that scored between 10 and 19 points, or from 10 to 19 points."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And experimental probability, we're really just trying to get an estimate of something happening based on data and experience that we've had in the past. So for example, let's say that you had, this is data from your football team and it's a couple of games or many games into the season, and you've been tabulating the number of points. You have a histogram of the number of games that scored between zero and nine points. You had two games that scored between zero and nine points, four games that scored between 10 and 19 points, or from 10 to 19 points. You have five games that went from 20 to 29 points. You had three games that went from 30 to 39 points. And then you had two games that go from 40 to 49 points."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You had two games that scored between zero and nine points, four games that scored between 10 and 19 points, or from 10 to 19 points. You have five games that went from 20 to 29 points. You had three games that went from 30 to 39 points. And then you had two games that go from 40 to 49 points. And let's say for your next game, and let's see, you've already had two, let's see how many games you've had so far. This is two, let me write it down. The game so far is two plus four plus five plus three plus two, so this is what, six, plus, this is six plus five is 11."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And then you had two games that go from 40 to 49 points. And let's say for your next game, and let's see, you've already had two, let's see how many games you've had so far. This is two, let me write it down. The game so far is two plus four plus five plus three plus two, so this is what, six, plus, this is six plus five is 11. 11 plus five is 16. So you've had 16 games so far this season. And you're curious, for your 17th game, so let me write this, game 17, you want to figure out what is the probability of scoring, scoring, scoring, let's say, score, let's say your score, let's say your points, your points are greater than or equal to 30."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "The game so far is two plus four plus five plus three plus two, so this is what, six, plus, this is six plus five is 11. 11 plus five is 16. So you've had 16 games so far this season. And you're curious, for your 17th game, so let me write this, game 17, you want to figure out what is the probability of scoring, scoring, scoring, let's say, score, let's say your score, let's say your points, your points are greater than or equal to 30. Your points are greater than or equal to 30 for game 17. So once again, this is very hard to find the exact theoretical probability. You don't know exactly, you can't predict the future."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And you're curious, for your 17th game, so let me write this, game 17, you want to figure out what is the probability of scoring, scoring, scoring, let's say, score, let's say your score, let's say your points, your points are greater than or equal to 30. Your points are greater than or equal to 30 for game 17. So once again, this is very hard to find the exact theoretical probability. You don't know exactly, you can't predict the future. You don't know who's gonna show up sick, how humans are going to interact with each other. Maybe someone screams something in the stand that just phases the quarterback in exactly the right or the wrong way. You don't know, this is an incredibly, incredibly complex system, how what might happen over the course of an entire football game."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You don't know exactly, you can't predict the future. You don't know who's gonna show up sick, how humans are going to interact with each other. Maybe someone screams something in the stand that just phases the quarterback in exactly the right or the wrong way. You don't know, this is an incredibly, incredibly complex system, how what might happen over the course of an entire football game. But you can estimate what'll happen based on what you'll see in your past experience. And it depends on the defense of the team you're facing and all that. So it's not going to be, you know, it's not going to be super exact, but you can estimate based on experiments, based on what you've seen in the past."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You don't know, this is an incredibly, incredibly complex system, how what might happen over the course of an entire football game. But you can estimate what'll happen based on what you'll see in your past experience. And it depends on the defense of the team you're facing and all that. So it's not going to be, you know, it's not going to be super exact, but you can estimate based on experiments, based on what you've seen in the past. Here, the experimental probability, and I would say the probability, the estimate, because I would, you know, you shouldn't walk away saying this, okay, we absolutely know for sure that if we conducted this next game experiment n times, it's definitely gonna turn out the same. Because this might be the toughest defense that you play all year, this might be the easiest defense that you play all year. But if you look at what's happened in the past, out of the 16 games so far, there have been three games, these three plus these two games, where you scored more than, greater than or equal to 30 points."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So it's not going to be, you know, it's not going to be super exact, but you can estimate based on experiments, based on what you've seen in the past. Here, the experimental probability, and I would say the probability, the estimate, because I would, you know, you shouldn't walk away saying this, okay, we absolutely know for sure that if we conducted this next game experiment n times, it's definitely gonna turn out the same. Because this might be the toughest defense that you play all year, this might be the easiest defense that you play all year. But if you look at what's happened in the past, out of the 16 games so far, there have been three games, these three plus these two games, where you scored more than, greater than or equal to 30 points. So five out of the 16 situations, you scored more than that. So an estimate of your probability, and you could view this as maybe your experimental probability of scoring more than 30 points based on past experience, is five, five out of the 16 games you've done this in the past. So you'd say it's 5 16ths."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "But if you look at what's happened in the past, out of the 16 games so far, there have been three games, these three plus these two games, where you scored more than, greater than or equal to 30 points. So five out of the 16 situations, you scored more than that. So an estimate of your probability, and you could view this as maybe your experimental probability of scoring more than 30 points based on past experience, is five, five out of the 16 games you've done this in the past. So you'd say it's 5 16ths. Now I wanna really have you take this with a grain of salt. You should not, you know, then go, say okay, I know for sure there's a 5 16ths probability of us winning this game. Because you only have some data points, every team you play is going to be different, it's gonna be different weather conditions, people are gonna be in different moods, et cetera, et cetera."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So you'd say it's 5 16ths. Now I wanna really have you take this with a grain of salt. You should not, you know, then go, say okay, I know for sure there's a 5 16ths probability of us winning this game. Because you only have some data points, every team you play is going to be different, it's gonna be different weather conditions, people are gonna be in different moods, et cetera, et cetera. This is really just an estimate. And I actually, I feel even a little bit of reservation is even calling it a probability. I would just say that this has been true of five out of 16 games in the past."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Because you only have some data points, every team you play is going to be different, it's gonna be different weather conditions, people are gonna be in different moods, et cetera, et cetera. This is really just an estimate. And I actually, I feel even a little bit of reservation is even calling it a probability. I would just say that this has been true of five out of 16 games in the past. So it's an indicator of what might be, what you might say, okay, based on experience, it's more likely than not that we don't score more than 30 points. But it's really just based on experiential data, what's happened in the season. You know, even the makeup of your football team might have changed."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "I would just say that this has been true of five out of 16 games in the past. So it's an indicator of what might be, what you might say, okay, based on experience, it's more likely than not that we don't score more than 30 points. But it's really just based on experiential data, what's happened in the season. You know, even the makeup of your football team might have changed. You might have gotten a different coach. You might have learned to train better. Who knows, one of your team members might have grown by three inches."}, {"video_title": "Experimental probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You know, even the makeup of your football team might have changed. You might have gotten a different coach. You might have learned to train better. Who knows, one of your team members might have grown by three inches. All of these things. So all of this has to be taken with a grain of salt. But this is one way of thinking about it, at least having a sense of what may happen."}, {"video_title": "Ways to represent data   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I kind of liked it I didn't like it or they might have rated it on a scale of zero to five which would have been numbers But it's numbers that are measuring people's opinions as opposed to here We have numbers that are measuring their actual scores So there's all different types of data, and I don't want to get into all of that But let's just start thinking about different ways to represent this data So this is one way you could view this as a table where you have the name let me and then you have the score So you have your name column, and then you have your score column, and I could construct it as a table so clearly Looks like a table like that. That's one way one very common way of representing of representing data just like that That's actually how most traditional databases record data in the tables like this, but you could also do it in other ways so you could record it as a as a Oftentimes called a bar graph or sometimes a histogram So you could put score on The vertical axis here, and then you could have your names over here, and let's see the scores. Let's see maybe we'll make this a 50 actually let me just Mark them off, so this is 10 20 30 that's too big 10 20 30 40 50 60 70 80 90 so that's and then a hundred so that's a hundred one two three four five That would be 50 right over there, and then you could go person by person so Amy Amy record got a 90 on the exam so the bar will go up to 90 So that is Amy and then you have bill you got in 95 So it's going to be between 90 and 100 so it's going to be right over there Bill got a 95 And so it would look like this Bill So that is bill, and then you have cam who got a hundred on the exam so Make sure I'm hand drawing it So it's not as it's not as precise as if I were to do it on a computer So this right over there that is cams score Effort got the same score as cam so her score is going to be let me do that in the color in a first color That's a phrase score right now there. She also got a hundred so Efra Efra and then finally Farah got an 80 so 60 70 80 so Farah got Got an 80 so this is Farah score right over here, so this is another way of representing the data and Here we see it in visual form, but it has the same information you can look up someone's name And then figure out their score Amy scored a 90 bill scored a 95 cam scored 100 effort Also scored 100 Farah scored an 80 and there's even other ways you can have some of this information In fact sometimes you might not even know their names and so then it would be less information But it might just a list of scores the professor must say here are the five scores that were That people got on the exam and they were list 90 95 95 100 100 100 and 80 now if it was listed and if this was all of the data you got this would be less information than the data That's in this bar graph or this histogram And or the data that's given in this table right over here because here not only do we know the scores But we know who got what score here. We only know the list of scores But there's even other ways and I this is this isn't and this is not an exhaustive video of all of the different ways you Can represent data you could also represent data by looking at the frequency of scores? So the frequency of scores right over here, so instead of writing the people you could write the scores So let's see you could say this is 80 85 90 95 and 100 and then you could record the frequency that people got these scores So how many times do you have a score of an 80? Well Farah got a is the only person with the score of 80 so you put one data point there No one got an 85 one person got a 90 so you put a data point there one person got a 95 So you could put that data point right over there, and then two people got a hundred so this is one and two Let's see the other hundred is in this color So I'm just doing the color you wouldn't necessarily have to color code it like this So this is another way to represent, and this is you know in this axis you could just view This is the number so this tells you how many 80s there were how many?"}, {"video_title": "Ways to represent data   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "She also got a hundred so Efra Efra and then finally Farah got an 80 so 60 70 80 so Farah got Got an 80 so this is Farah score right over here, so this is another way of representing the data and Here we see it in visual form, but it has the same information you can look up someone's name And then figure out their score Amy scored a 90 bill scored a 95 cam scored 100 effort Also scored 100 Farah scored an 80 and there's even other ways you can have some of this information In fact sometimes you might not even know their names and so then it would be less information But it might just a list of scores the professor must say here are the five scores that were That people got on the exam and they were list 90 95 95 100 100 100 and 80 now if it was listed and if this was all of the data you got this would be less information than the data That's in this bar graph or this histogram And or the data that's given in this table right over here because here not only do we know the scores But we know who got what score here. We only know the list of scores But there's even other ways and I this is this isn't and this is not an exhaustive video of all of the different ways you Can represent data you could also represent data by looking at the frequency of scores? So the frequency of scores right over here, so instead of writing the people you could write the scores So let's see you could say this is 80 85 90 95 and 100 and then you could record the frequency that people got these scores So how many times do you have a score of an 80? Well Farah got a is the only person with the score of 80 so you put one data point there No one got an 85 one person got a 90 so you put a data point there one person got a 95 So you could put that data point right over there, and then two people got a hundred so this is one and two Let's see the other hundred is in this color So I'm just doing the color you wouldn't necessarily have to color code it like this So this is another way to represent, and this is you know in this axis you could just view This is the number so this tells you how many 80s there were how many? 90s there are how many 95s and how many hundreds so this right over here has the same data as this list of numbers? It's just another way of looking at it and once you have your data arranged in any of these ways We can start to ask Interesting questions we can ask ourselves things like well. What is the range of data?"}, {"video_title": "Ways to represent data   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well Farah got a is the only person with the score of 80 so you put one data point there No one got an 85 one person got a 90 so you put a data point there one person got a 95 So you could put that data point right over there, and then two people got a hundred so this is one and two Let's see the other hundred is in this color So I'm just doing the color you wouldn't necessarily have to color code it like this So this is another way to represent, and this is you know in this axis you could just view This is the number so this tells you how many 80s there were how many? 90s there are how many 95s and how many hundreds so this right over here has the same data as this list of numbers? It's just another way of looking at it and once you have your data arranged in any of these ways We can start to ask Interesting questions we can ask ourselves things like well. What is the range of data? What is the range in? The data and the range is just the spread between the lowest point and the highest point so the range in this data It's going to be the difference between the highest score and the highest scores are hundred and the lowest score and 80 So the range is going to be the difference between the max Minus the min the maximum score minus the minimum score so it's going to be 100 Minus 80 is equal to 20 so that gives you a sense of things it kind of gives you a sense of spread You could also ask yourself. Well."}, {"video_title": "Ways to represent data   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "What is the range of data? What is the range in? The data and the range is just the spread between the lowest point and the highest point so the range in this data It's going to be the difference between the highest score and the highest scores are hundred and the lowest score and 80 So the range is going to be the difference between the max Minus the min the maximum score minus the minimum score so it's going to be 100 Minus 80 is equal to 20 so that gives you a sense of things it kind of gives you a sense of spread You could also ask yourself. Well. How many how many people scored below 100 these are just interesting questions below 100 and you can actually answer that question well actually you could have answered either of these questions with any of these different ways of Looking at the data if you say how many people scored below 100 well one two three how many people scored below 100 well 100 is up here So it's going to be one two three how many people scored below 100? One two three how many people scored below 100? One two three and so anyway you look at it you would have gotten three And you could also ask yourself."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We hopefully now have a respectable working knowledge of the sampling distribution of the sample mean. And what I want to do in this video is explore a little bit more on how that distribution changes as we change our sample size n. I'll write n down right here, our sample size n. So just as a bit of review, we saw before, we could just start off with any crazy distribution. Maybe it looks something like this. I'll do discrete distribution. Really, to model anything, at some point you have to make it discrete. It could be a very granular discrete distribution. But let's say something crazy that looks like this."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll do discrete distribution. Really, to model anything, at some point you have to make it discrete. It could be a very granular discrete distribution. But let's say something crazy that looks like this. This is clearly not a normal distribution. But we saw in the first video, if you take, let's say, sample sizes of four. So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "But let's say something crazy that looks like this. This is clearly not a normal distribution. But we saw in the first video, if you take, let's say, sample sizes of four. So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took four numbers at a time and averaged them, let me do that here. If you took four numbers at a time, let's say we use this distribution to generate four random numbers. We're very likely to get a 9."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took four numbers at a time and averaged them, let me do that here. If you took four numbers at a time, let's say we use this distribution to generate four random numbers. We're very likely to get a 9. We're definitely not going to get any 7 or 8's. We're definitely not going to get a 4. We might get a 1 or 2."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're very likely to get a 9. We're definitely not going to get any 7 or 8's. We're definitely not going to get a 4. We might get a 1 or 2. 3 is also very likely. 5 is very likely. So we use this function to essentially generate random numbers for us."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We might get a 1 or 2. 3 is also very likely. 5 is very likely. So we use this function to essentially generate random numbers for us. And we take samples of four, and then we average them up. So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1. So what is that?"}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we use this function to essentially generate random numbers for us. And we take samples of four, and then we average them up. So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1. So what is that? That's 14 plus 10, 24 divided by 4. The average for this first trial, for this first sample of four, is going to be 6. They add up to 24 divided by 4."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is that? That's 14 plus 10, 24 divided by 4. The average for this first trial, for this first sample of four, is going to be 6. They add up to 24 divided by 4. So we would plot it right here. Our average was 6 that time. Just like that, and we'll just keep doing it."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "They add up to 24 divided by 4. So we would plot it right here. Our average was 6 that time. Just like that, and we'll just keep doing it. And we've seen in the past that if you just keep doing this, this is going to start looking something like a normal distribution. So maybe we do it again. The average is 6 again."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just like that, and we'll just keep doing it. And we've seen in the past that if you just keep doing this, this is going to start looking something like a normal distribution. So maybe we do it again. The average is 6 again. Maybe we do it again. The average is 5. We do it again."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "The average is 6 again. Maybe we do it again. The average is 5. We do it again. The average is 7. We do it again. The average is 6."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We do it again. The average is 7. We do it again. The average is 6. And then if you just do this a ton, a ton of times, your distribution might look something that looks very much like a normal distribution. So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution. Obviously, there's some average values."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "The average is 6. And then if you just do this a ton, a ton of times, your distribution might look something that looks very much like a normal distribution. So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution. Obviously, there's some average values. It won't be a perfect normal distribution, because you can't ever get anything less than 0, or anything less than 1, really, as an average. You can't get 0 as an average. And you can't get anything more than 9."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Obviously, there's some average values. It won't be a perfect normal distribution, because you can't ever get anything less than 0, or anything less than 1, really, as an average. You can't get 0 as an average. And you can't get anything more than 9. So it's not going to have infinitely long tails. But at least for the middle part of it, a normal distribution might be a good approximation. In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you can't get anything more than 9. So it's not going to have infinitely long tails. But at least for the middle part of it, a normal distribution might be a good approximation. In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. Every time we do a trial, we took 4. And we took their average, and we plotted it. We could have had n equal 10."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. Every time we do a trial, we took 4. And we took their average, and we plotted it. We could have had n equal 10. We could have taken 10 samples from this population, you could say, or from this random variable, averaged them, and then plotted them here. And in the last video, we ran the simulation. I'm going to go back to that simulation in a second."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could have had n equal 10. We could have taken 10 samples from this population, you could say, or from this random variable, averaged them, and then plotted them here. And in the last video, we ran the simulation. I'm going to go back to that simulation in a second. We saw a couple of things. And I'll show it to you in a little bit more depth this time. When n is pretty small, it doesn't approach a normal distribution that well."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to go back to that simulation in a second. We saw a couple of things. And I'll show it to you in a little bit more depth this time. When n is pretty small, it doesn't approach a normal distribution that well. So when n is small, I mean, let's take the extreme case. What happens when n is equal to 1? That literally just means I take one instance of this random variable and average it."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "When n is pretty small, it doesn't approach a normal distribution that well. So when n is small, I mean, let's take the extreme case. What happens when n is equal to 1? That literally just means I take one instance of this random variable and average it. Well, it's just going to be that thing. So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like? Well, it's definitely not going to look like a normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "That literally just means I take one instance of this random variable and average it. Well, it's just going to be that thing. So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like? Well, it's definitely not going to look like a normal distribution. It's going to look as if you're going to have a couple of ones, you're going to have a couple of twos, you're going to have more threes like that, you're going to have no fours, you're going to have a bunch of fives, you're going to have some sixes that look like that, and then you're going to have a bunch of nines. So there your sampling distribution of the sample mean for an n of 1 is going to look, I don't care how many trials you do, it's not going to look like a normal distribution. So the central limit theorem, although, I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's definitely not going to look like a normal distribution. It's going to look as if you're going to have a couple of ones, you're going to have a couple of twos, you're going to have more threes like that, you're going to have no fours, you're going to have a bunch of fives, you're going to have some sixes that look like that, and then you're going to have a bunch of nines. So there your sampling distribution of the sample mean for an n of 1 is going to look, I don't care how many trials you do, it's not going to look like a normal distribution. So the central limit theorem, although, I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. As n gets larger, though, it starts to make sense. Let's say if we got n equals 2, and I'm all just doing this in my head, I don't know what the actual distributions would look like, but then it still would be difficult for it to become an exact normal distribution, but then you can get more instances, you could get more, you know, you might get things from all of the above, but you can only get 2 in each of your baskets that you're averaging, you're only going to get 2 numbers, right? So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the central limit theorem, although, I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. As n gets larger, though, it starts to make sense. Let's say if we got n equals 2, and I'm all just doing this in my head, I don't know what the actual distributions would look like, but then it still would be difficult for it to become an exact normal distribution, but then you can get more instances, you could get more, you know, you might get things from all of the above, but you can only get 2 in each of your baskets that you're averaging, you're only going to get 2 numbers, right? So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. So you're never going to get 7.5 as, so maybe when you plot, when you plot it, maybe, you know, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that. So it still won't be a normal distribution when n is equal to 2. So there's a couple of interesting things here."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. So you're never going to get 7.5 as, so maybe when you plot, when you plot it, maybe, you know, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that. So it still won't be a normal distribution when n is equal to 2. So there's a couple of interesting things here. So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is. The central limit theorem says as n approaches, really as it approaches infinity, then is when you get the real normal distribution. Normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a couple of interesting things here. So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is. The central limit theorem says as n approaches, really as it approaches infinity, then is when you get the real normal distribution. Normal distribution. But in kind of everyday practice, you don't have to get that much beyond n equals 2. If you get to n equals 10 or n equals 15, you're getting very close to a normal distribution. So this converges to a normal distribution very quickly."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Normal distribution. But in kind of everyday practice, you don't have to get that much beyond n equals 2. If you get to n equals 10 or n equals 15, you're getting very close to a normal distribution. So this converges to a normal distribution very quickly. Distribution. Now the other thing is you obviously want many, many trials. So this is your sample size."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this converges to a normal distribution very quickly. Distribution. Now the other thing is you obviously want many, many trials. So this is your sample size. That is your sample size. That's the size of each of your baskets. In the very first video I did on this, I took a sample as a size of 4."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is your sample size. That is your sample size. That's the size of each of your baskets. In the very first video I did on this, I took a sample as a size of 4. In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else. This is a sample size of 1. So that's our sample size."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the very first video I did on this, I took a sample as a size of 4. In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else. This is a sample size of 1. So that's our sample size. So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution. Now in order to actually see that normal distribution and actually prove it to yourself, you would have to do this many, many times. Remember, the normal distribution happens..."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's our sample size. So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution. Now in order to actually see that normal distribution and actually prove it to yourself, you would have to do this many, many times. Remember, the normal distribution happens... This is essentially... This is kind of the population or this is the random variable. That tells you all of the possibilities."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, the normal distribution happens... This is essentially... This is kind of the population or this is the random variable. That tells you all of the possibilities. In real life, we seldom know all of the possibilities. In fact, in real life, we seldom know the pure probability generating function, only if we're kind of writing it or if we're writing a computer program. Normally, we're doing samples and we're trying to estimate things."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "That tells you all of the possibilities. In real life, we seldom know all of the possibilities. In fact, in real life, we seldom know the pure probability generating function, only if we're kind of writing it or if we're writing a computer program. Normally, we're doing samples and we're trying to estimate things. So normally, there's some random variable and then maybe we'll do a bunch of... We take a bunch of samples, we take their means and we plot them and then we're going to get some type of normal distribution. Let's say we take samples of 100 and we average them. We're going to get some normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Normally, we're doing samples and we're trying to estimate things. So normally, there's some random variable and then maybe we'll do a bunch of... We take a bunch of samples, we take their means and we plot them and then we're going to get some type of normal distribution. Let's say we take samples of 100 and we average them. We're going to get some normal distribution. And in theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean. This thing is a real distribution. It's a real distribution with a real mean."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to get some normal distribution. And in theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean. This thing is a real distribution. It's a real distribution with a real mean. Its mean, it has a pure mean. Its mean, so the mean of the sampling distribution of the sample mean, we'll write it like that. Notice I didn't write it as just the X."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's a real distribution with a real mean. Its mean, it has a pure mean. Its mean, so the mean of the sampling distribution of the sample mean, we'll write it like that. Notice I didn't write it as just the X. What this is, this is actually saying that this is a real population mean. This is a real random variable mean. If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say sample size, let's say we're dealing with a world where our sample size is 10."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Notice I didn't write it as just the X. What this is, this is actually saying that this is a real population mean. This is a real random variable mean. If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say sample size, let's say we're dealing with a world where our sample size is 10. If you took all of the combinations of 10 samples from some original distribution and you averaged them out, this would describe that function. Of course, in reality, if you don't know the original distribution, you can't take an infinite samples from it. So you won't know every combination."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say sample size, let's say we're dealing with a world where our sample size is 10. If you took all of the combinations of 10 samples from some original distribution and you averaged them out, this would describe that function. Of course, in reality, if you don't know the original distribution, you can't take an infinite samples from it. So you won't know every combination. But if you did it with a thousand, if you did the trial a thousand times, so a thousand times you took 10 samples from some distribution and took a thousand averages and then plotted them, you're gonna get pretty close. You're gonna get pretty close. Now, the next thing I wanna touch on is, what happens is, we know as n approaches infinity, it becomes more of a normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you won't know every combination. But if you did it with a thousand, if you did the trial a thousand times, so a thousand times you took 10 samples from some distribution and took a thousand averages and then plotted them, you're gonna get pretty close. You're gonna get pretty close. Now, the next thing I wanna touch on is, what happens is, we know as n approaches infinity, it becomes more of a normal distribution. But as I said already, n equals 10 is pretty good and n equals 20 is even better. But we saw something in the last video that at least I find pretty interesting. Let's say we start with this crazy distribution up here."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, the next thing I wanna touch on is, what happens is, we know as n approaches infinity, it becomes more of a normal distribution. But as I said already, n equals 10 is pretty good and n equals 20 is even better. But we saw something in the last video that at least I find pretty interesting. Let's say we start with this crazy distribution up here. It really doesn't matter what distribution we're starting with. We saw in the simulation that when n is equal, let's say n is equal to five, our graph after we try, we take samples of five, average them, and we do it 10,000 times, our graph looks something like this. It's kind of wide like that."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say we start with this crazy distribution up here. It really doesn't matter what distribution we're starting with. We saw in the simulation that when n is equal, let's say n is equal to five, our graph after we try, we take samples of five, average them, and we do it 10,000 times, our graph looks something like this. It's kind of wide like that. And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that, a little bit more. So not only was it more normal, that's what the central limit theorem tells us because we're taking larger sample sizes, but it had a smaller standard deviation or a smaller variance, right? The mean is gonna be the same either case."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's kind of wide like that. And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that, a little bit more. So not only was it more normal, that's what the central limit theorem tells us because we're taking larger sample sizes, but it had a smaller standard deviation or a smaller variance, right? The mean is gonna be the same either case. But when our sample size was larger, our standard deviation became smaller. In fact, our standard deviation became smaller than our original population distribution, so let me, or our original probability density function. Let me show you that with a simulation."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean is gonna be the same either case. But when our sample size was larger, our standard deviation became smaller. In fact, our standard deviation became smaller than our original population distribution, so let me, or our original probability density function. Let me show you that with a simulation. So let me clear everything. And this simulation is as good as any. So the first thing I wanna show, or this distribution is as good as any."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me show you that with a simulation. So let me clear everything. And this simulation is as good as any. So the first thing I wanna show, or this distribution is as good as any. The first thing I wanna show you is that n of two is really not that good. So let's compare an n of two to, let's say an n of 16. So when you compare an n of two to an n of 16, you know, let's do it once."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the first thing I wanna show, or this distribution is as good as any. The first thing I wanna show you is that n of two is really not that good. So let's compare an n of two to, let's say an n of 16. So when you compare an n of two to an n of 16, you know, let's do it once. So you got one, two trials, you average them, and then it's gonna do it 16. And then it's gonna plot it down here, and average there. Let's do that 10,000 times."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when you compare an n of two to an n of 16, you know, let's do it once. So you got one, two trials, you average them, and then it's gonna do it 16. And then it's gonna plot it down here, and average there. Let's do that 10,000 times. So notice, when you took an n of two, even though we did it 10,000 times, this is not approaching a normal distribution. You can actually see it in the skew and kurtosis numbers. It has a rightward positive skew, which means it has a longer tail to the right than to the left."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do that 10,000 times. So notice, when you took an n of two, even though we did it 10,000 times, this is not approaching a normal distribution. You can actually see it in the skew and kurtosis numbers. It has a rightward positive skew, which means it has a longer tail to the right than to the left. And then it has a negative kurtosis, which means that it's a little bit, it has shorter tails and smaller peaks than a standard normal distribution. Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times. Here, and notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller, and our skew is much smaller, so we are more normal in this situation."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "It has a rightward positive skew, which means it has a longer tail to the right than to the left. And then it has a negative kurtosis, which means that it's a little bit, it has shorter tails and smaller peaks than a standard normal distribution. Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times. Here, and notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller, and our skew is much smaller, so we are more normal in this situation. But even a more interesting thing is our standard deviation is smaller, right? This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution. Now, let me do it with two, let me clear everything again."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here, and notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller, and our skew is much smaller, so we are more normal in this situation. But even a more interesting thing is our standard deviation is smaller, right? This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution. Now, let me do it with two, let me clear everything again. I like this distribution, because it's a very non-normal distribution. It looks like a bimodal distribution of some kind. And let's take a scenario where I take an n of, let's take two good n's, let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, let me do it with two, let me clear everything again. I like this distribution, because it's a very non-normal distribution. It looks like a bimodal distribution of some kind. And let's take a scenario where I take an n of, let's take two good n's, let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. Let's compare them a little bit. So if we, let's just do one trial animated, just to, it's always nice to see it. So first it's gonna do 16 of these trials and average them, and there we go."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's take a scenario where I take an n of, let's take two good n's, let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. Let's compare them a little bit. So if we, let's just do one trial animated, just to, it's always nice to see it. So first it's gonna do 16 of these trials and average them, and there we go. Then it's gonna do 25 of these trials, and then average them, and then there we go. Now let's do that, what I just did, animated, let's do it 10,000 times, miracles of computers. Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So first it's gonna do 16 of these trials and average them, and there we go. Then it's gonna do 25 of these trials, and then average them, and then there we go. Now let's do that, what I just did, animated, let's do it 10,000 times, miracles of computers. Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. The n is equal to 25 is more normal, it has less skew, slightly less skew than n is equal to 16. It has slightly less kurtosis, which means it's closer to being a normal distribution than n is equal to 16. But even more interesting, it's more squeezed in, it has a lower standard deviation."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. The n is equal to 25 is more normal, it has less skew, slightly less skew than n is equal to 16. It has slightly less kurtosis, which means it's closer to being a normal distribution than n is equal to 16. But even more interesting, it's more squeezed in, it has a lower standard deviation. The standard deviation here is 2.1, and the standard deviation here is 2.64. So that's another, I mean, you know, and I kind of touched on that in the last video, and it kind of makes sense. For every sample you do for your average, the more you put in that sample, the less standard deviation."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "But even more interesting, it's more squeezed in, it has a lower standard deviation. The standard deviation here is 2.1, and the standard deviation here is 2.64. So that's another, I mean, you know, and I kind of touched on that in the last video, and it kind of makes sense. For every sample you do for your average, the more you put in that sample, the less standard deviation. Think of the extreme case. If instead of taking 16 samples from our distribution every time, or instead of taking 25, if I were to take a million samples from this distribution every time, if I were to take a million samples from this distribution every time, that sample mean is always gonna be pretty darn close to my mean. If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm gonna get a pretty good estimate of that mean."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "For every sample you do for your average, the more you put in that sample, the less standard deviation. Think of the extreme case. If instead of taking 16 samples from our distribution every time, or instead of taking 25, if I were to take a million samples from this distribution every time, if I were to take a million samples from this distribution every time, that sample mean is always gonna be pretty darn close to my mean. If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm gonna get a pretty good estimate of that mean. The probability that a bunch of the, a million numbers are all out here is very low. So if n is a million, of course, all of my sample means when I average them are all gonna be really tightly focused around the mean itself. And it actually, and so hopefully that kind of makes sense to you as well."}, {"video_title": "Sampling distribution of the sample mean 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm gonna get a pretty good estimate of that mean. The probability that a bunch of the, a million numbers are all out here is very low. So if n is a million, of course, all of my sample means when I average them are all gonna be really tightly focused around the mean itself. And it actually, and so hopefully that kind of makes sense to you as well. If it doesn't, just think about it, or even use this tool and experiment with it, just so you can trust that that is really the case. And it actually turns out that there's a very clean formula that relates the standard deviation of the original probability distribution function to the standard deviation of the sampling distribution of the sample mean. And as you can imagine, it is a function of your sample size of how many samples you take out in every basket before you average them."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then in the next few videos, we'll actually use it to really test how well theoretical distributions explain observed ones, or how good a fit observed results are for theoretical distributions. So let's just think about it a little bit. So let's say I have some random variables. And each of them are independent, standard, normally distributed random variables. Let me just remind you what that means. So let's say I have the random variable X. If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And each of them are independent, standard, normally distributed random variables. Let me just remind you what that means. So let's say I have the random variable X. If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1. Or you could say that the expected value of X is equal to 0, or in that the variance of our random variable X is equal to 1. Or just to visualize it, is that we're sampling, when we take an instantiation of this variable, we're sampling from a normal distribution, a standardized normal distribution that looks like this. Mean of 0, and then a variance of 1, which would also mean, of course, a standard deviation of 1."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1. Or you could say that the expected value of X is equal to 0, or in that the variance of our random variable X is equal to 1. Or just to visualize it, is that we're sampling, when we take an instantiation of this variable, we're sampling from a normal distribution, a standardized normal distribution that looks like this. Mean of 0, and then a variance of 1, which would also mean, of course, a standard deviation of 1. So that could be the standard deviation, or the variance, or the standard deviation, that would be equal to 1. So a chi-squared distribution, if you just take one of these random variables, and let me define it this way. Let me define a new random variable, q, that is equal to you essentially sampling from this standard normal distribution, and then squaring whatever number you got."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Mean of 0, and then a variance of 1, which would also mean, of course, a standard deviation of 1. So that could be the standard deviation, or the variance, or the standard deviation, that would be equal to 1. So a chi-squared distribution, if you just take one of these random variables, and let me define it this way. Let me define a new random variable, q, that is equal to you essentially sampling from this standard normal distribution, and then squaring whatever number you got. So it is equal to this random variable X squared. The distribution for this random variable right here is going to be an example of the chi-squared distribution. Actually, what we're going to see in this video is that the chi-squared distribution is actually a set of distributions, depending on how many sums you have."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me define a new random variable, q, that is equal to you essentially sampling from this standard normal distribution, and then squaring whatever number you got. So it is equal to this random variable X squared. The distribution for this random variable right here is going to be an example of the chi-squared distribution. Actually, what we're going to see in this video is that the chi-squared distribution is actually a set of distributions, depending on how many sums you have. Right now, we only have one random variable that we're squaring. So this is just one of the examples, and we'll talk more about them in a second. So this right here, we could write that q is a chi-squared distributed random variable, or that we could use this notation right here."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, what we're going to see in this video is that the chi-squared distribution is actually a set of distributions, depending on how many sums you have. Right now, we only have one random variable that we're squaring. So this is just one of the examples, and we'll talk more about them in a second. So this right here, we could write that q is a chi-squared distributed random variable, or that we could use this notation right here. q is, we could write it like this. So this isn't an X anymore, this is the Greek letter chi, although it looks a lot like a curvy X. So it's a member of chi-squared, and since we're only taking one sum over here, we're only taking the sum of one independent, standard normally distributed variable, we say that this only has one degree of freedom."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right here, we could write that q is a chi-squared distributed random variable, or that we could use this notation right here. q is, we could write it like this. So this isn't an X anymore, this is the Greek letter chi, although it looks a lot like a curvy X. So it's a member of chi-squared, and since we're only taking one sum over here, we're only taking the sum of one independent, standard normally distributed variable, we say that this only has one degree of freedom. And we write that over here. So this right here is our degree of freedom. We have one degree of freedom right over there."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's a member of chi-squared, and since we're only taking one sum over here, we're only taking the sum of one independent, standard normally distributed variable, we say that this only has one degree of freedom. And we write that over here. So this right here is our degree of freedom. We have one degree of freedom right over there. Now, if we defined, so let's call this q1. Let's say I have another random variable. Let's call this q, let me do it in a different color."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have one degree of freedom right over there. Now, if we defined, so let's call this q1. Let's say I have another random variable. Let's call this q, let me do it in a different color. Let me do q2 in blue. Let's say I have another random variable q2. That is defined as, let's say I have one independent, standard normally distributed variable."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's call this q, let me do it in a different color. Let me do q2 in blue. Let's say I have another random variable q2. That is defined as, let's say I have one independent, standard normally distributed variable. I'll call that x1, and I square it. And then I have another independent, standard normally distributed variable x2, and I square it. So you can imagine, both of these guys have distributions like this, and they're independent."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is defined as, let's say I have one independent, standard normally distributed variable. I'll call that x1, and I square it. And then I have another independent, standard normally distributed variable x2, and I square it. So you can imagine, both of these guys have distributions like this, and they're independent. So to get to sample q2, you essentially sample x1 from this distribution, square that value, sample x2 from the same distribution essentially, square that value, and then add the two, and you're going to get q2. This over here, we would write, so this is q1. q2 here is a chi-squared distributed random variable with two degrees of freedom."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you can imagine, both of these guys have distributions like this, and they're independent. So to get to sample q2, you essentially sample x1 from this distribution, square that value, sample x2 from the same distribution essentially, square that value, and then add the two, and you're going to get q2. This over here, we would write, so this is q1. q2 here is a chi-squared distributed random variable with two degrees of freedom. And just to visualize the set of chi-squared distributions, let's look at this over here. So I got this off of Wikipedia. This shows us some of the probability density functions for some of the chi-squared distributions."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "q2 here is a chi-squared distributed random variable with two degrees of freedom. And just to visualize the set of chi-squared distributions, let's look at this over here. So I got this off of Wikipedia. This shows us some of the probability density functions for some of the chi-squared distributions. This first one over here, for k equal to 1, that's the degrees of freedom. So this is essentially our q1. This is our probability density function for q1."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "This shows us some of the probability density functions for some of the chi-squared distributions. This first one over here, for k equal to 1, that's the degrees of freedom. So this is essentially our q1. This is our probability density function for q1. And notice, it really spikes close to 0, and that makes sense, because if you're sampling just once from this standard normal distribution, there's a very high likelihood that you're going to get something pretty close to 0. And then if you square something close to 0, remember, these are decimals, they're going to be less than 1, pretty close to 0, it's going to become even smaller. So you have a high probability of getting a very small value."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our probability density function for q1. And notice, it really spikes close to 0, and that makes sense, because if you're sampling just once from this standard normal distribution, there's a very high likelihood that you're going to get something pretty close to 0. And then if you square something close to 0, remember, these are decimals, they're going to be less than 1, pretty close to 0, it's going to become even smaller. So you have a high probability of getting a very small value. You have high probabilities of getting values less than some threshold, this right here, less than 1 right here, so less than 1 half. And you have a very low probability of getting a large number. I mean, to get a 4, you would have to sample a 2 from this distribution, and we know that 2 is, actually, it's 2 variances, or 2 standard deviations from the mean."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have a high probability of getting a very small value. You have high probabilities of getting values less than some threshold, this right here, less than 1 right here, so less than 1 half. And you have a very low probability of getting a large number. I mean, to get a 4, you would have to sample a 2 from this distribution, and we know that 2 is, actually, it's 2 variances, or 2 standard deviations from the mean. So it's less likely. And actually, that's to get a 4. So to get even larger numbers are going to be even less likely, so that's why you see this shape over here."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "I mean, to get a 4, you would have to sample a 2 from this distribution, and we know that 2 is, actually, it's 2 variances, or 2 standard deviations from the mean. So it's less likely. And actually, that's to get a 4. So to get even larger numbers are going to be even less likely, so that's why you see this shape over here. Now, when you have 2 degrees of freedom, it moderates a little bit. This is the shape, this blue line right here, is the shape of q2. And notice, you're a little bit less likely to get values close to 0, and a little bit more likely to get numbers further out, but it still is kind of shifted, or heavily weighted, towards small numbers."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So to get even larger numbers are going to be even less likely, so that's why you see this shape over here. Now, when you have 2 degrees of freedom, it moderates a little bit. This is the shape, this blue line right here, is the shape of q2. And notice, you're a little bit less likely to get values close to 0, and a little bit more likely to get numbers further out, but it still is kind of shifted, or heavily weighted, towards small numbers. And then if we had another random variable, another chi squared distributed random variable, so then we have, let's say, q3, and let's define it as the sum of 3 of these independent variables, each of them that have a standard normal distribution. So x1, x2 squared, plus x3 squared. Then all of a sudden, our q3, this is q2 right here, has a chi squared distribution with 3 degrees of freedom."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And notice, you're a little bit less likely to get values close to 0, and a little bit more likely to get numbers further out, but it still is kind of shifted, or heavily weighted, towards small numbers. And then if we had another random variable, another chi squared distributed random variable, so then we have, let's say, q3, and let's define it as the sum of 3 of these independent variables, each of them that have a standard normal distribution. So x1, x2 squared, plus x3 squared. Then all of a sudden, our q3, this is q2 right here, has a chi squared distribution with 3 degrees of freedom. And so this guy right over here, that will be this green line, maybe I should have done this in green. This will be this green line over here. And then notice, now it's starting to become a little bit more likely that you get values in this range over here, because you're taking the sum."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then all of a sudden, our q3, this is q2 right here, has a chi squared distribution with 3 degrees of freedom. And so this guy right over here, that will be this green line, maybe I should have done this in green. This will be this green line over here. And then notice, now it's starting to become a little bit more likely that you get values in this range over here, because you're taking the sum. Each of these are going to be pretty small values, but you're taking the sum, so it starts to shift it a little over to the right. And so the more degrees of freedom you have, the further this lump starts to move to the right, and to some degree, the more symmetric it gets. And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0, because we're always just squaring these values."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then notice, now it's starting to become a little bit more likely that you get values in this range over here, because you're taking the sum. Each of these are going to be pretty small values, but you're taking the sum, so it starts to shift it a little over to the right. And so the more degrees of freedom you have, the further this lump starts to move to the right, and to some degree, the more symmetric it gets. And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0, because we're always just squaring these values. Each of these guys can have values below 0. They're normally distributed. They could have negative values."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0, because we're always just squaring these values. Each of these guys can have values below 0. They're normally distributed. They could have negative values. But since we're squaring and taking the sum of squares, this is always going to be positive. And the place that this is going to be useful, and we're going to see in the next few videos, is in measuring essentially error from an expected value. And if you take this total error, you can figure out the probability of getting that total error if you hold some parameters, or if you assume some parameters."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "They could have negative values. But since we're squaring and taking the sum of squares, this is always going to be positive. And the place that this is going to be useful, and we're going to see in the next few videos, is in measuring essentially error from an expected value. And if you take this total error, you can figure out the probability of getting that total error if you hold some parameters, or if you assume some parameters. And we'll talk more about it in the next video. Now with that said, I just want to show you how to read a chi squared distribution table. So if I were to ask you if this is our distribution, let me pick this blue one right here."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you take this total error, you can figure out the probability of getting that total error if you hold some parameters, or if you assume some parameters. And we'll talk more about it in the next video. Now with that said, I just want to show you how to read a chi squared distribution table. So if I were to ask you if this is our distribution, let me pick this blue one right here. So over here we have 2 degrees of freedom, because we're adding 2 of these guys right here. If I were to ask you what is the probability of Q2 being greater than 2.41? And I'm picking that value for a reason."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I were to ask you if this is our distribution, let me pick this blue one right here. So over here we have 2 degrees of freedom, because we're adding 2 of these guys right here. If I were to ask you what is the probability of Q2 being greater than 2.41? And I'm picking that value for a reason. So I want the probability of Q2 being greater than 2.41. What I want to do is I'll look at a chi squared table like this. Q2 is a version of chi squared with 2 degrees of freedom."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm picking that value for a reason. So I want the probability of Q2 being greater than 2.41. What I want to do is I'll look at a chi squared table like this. Q2 is a version of chi squared with 2 degrees of freedom. So I look at this row right here under 2 degrees of freedom, and I want the probability of getting a value above 2.41. And I picked 2.41 because it's actually at this table. And so most of these chi squared, the reason why we have these weird numbers like this instead of whole numbers or easy to read fractions, is it's actually driven by the p-value."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Q2 is a version of chi squared with 2 degrees of freedom. So I look at this row right here under 2 degrees of freedom, and I want the probability of getting a value above 2.41. And I picked 2.41 because it's actually at this table. And so most of these chi squared, the reason why we have these weird numbers like this instead of whole numbers or easy to read fractions, is it's actually driven by the p-value. It's driven by the probability of getting something larger than that value. So normally you would look at it the other way. You'd say, OK, if I want to see what chi squared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that, then I would look up 2.41."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so most of these chi squared, the reason why we have these weird numbers like this instead of whole numbers or easy to read fractions, is it's actually driven by the p-value. It's driven by the probability of getting something larger than that value. So normally you would look at it the other way. You'd say, OK, if I want to see what chi squared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that, then I would look up 2.41. But I'm doing it the other way just for the sake of this video. So if I want the probability of this random variable right here being greater than 2.41, or its p-value, we read it right here, it is 30%. And just to visualize it on this chart, this chi squared distribution, this was q2, the blue one over here, 2.41 is going to sit, let's see, this is 3, this is 2.5, so 2.41 is going to be someplace right around here."}, {"video_title": "Chi-square distribution introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "You'd say, OK, if I want to see what chi squared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that, then I would look up 2.41. But I'm doing it the other way just for the sake of this video. So if I want the probability of this random variable right here being greater than 2.41, or its p-value, we read it right here, it is 30%. And just to visualize it on this chart, this chi squared distribution, this was q2, the blue one over here, 2.41 is going to sit, let's see, this is 3, this is 2.5, so 2.41 is going to be someplace right around here. So essentially what that table is telling us is this entire area under this blue line right here, what is that? And that right there is going to be 30% of, well, it's going to be 0.3. Or you could view it as 30% of the entire area under this curve, because obviously all the probabilities have to add up to 1."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "At the Olympic Games, many events have several rounds of competition. One of these is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the semi-final round at the 2012 Olympics. The lower dot plot shows the times of the same eight swimmers, but in the final round. Which pieces of information can be gathered from these dot plots? In the semi-final round, we see that these are the eight times of the eight swimmers. 53 swimmers finished in exactly 53.5 seconds."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "The lower dot plot shows the times of the same eight swimmers, but in the final round. Which pieces of information can be gathered from these dot plots? In the semi-final round, we see that these are the eight times of the eight swimmers. 53 swimmers finished in exactly 53.5 seconds. One swimmer finished in 53.7 seconds right here. And one swimmer right over here finished in 52.7 seconds. We can think about similar things for each of these dots."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "53 swimmers finished in exactly 53.5 seconds. One swimmer finished in 53.7 seconds right here. And one swimmer right over here finished in 52.7 seconds. We can think about similar things for each of these dots. Now in the final round, one swimmer here went much, much, much faster. So this is in 52.2 seconds. While this swimmer right over here went slower."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "We can think about similar things for each of these dots. Now in the final round, one swimmer here went much, much, much faster. So this is in 52.2 seconds. While this swimmer right over here went slower. We don't know which dot he was up here, but regardless of which dot he was up here, this dot took more time than all of these dots. So his time definitely got worse. This is at 53.8 seconds."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "While this swimmer right over here went slower. We don't know which dot he was up here, but regardless of which dot he was up here, this dot took more time than all of these dots. So his time definitely got worse. This is at 53.8 seconds. Let's look at the statements and see which of these apply. The swimmers had faster times on average in the finals. Is this true?"}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "This is at 53.8 seconds. Let's look at the statements and see which of these apply. The swimmers had faster times on average in the finals. Is this true? Faster times on average in the finals. If we look at the finals right over here, we could take each of these times, add them up and then divide by eight the number of times we have. Let's see if we can get an intuition for where this is."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "Is this true? Faster times on average in the finals. If we look at the finals right over here, we could take each of these times, add them up and then divide by eight the number of times we have. Let's see if we can get an intuition for where this is. We're really just comparing these two plots or these two distributions. Let's see. If all the data was these three points and these three points, we could intuit that the mean would be right around there."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "Let's see if we can get an intuition for where this is. We're really just comparing these two plots or these two distributions. Let's see. If all the data was these three points and these three points, we could intuit that the mean would be right around there. It would be around 53.2 or 53.3 seconds, right around there. Then we have this point and this point. If you just found the mean of that point and that point, so halfway between that point and that point would get you right around there."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "If all the data was these three points and these three points, we could intuit that the mean would be right around there. It would be around 53.2 or 53.3 seconds, right around there. Then we have this point and this point. If you just found the mean of that point and that point, so halfway between that point and that point would get you right around there. The mean of those two points would bring down the mean a little bit. Once again, I'm not figuring out the exact number, but maybe it would be around 53.1 or 53.2 seconds. That's my intuition for the mean of the final round."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "If you just found the mean of that point and that point, so halfway between that point and that point would get you right around there. The mean of those two points would bring down the mean a little bit. Once again, I'm not figuring out the exact number, but maybe it would be around 53.1 or 53.2 seconds. That's my intuition for the mean of the final round. Now let's think about the mean of the semifinal round. Let's just look at these bottom five dots. If you find the mean, you could intuit it would be maybe right around, someplace around here, pretty close to around 53.3 seconds."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "That's my intuition for the mean of the final round. Now let's think about the mean of the semifinal round. Let's just look at these bottom five dots. If you find the mean, you could intuit it would be maybe right around, someplace around here, pretty close to around 53.3 seconds. Then you have all these other ones that are at 53.5 and 53.3, which will bring the mean even higher. I think it's fair to say that the mean in the final round, the time, is less than the mean up here. You could calculate it yourself, but I'm just trying to look at the distributions and get an intuition here."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "If you find the mean, you could intuit it would be maybe right around, someplace around here, pretty close to around 53.3 seconds. Then you have all these other ones that are at 53.5 and 53.3, which will bring the mean even higher. I think it's fair to say that the mean in the final round, the time, is less than the mean up here. You could calculate it yourself, but I'm just trying to look at the distributions and get an intuition here. At least in this case, it looks pretty clear that the swimmers had faster times on average in the finals. It took them less time. One of the swimmers was disqualified from the finals."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "You could calculate it yourself, but I'm just trying to look at the distributions and get an intuition here. At least in this case, it looks pretty clear that the swimmers had faster times on average in the finals. It took them less time. One of the swimmers was disqualified from the finals. That's not true. We have eight swimmers in the semifinal round and we have eight swimmers in the final round, so that one's not true. The times in the finals vary noticeably more than the times in the semifinals."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "One of the swimmers was disqualified from the finals. That's not true. We have eight swimmers in the semifinal round and we have eight swimmers in the final round, so that one's not true. The times in the finals vary noticeably more than the times in the semifinals. That does look to be true. We see in the semifinals a lot of the times were clumped up right around here, at 53.3 seconds and 53.5 seconds. The high time isn't as high as this time."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "The times in the finals vary noticeably more than the times in the semifinals. That does look to be true. We see in the semifinals a lot of the times were clumped up right around here, at 53.3 seconds and 53.5 seconds. The high time isn't as high as this time. The low time isn't as low there. The final round is definitely, definitely, they vary noticeably more. Individually, the swimmers all swam faster in the finals than they did in the semifinals."}, {"video_title": "Comparing distributions with dot plots (example problem)   7th grade   Khan Academy.mp3", "Sentence": "The high time isn't as high as this time. The low time isn't as low there. The final round is definitely, definitely, they vary noticeably more. Individually, the swimmers all swam faster in the finals than they did in the semifinals. That's not true. Whoever this was, clearly they were one of these data points up here. This data point took more time than all of these data points."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say that we have a random variable x. Maybe it represents the height of a randomly selected person walking out of the mall or something like that. And right over here, we have its probability distribution. And I've drawn it as a bell curve, as a normal distribution right over here, but it could have many other distributions, but for the visualization's sake, it's a normal one in this example. And I've also drawn the mean of this distribution right over here, and I've also drawn one standard deviation above the mean and one standard deviation below the mean. What we're going to do in this video is think about how does this distribution, and in particular, how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable. So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And I've drawn it as a bell curve, as a normal distribution right over here, but it could have many other distributions, but for the visualization's sake, it's a normal one in this example. And I've also drawn the mean of this distribution right over here, and I've also drawn one standard deviation above the mean and one standard deviation below the mean. What we're going to do in this video is think about how does this distribution, and in particular, how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable. So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. So let's say we add, so we're gonna add some constant here. I'll do a lowercase k. This is not a random variable. This is a constant."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. So let's say we add, so we're gonna add some constant here. I'll do a lowercase k. This is not a random variable. This is a constant. It could be the number 10. So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason. Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x?"}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This is a constant. It could be the number 10. So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason. Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? So we could visualize that. So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example. And maybe k is quite large."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? So we could visualize that. So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example. And maybe k is quite large. Maybe it looks something like that. This is my distribution for my random variable y here. And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And maybe k is quite large. Maybe it looks something like that. This is my distribution for my random variable y here. And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. The mean is going to now be k larger. So we can write that down. We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. The mean is going to now be k larger. So we can write that down. We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. But has the standard deviation changed? Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change. So for our random variable x, this length right over here is one standard deviation."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. But has the standard deviation changed? Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change. So for our random variable x, this length right over here is one standard deviation. Well, that's also going to be the same as one standard deviation here. This is one standard deviation here. This is going to be the same as our standard deviation for our random variable y."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So for our random variable x, this length right over here is one standard deviation. Well, that's also going to be the same as one standard deviation here. This is one standard deviation here. This is going to be the same as our standard deviation for our random variable y. And so we can say the standard deviation of y, of our random variable y, is equal to the standard deviation of our random variable x. So if you just add to a random variable, it would change the mean, but not the standard deviation. You see it visually here."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This is going to be the same as our standard deviation for our random variable y. And so we can say the standard deviation of y, of our random variable y, is equal to the standard deviation of our random variable x. So if you just add to a random variable, it would change the mean, but not the standard deviation. You see it visually here. Now, what if you were to scale a random variable? So what if I have another random variable? I don't know, let's call it z."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You see it visually here. Now, what if you were to scale a random variable? So what if I have another random variable? I don't know, let's call it z. And let's say z is equal to some constant, some constant times x. And so remember, this isn't, the k is not a random variable, it's just gonna be a number. It could be, say, the number two."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "I don't know, let's call it z. And let's say z is equal to some constant, some constant times x. And so remember, this isn't, the k is not a random variable, it's just gonna be a number. It could be, say, the number two. Well, let's think about what would happen. So let me redraw the distribution for our random variable x. So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It could be, say, the number two. Well, let's think about what would happen. So let me redraw the distribution for our random variable x. So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. So I can do that with my little drawing tool here. Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two. So let me align the axes here so that we can appreciate this."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. So I can do that with my little drawing tool here. Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two. So let me align the axes here so that we can appreciate this. So it's going to look something like this. It's going to look something like this when you scale the random variable. This is what the distribution of our random variable z is going to look like."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let me align the axes here so that we can appreciate this. So it's going to look something like this. It's going to look something like this when you scale the random variable. This is what the distribution of our random variable z is going to look like. I'll do it in the z's color so that it's clear. And so you can see two things. One, the mean for sure shifted."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This is what the distribution of our random variable z is going to look like. I'll do it in the z's color so that it's clear. And so you can see two things. One, the mean for sure shifted. The mean here for sure got pushed out. It definitely got scaled up. But also, we see that the standard deviations got scaled, that the standard deviation right over here of z, that this has been scaled, it actually turns out that it's been scaled by a factor of k. So this is going to be equal to k times the standard deviation of our random variable x."}, {"video_title": "Impact of transforming (scaling and shifting) random variables   AP Statistics   Khan Academy.mp3", "Sentence": "One, the mean for sure shifted. The mean here for sure got pushed out. It definitely got scaled up. But also, we see that the standard deviations got scaled, that the standard deviation right over here of z, that this has been scaled, it actually turns out that it's been scaled by a factor of k. So this is going to be equal to k times the standard deviation of our random variable x. And it turns out that our mean right over here, so let me write that too, that our mean of our random variable z is going to be equal to, that's also going to be scaled up, times, or it's gonna be k times the mean of our random variable x. So the big takeaways here, if you have one random variable that's constructed by adding a constant to another random variable, it's going to shift the mean by that constant, but it's not going to affect the standard deviation. If you try to scale, if you multiply one random variable to get another one by some constant, then that's going to affect both the standard deviation, it's gonna scale that, and it's going to affect the mean."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So they took a sample of 200 residents to test the null hypothesis, is that the unemployment rate is the same as the national one versus the alternative hypothesis, which is that the unemployment rate is not the same as the national, where P is the proportion of residents in the town that are unemployed. The sample included 22 residents who were unemployed. Assuming that the conditions for inference have been met, and so that's the random, normal, and independence conditions that we've talked about in previous videos, identify the correct test statistic for this significance test. So let me just, I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town, that's what this P represents, is the same as the national unemployment. And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak. And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So let me just, I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town, that's what this P represents, is the same as the national unemployment. And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak. And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. And so what we would do is we would set some type of a significance level. We would assume that the mayor of the town sets it. Let's say he sets or she sets a significance level of 0.5."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. And so what we would do is we would set some type of a significance level. We would assume that the mayor of the town sets it. Let's say he sets or she sets a significance level of 0.5. And then what we wanna do is conduct the experiment. So this is the entire population of the town. They take a sample of 200 people."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say he sets or she sets a significance level of 0.5. And then what we wanna do is conduct the experiment. So this is the entire population of the town. They take a sample of 200 people. So this is our sample. N is equal to 200. Since it met the independence condition, we'll assume that this is less than 10% of the population."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "They take a sample of 200 people. So this is our sample. N is equal to 200. Since it met the independence condition, we'll assume that this is less than 10% of the population. And we calculate a sample statistic here. And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion. And we figure out that it is 22 out of the 200 people in the sample are unemployed."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Since it met the independence condition, we'll assume that this is less than 10% of the population. And we calculate a sample statistic here. And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion. And we figure out that it is 22 out of the 200 people in the sample are unemployed. So this is 0.11. Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And we figure out that it is 22 out of the 200 people in the sample are unemployed. So this is 0.11. Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. But how do you figure out this probability? Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it? And then we could say, what's the probability of getting that many standard deviations or further from the true proportion?"}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. But how do you figure out this probability? Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it? And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? We could use a z-table to do that. And so what we wanna do is figure out the number of standard deviations. And so that would be a z-statistic."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? We could use a z-table to do that. And so what we wanna do is figure out the number of standard deviations. And so that would be a z-statistic. And so how do we figure it out? Well, we can figure out the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And so that would be a z-statistic. And so how do we figure it out? Well, we can figure out the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. And we can figure that out. Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is. But here we're assuming a population proportion."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. And we can figure that out. Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is. But here we're assuming a population proportion. So we're assuming it is 0.08. And then we'll multiply that times one minus 0.08. So we'll multiply that times 0.92."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "But here we're assuming a population proportion. So we're assuming it is 0.08. And then we'll multiply that times one minus 0.08. So we'll multiply that times 0.92. And this comes straight from, we've seen it in previous videos, the standard deviation of the sampling distribution of sample proportions. And then you divide that by n, which is 200 right over here. And we could get a calculator out to figure this out."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So we'll multiply that times 0.92. And this comes straight from, we've seen it in previous videos, the standard deviation of the sampling distribution of sample proportions. And then you divide that by n, which is 200 right over here. And we could get a calculator out to figure this out. But this will give us some value, which just says how many standard deviations away from 0.08 is 0.11. And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion. And then that will give us our p-value, which we can compare to significance level."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And we could get a calculator out to figure this out. But this will give us some value, which just says how many standard deviations away from 0.08 is 0.11. And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion. And then that will give us our p-value, which we can compare to significance level. Sometimes you will see a formula that looks something like this. And you say, hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis."}, {"video_title": "Calculating a z statistic in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And then that will give us our p-value, which we can compare to significance level. Sometimes you will see a formula that looks something like this. And you say, hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis. That's what this little zero says. Now, this is the assumed population proportion from the null hypothesis. And you divide that by the standard deviation, the assumed standard deviation of the sampling distribution of the sample proportions."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "The government created the following stem and leaf plot, showing the number of turtles at each major zoo in the country. How many zoos have fewer than 46 turtles? So what the stem and leaf plot does is it gives us the first digit in each number, and essentially you could say it called us the tens place, and then it gives us the ones place. So there was only one zoo that had four turtles, so you could view this as zero, zero, four, or four turtles. Then there's, let's see, so everything here, the tens place is a one, so this number right over here is really an 11, this is a 14, this right over here would be a 16, that's a 16, and so forth and so on, this would be a 17, 18. All of this, this is 23, this is 23, this is 26, because we have our tens place right over here, this is the first digit. So let's go ahead and answer the question, how many zoos had fewer than 46 turtles?"}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So there was only one zoo that had four turtles, so you could view this as zero, zero, four, or four turtles. Then there's, let's see, so everything here, the tens place is a one, so this number right over here is really an 11, this is a 14, this right over here would be a 16, that's a 16, and so forth and so on, this would be a 17, 18. All of this, this is 23, this is 23, this is 26, because we have our tens place right over here, this is the first digit. So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? So there are no zoos that had 40 anything turtles, and so all of these zoos here, so all of these had 30 something turtles, this, these had 20 something turtles, these have, in the teens, this has single digits. So it's literally as many zoos as we have listed here. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? So there are no zoos that had 40 anything turtles, and so all of these zoos here, so all of these had 30 something turtles, this, these had 20 something turtles, these have, in the teens, this has single digits. So it's literally as many zoos as we have listed here. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. 17 zoos have fewer than 46 turtles. Let's do another one. The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. 17 zoos have fewer than 46 turtles. Let's do another one. The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. What was the smallest number of coconuts at any one grocery store? So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores. So at any one grocery store, the smallest number, well, that's this one right over here."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. What was the smallest number of coconuts at any one grocery store? So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores. So at any one grocery store, the smallest number, well, that's this one right over here. And remember, it's not two, we have our tens places right over here, it's a one. So this right over here represents 12 coconuts at that store. So we'll put 12 right over here."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So at any one grocery store, the smallest number, well, that's this one right over here. And remember, it's not two, we have our tens places right over here, it's a one. So this right over here represents 12 coconuts at that store. So we'll put 12 right over here. Let's try out another one. A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores. How many department stores have exactly seven watches?"}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So we'll put 12 right over here. Let's try out another one. A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores. How many department stores have exactly seven watches? Well, that's only this one right over here, zero, seven, watches, this one right, this, this, and that one are not seven. This is representing 17 because it's in the row with one at the beginning. This right over here represents 27 because it's in the row with the two at the beginning."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "How many department stores have exactly seven watches? Well, that's only this one right over here, zero, seven, watches, this one right, this, this, and that one are not seven. This is representing 17 because it's in the row with one at the beginning. This right over here represents 27 because it's in the row with the two at the beginning. So there's only one store that has exactly seven watches. Let's do one more, this is kind of fun. A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "This right over here represents 27 because it's in the row with the two at the beginning. So there's only one store that has exactly seven watches. Let's do one more, this is kind of fun. A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. How many zoos have more than 24 tigers? So we can ignore the zeros and the teens and we get into the 20s. This is 25, so that meets the criteria, and then you go to 28, 29."}, {"video_title": "Reading stem and leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. How many zoos have more than 24 tigers? So we can ignore the zeros and the teens and we get into the 20s. This is 25, so that meets the criteria, and then you go to 28, 29. So all of these, all of these in the 30s, and all of these right over here, this three zero, this doesn't mean zero tigers, this is 30 tigers. This is 40 tigers. So we count one, two, three, four, five, six, seven, eight, nine, nine zoos have more than 24 tigers."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I suggested that, hey, why don't you visualize this, draw, graph this probability distribution, this binomial probability distribution. And when I thought about it, I said, well, you know, I too would enjoy graphing it, and we might as well do it together, because whenever you graph these things, it makes it very visual, and kind of the shape of a binomial distribution like this. So let's do that. So let me maybe move over to the right a little bit. I really just need to be able to keep track of these things right over here. So let me draw some lines. So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me maybe move over to the right a little bit. I really just need to be able to keep track of these things right over here. So let me draw some lines. So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. So let's do that. See, the highest one is a little over 30%, 32%. So maybe we'll go as high as 40% here."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. So let's do that. See, the highest one is a little over 30%, 32%. So maybe we'll go as high as 40% here. 40%, and then this would be 20%. 20, that looks about halfway, 20%. This would be 10%."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So maybe we'll go as high as 40% here. 40%, and then this would be 20%. 20, that looks about halfway, 20%. This would be 10%. 10%, and this would be 30%. 30%. And then in this axis, let's do the different values that the random variable could take on."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would be 10%. 10%, and this would be 30%. 30%. And then in this axis, let's do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable taking on the value one. Zero, one."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then in this axis, let's do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable taking on the value one. Zero, one. The random variable taking on two. Two, we're almost there. Let's see, three."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Zero, one. The random variable taking on two. Two, we're almost there. Let's see, three. And then four. Four, and then five. Five, and then finally six."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see, three. And then four. Four, and then five. Five, and then finally six. Take x equals six. And then six, and now let's just graph all of these. So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Five, and then finally six. Take x equals six. And then six, and now let's just graph all of these. So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. Actually, let me do it in that green color. So let me make sure, so in that green color, you're gonna have just a little bit of a showing. One as well is kind of barely a showing, so it shows up a little bit more."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. Actually, let me do it in that green color. So let me make sure, so in that green color, you're gonna have just a little bit of a showing. One as well is kind of barely a showing, so it shows up a little bit more. So let me draw it like that. That is 1% right over there. Now two is 6%, which on this scale is gonna be about, it's about that high."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "One as well is kind of barely a showing, so it shows up a little bit more. So let me draw it like that. That is 1% right over there. Now two is 6%, which on this scale is gonna be about, it's about that high. So we draw it like that. So that is two, so that is 6%, right there. X equaling three, 18.5% shot of that happening."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now two is 6%, which on this scale is gonna be about, it's about that high. So we draw it like that. So that is two, so that is 6%, right there. X equaling three, 18.5% shot of that happening. So 18.5, it gets us right about, it's a hand-drawn chart, or histogram, so you have to bear with me. So it's roughly there. And then four was 32.4%."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "X equaling three, 18.5% shot of that happening. So 18.5, it gets us right about, it's a hand-drawn chart, or histogram, so you have to bear with me. So it's roughly there. And then four was 32.4%. So that is up here. So 32.4% is right, looks like that. So let me shade that in."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then four was 32.4%. So that is up here. So 32.4% is right, looks like that. So let me shade that in. 32.4%, and then five was 30.3%. So 30.3%, and slightly lower, just like that. And it looks like this."}, {"video_title": "Graphing basketball binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me shade that in. 32.4%, and then five was 30.3%. So 30.3%, and slightly lower, just like that. And it looks like this. 30.3%, and finally six is 11.8%. So really this whole video was just an exercise in making a histogram, but it's useful because to actually visualize what the distribution looks like, and what's really interesting is to think about, well, how does this change as you change the free throw percentage, or as you change the number of shots you take, how does this change this binomial distribution? And you could do that on a spreadsheet and actually see how that all works out."}, {"video_title": "Interpreting two-way tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "They give us this, as they say, the two-way table of column relative frequencies. So for example, this column right over here is men. The column total is one, or you could say 100%. And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other."}, {"video_title": "Interpreting two-way tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see."}, {"video_title": "Interpreting two-way tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney."}, {"video_title": "Interpreting two-way tables   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney. So a man was more likely. There's a, if you randomly picked a man who voted, there was a 52% chance they voted for Romney, while if you randomly picked a woman, there was a 43%, a woman who voted, there was a 43% chance that she voted for Romney. So yes, male voters were more likely to vote for Romney than female voters."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Consider the density curve below. And so we have a density curve that describes the probability distribution for a continuous random variable. This random variable can take on values from one to five and has an equal probability of taking on any of these values from one to five. Find the probability that x is less than four. So x can go from one to four, there's no probability that it'll be less than one. So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Find the probability that x is less than four. So x can go from one to four, there's no probability that it'll be less than one. So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 centimeters."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170?"}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that out?"}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is roughly 68%. It's closer to 68.2%. For our purposes, 68 will work fine."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is roughly 68%. It's closer to 68.2%. For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is going to be equal to one. And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately."}, {"video_title": "Probabilities from density curves   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is going to be equal to one. And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately. And so that's going to be approximately 0.16. If you want a slightly more precise value, you could use a z-table. The area below one standard deviation above the mean will be closer to about 84.1%, in which case this would be about 15.9%, or 0.159."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Exclude the median when computing the quartiles. All right, let's see if we can do this. So we have a bunch of data here, and they say if it helps, you might drag the numbers to put them in a different order so we can drag these numbers around, which is useful because we will want to order them. The order isn't checked with your answer. I'm doing this off of the Khan Academy exercises, so I don't have my drawing tablet here. I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with. Anyway, so let's do this."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The order isn't checked with your answer. I'm doing this off of the Khan Academy exercises, so I don't have my drawing tablet here. I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with. Anyway, so let's do this. Let's order this thing so we can figure out the range of numbers. What's the lowest and what's the highest? So let's see, there's a seven here."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Anyway, so let's do this. Let's order this thing so we can figure out the range of numbers. What's the lowest and what's the highest? So let's see, there's a seven here. Then let's see, we have some eights. We've got some eights going on. And then we have some nines."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's see, there's a seven here. Then let's see, we have some eights. We've got some eights going on. And then we have some nines. Actually, we have a bunch of nines. We have four nines here. We have some nines."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then we have some nines. Actually, we have a bunch of nines. We have four nines here. We have some nines. And then let's see, 13 is the largest number. There we go, we've ordered the numbers. So our smallest number is seven."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have some nines. And then let's see, 13 is the largest number. There we go, we've ordered the numbers. So our smallest number is seven. And this is what the whiskers are useful for, for helping us figure out the entire range of numbers. Our smallest number is seven. Our largest number is 13."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So our smallest number is seven. And this is what the whiskers are useful for, for helping us figure out the entire range of numbers. Our smallest number is seven. Our largest number is 13. So we know the range. Now let's plot the median. And this will help us."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Our largest number is 13. So we know the range. Now let's plot the median. And this will help us. One's getting this center line of our box, but then also we need to do that to figure out what these other lines are that kind of define the box, to define the middle two fourths of our number, of our data, or the middle two quartiles, roughly the middle two quartiles. It depends how some of the numbers work out. But this middle number, this middle line is going to be the median of our entire data set."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And this will help us. One's getting this center line of our box, but then also we need to do that to figure out what these other lines are that kind of define the box, to define the middle two fourths of our number, of our data, or the middle two quartiles, roughly the middle two quartiles. It depends how some of the numbers work out. But this middle number, this middle line is going to be the median of our entire data set. Now the median is just the middle number. If we sort them in order, median is just the middle number. We have 11 numbers here."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But this middle number, this middle line is going to be the median of our entire data set. Now the median is just the middle number. If we sort them in order, median is just the middle number. We have 11 numbers here. So the middle one is gonna have five on either side. So it's gonna be this nine. If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have 11 numbers here. So the middle one is gonna have five on either side. So it's gonna be this nine. If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. And then to find the median, you would have found the mean of those two. If that last sentence was confusing, watch the videos on Khan Academy on median. And I go into much more detail on that."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. And then to find the median, you would have found the mean of those two. If that last sentence was confusing, watch the videos on Khan Academy on median. And I go into much more detail on that. But here I have 11 numbers. So my median is going to be the middle one. It has five larger, five less."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I go into much more detail on that. But here I have 11 numbers. So my median is going to be the middle one. It has five larger, five less. It's this nine right over here. If I had my pen tablet, I would circle it. So it's this nine."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It has five larger, five less. It's this nine right over here. If I had my pen tablet, I would circle it. So it's this nine. That is the median. And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half? And they told us to exclude the median when we compute the quartiles."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it's this nine. That is the median. And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half? And they told us to exclude the median when we compute the quartiles. So this was the median. Let's ignore that. So let's look at all the numbers below that."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And they told us to exclude the median when we compute the quartiles. So this was the median. Let's ignore that. So let's look at all the numbers below that. So this nine, eight, eight, eight, and seven. So we have five numbers. What's the median of these five numbers?"}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's look at all the numbers below that. So this nine, eight, eight, eight, and seven. So we have five numbers. What's the median of these five numbers? Well, the median's the middle number. That is eight. So the beginning of our second quartile is gonna be at eight right over there."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "What's the median of these five numbers? Well, the median's the middle number. That is eight. So the beginning of our second quartile is gonna be at eight right over there. And we do the same thing for our third quartile. Remember, this was our median of our entire data set. Let's exclude it."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the beginning of our second quartile is gonna be at eight right over there. And we do the same thing for our third quartile. Remember, this was our median of our entire data set. Let's exclude it. Let's look at the top half of the numbers, so to speak. And there's five numbers here in order. So the middle one, the median of this, is 10."}, {"video_title": "Box and whisker plot exercise example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's exclude it. Let's look at the top half of the numbers, so to speak. And there's five numbers here in order. So the middle one, the median of this, is 10. So that's gonna be the top of our second quartile. And just like that, we're done. We have constructed our box-end Whitaker plot."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "He asks his listeners to visit his website and participate in the poll. The poll shows that 89% of about 200 respondents love his show. What is the most concerning source of bias in this scenario? And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer?"}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week?"}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked?"}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario?"}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here."}, {"video_title": "Examples of bias in surveys   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here. There were three people who didn't respond, but that's not as big of a deal. Voluntary response sampling, well, he didn't ask people, like in the last example, like, hey, if you can go here and fill it out, I guess there is actually, actually, no, take that back. There is a little bit of voluntary response here, where he goes to these 100 people and he asks them to respond, and so you have the 97 people who choose to respond."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we see this first example. A statistician recorded the length of each of Pixar's first 14 films. The statistician made a dot plot. Each dot is a film, a histogram, and a box plot to display the running time data. Which display could be used to find the median? To find the median. All right, so let's look at these displays."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Each dot is a film, a histogram, and a box plot to display the running time data. Which display could be used to find the median? To find the median. All right, so let's look at these displays. So over here we see the 14, this is the dot plot. We have a dot for each of the 14 films. So one film had a running time of 81 minutes."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "All right, so let's look at these displays. So over here we see the 14, this is the dot plot. We have a dot for each of the 14 films. So one film had a running time of 81 minutes. We see that there. One film had a running time of 92. One had a running time of 93."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So one film had a running time of 81 minutes. We see that there. One film had a running time of 92. One had a running time of 93. We see one had a running time of 95. We see two had running times of 96 minutes. And so on and so forth."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "One had a running time of 93. We see one had a running time of 95. We see two had running times of 96 minutes. And so on and so forth. So I claim that I could use this to figure out the median because I could make a list of all of the running times of the films, I could order them, and then I could find the middle value. I could literally make a list. I could write down 81 and then write down 92, then write down 93, then write down 95, then I could write down 96 twice, and then I could write down 98, then I could write down 100."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And so on and so forth. So I claim that I could use this to figure out the median because I could make a list of all of the running times of the films, I could order them, and then I could find the middle value. I could literally make a list. I could write down 81 and then write down 92, then write down 93, then write down 95, then I could write down 96 twice, and then I could write down 98, then I could write down 100. I see where you, I think you see where this is going. I could write out the entire list and then I could find the middle value. So the dot plot I could definitely use to find the median."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I could write down 81 and then write down 92, then write down 93, then write down 95, then I could write down 96 twice, and then I could write down 98, then I could write down 100. I see where you, I think you see where this is going. I could write out the entire list and then I could find the middle value. So the dot plot I could definitely use to find the median. Now what about the histogram? This is the histogram right over here. And the key here is for median, to figure out a median I just need to figure out a list of numbers."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the dot plot I could definitely use to find the median. Now what about the histogram? This is the histogram right over here. And the key here is for median, to figure out a median I just need to figure out a list of numbers. I need to figure out a list of numbers. So here, I don't know, you know, they say I have one film that's between 80 and 85, but I don't know its exact running time. Its running time might have been 81 minutes."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And the key here is for median, to figure out a median I just need to figure out a list of numbers. I need to figure out a list of numbers. So here, I don't know, you know, they say I have one film that's between 80 and 85, but I don't know its exact running time. Its running time might have been 81 minutes. Its running time might have been 84 minutes. So I don't know here, and so I can't really make a list of the running times of the films and find the middle value. So I don't think I'm gonna be able to do it using the histogram."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Its running time might have been 81 minutes. Its running time might have been 84 minutes. So I don't know here, and so I can't really make a list of the running times of the films and find the middle value. So I don't think I'm gonna be able to do it using the histogram. Now, with the box plot right over here, so I'm not gonna click histogram, with the box plot over here, I might not be able to make a list of all the values, but the box plot explicitly tells us what the median is. This middle line in the middle of the box that tells us the median is, what is this? This median is, if this is 100, this is 99."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So I don't think I'm gonna be able to do it using the histogram. Now, with the box plot right over here, so I'm not gonna click histogram, with the box plot over here, I might not be able to make a list of all the values, but the box plot explicitly tells us what the median is. This middle line in the middle of the box that tells us the median is, what is this? This median is, if this is 100, this is 99. So this is 95, 96, 97, 98, 99. It explicitly tells us the median is 99. This is actually the easiest for calculating the median."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This median is, if this is 100, this is 99. So this is 95, 96, 97, 98, 99. It explicitly tells us the median is 99. This is actually the easiest for calculating the median. So I'll go with the box plot. So the histogram is of no use to me if I want to calculate the median. Let's do a couple more of these."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is actually the easiest for calculating the median. So I'll go with the box plot. So the histogram is of no use to me if I want to calculate the median. Let's do a couple more of these. Nam owns a used car lot. He checked the odometers of the cars and recorded how far they had driven. He then created both a histogram and a box plot to display the same data."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's do a couple more of these. Nam owns a used car lot. He checked the odometers of the cars and recorded how far they had driven. He then created both a histogram and a box plot to display the same data. Both diagrams are shown below. Which display can be used to find how many vehicles had driven more than 200,000 kilometers? So how many vehicles had driven more than 200,000 kilometers?"}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "He then created both a histogram and a box plot to display the same data. Both diagrams are shown below. Which display can be used to find how many vehicles had driven more than 200,000 kilometers? So how many vehicles had driven more than 200,000 kilometers? So it looks like here in this histogram, I have three vehicles that were between 200 and 250, and then I have two vehicles that are between 250 and 300. So it looks pretty clear that I have five vehicles. Three that had a mileage between 200,000 and 250,000, and then I had two that had mileage between 250,000 and 300,000."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So how many vehicles had driven more than 200,000 kilometers? So it looks like here in this histogram, I have three vehicles that were between 200 and 250, and then I have two vehicles that are between 250 and 300. So it looks pretty clear that I have five vehicles. Three that had a mileage between 200,000 and 250,000, and then I had two that had mileage between 250,000 and 300,000. So I'm able to answer the question. Five vehicles had a mileage more than 200,000. And so I would say that the histogram is pretty useful."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Three that had a mileage between 200,000 and 250,000, and then I had two that had mileage between 250,000 and 300,000. So I'm able to answer the question. Five vehicles had a mileage more than 200,000. And so I would say that the histogram is pretty useful. But let's verify that the box plot isn't so useful. So I want to know how many vehicles had a mileage more than 200,000. Well, I know that if I have a mileage more than 200,000, I'm going to be in the fourth quartile, but I don't know how many values I have sitting there in the fourth quartile, just looking at this data over here."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And so I would say that the histogram is pretty useful. But let's verify that the box plot isn't so useful. So I want to know how many vehicles had a mileage more than 200,000. Well, I know that if I have a mileage more than 200,000, I'm going to be in the fourth quartile, but I don't know how many values I have sitting there in the fourth quartile, just looking at this data over here. So that's not going to be useful for answering that question. Let's look at the second question. Which display can be used to find the median distance was approximately 140,000 kilometers?"}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, I know that if I have a mileage more than 200,000, I'm going to be in the fourth quartile, but I don't know how many values I have sitting there in the fourth quartile, just looking at this data over here. So that's not going to be useful for answering that question. Let's look at the second question. Which display can be used to find the median distance was approximately 140,000 kilometers? Well, to calculate the median, you essentially want to be able to list all of the numbers and then find the middle number. And over here, I can't list all of the numbers. I know that there's three values that are between zero and 50,000 kilometers, but I don't know what they are."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Which display can be used to find the median distance was approximately 140,000 kilometers? Well, to calculate the median, you essentially want to be able to list all of the numbers and then find the middle number. And over here, I can't list all of the numbers. I know that there's three values that are between zero and 50,000 kilometers, but I don't know what they are. It could be 10,000, 10,000, 10,000. It could be 10,000, 15,000, and 40,000. I don't know what they are."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I know that there's three values that are between zero and 50,000 kilometers, but I don't know what they are. It could be 10,000, 10,000, 10,000. It could be 10,000, 15,000, and 40,000. I don't know what they are. And so if I can't list all of these things and put them in order, I really am going to have trouble finding the middle value, the middle values going to be in this range right around here, but I don't know exactly what it's going to be. The histogram is not useful because it's throwing all the values into these buckets. While on the box plot, it explicitly, it directly tells me the median value."}, {"video_title": "Comparing dot plots, histograms, and box plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I don't know what they are. And so if I can't list all of these things and put them in order, I really am going to have trouble finding the middle value, the middle values going to be in this range right around here, but I don't know exactly what it's going to be. The histogram is not useful because it's throwing all the values into these buckets. While on the box plot, it explicitly, it directly tells me the median value. This line right over here, the middle of the box, this tells us the median value. And we see that the median value here, this is 140,000 kilometers. This is 100, 110, 120, 130, 140,000 kilometers is the median mileage for the cars."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "He filled a sample of 20 drinks to test his null hypothesis, which is the actual population mean for how much drink there was in the drinks, per drink is 530 milliliters, versus his alternative hypothesis is that the population mean is not 530 milliliters, where mu is the mean filling amount. The drinks in the sample contained a mean amount of 528 milliliters with a standard deviation of four milliliters. These results produced a test statistic of t is equal to negative 2.236 and a p-value of approximately 0.038. Assuming the conditions for inference were met, what is an appropriate conclusion at the alpha equals 0.05 significance level? And they give us some choices here. And like always, I encourage you to pause this video and see if you can figure it out on your own. All right, so now let's work through this together."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "Assuming the conditions for inference were met, what is an appropriate conclusion at the alpha equals 0.05 significance level? And they give us some choices here. And like always, I encourage you to pause this video and see if you can figure it out on your own. All right, so now let's work through this together. So let's just remind ourselves what's going on. So you have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "All right, so now let's work through this together. So let's just remind ourselves what's going on. So you have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks. And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks. And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. They tell us these things right over here. And then using this information and actually our sample size, you're able to calculate a t-statistic. You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. They tell us these things right over here. And then using this information and actually our sample size, you're able to calculate a t-statistic. You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true? And if that probability is lower than our significance level, then we say, hey, that's a very low probability. We're going to reject our null hypothesis, which would suggest our alternative."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true? And if that probability is lower than our significance level, then we say, hey, that's a very low probability. We're going to reject our null hypothesis, which would suggest our alternative. So the key to this question is just to compare this p-value right over here to our significance level. And as we see, the p-value, 0.038, is indeed less than 0.05. And so because of this, we would reject the null hypothesis."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "We're going to reject our null hypothesis, which would suggest our alternative. So the key to this question is just to compare this p-value right over here to our significance level. And as we see, the p-value, 0.038, is indeed less than 0.05. And so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL. And so if we look at our choices here, so the first choice says reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 mL."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "And so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL. And so if we look at our choices here, so the first choice says reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 mL. Yeah, that one looks good. This suggests, this is strong evidence. This suggests the alternative hypothesis, which is that right over there."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "This is strong evidence that the mean filling amount is different than 530 mL. Yeah, that one looks good. This suggests, this is strong evidence. This suggests the alternative hypothesis, which is that right over there. But let's read the other one, just to make sure that they don't make sense. So this is rejecting the null hypothesis. That looks true so far."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "This suggests the alternative hypothesis, which is that right over there. But let's read the other one, just to make sure that they don't make sense. So this is rejecting the null hypothesis. That looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 mL. No, not, the first one is definitely much stronger. Fail to reject the null hypothesis."}, {"video_title": "Comparing P-value from t statistic to significance level   AP Statistics   Khan Academy.mp3", "Sentence": "That looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 mL. No, not, the first one is definitely much stronger. Fail to reject the null hypothesis. No, we are rejecting the null hypothesis because our p-value is lower than our significance level. Fail to reject. We'd rule that one out as well."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have the men, and the men, some proportion are going to vote for the candidate. We'll call that P1. So this is the proportion that will vote for the candidate. And the rest of the men will not vote for the candidate. So 1 minus P1 will not vote for the candidate. And then for the women, you're going to see something similar. So this is the women right over here."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the rest of the men will not vote for the candidate. So 1 minus P1 will not vote for the candidate. And then for the women, you're going to see something similar. So this is the women right over here. And some proportion will vote for the candidate. We don't know if it's the same as P1. We don't know if it's the same as the men."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the women right over here. And some proportion will vote for the candidate. We don't know if it's the same as P1. We don't know if it's the same as the men. So we'll call it P2. And then the rest of the women will not vote for the candidate. 1 minus P2."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't know if it's the same as the men. So we'll call it P2. And then the rest of the women will not vote for the candidate. 1 minus P2. So the not voting are 0s. The ones that are voting are 1s. And these are both Bernoulli distributions."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 minus P2. So the not voting are 0s. The ones that are voting are 1s. And these are both Bernoulli distributions. And we know, just because this will be useful later on, we know that the means of this distribution are the same as the proportion that will vote for it. So you can see the mean of the men, or the proportion of the men that will vote. So we'll call that mean 1 is equal to P1."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And these are both Bernoulli distributions. And we know, just because this will be useful later on, we know that the means of this distribution are the same as the proportion that will vote for it. So you can see the mean of the men, or the proportion of the men that will vote. So we'll call that mean 1 is equal to P1. And the variance here, the variance of this first distribution, I should have done everything in yellow actually. I'll do everything in yellow. So the mean of this distribution is P1."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we'll call that mean 1 is equal to P1. And the variance here, the variance of this first distribution, I should have done everything in yellow actually. I'll do everything in yellow. So the mean of this distribution is P1. The variance of this distribution, we'll call that variance 1, is just these two proportions multiplied by each other. So it's P1 times 1 minus P1. And we saw this many, many videos ago when we learned about Bernoulli distributions."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the mean of this distribution is P1. The variance of this distribution, we'll call that variance 1, is just these two proportions multiplied by each other. So it's P1 times 1 minus P1. And we saw this many, many videos ago when we learned about Bernoulli distributions. And we're going to see the exact same thing with the women. The mean of this Bernoulli distribution is going to be P2. And then the variance of this Bernoulli distribution is going to be these two proportions multiplied."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we saw this many, many videos ago when we learned about Bernoulli distributions. And we're going to see the exact same thing with the women. The mean of this Bernoulli distribution is going to be P2. And then the variance of this Bernoulli distribution is going to be these two proportions multiplied. So P2 times 1 minus P2. Now, what I want to do, and I think I said this at the beginning of the video, is I want to figure out if there's a meaningful difference between the way that the men will vote and the women will vote. I want to figure out is this meaningful?"}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the variance of this Bernoulli distribution is going to be these two proportions multiplied. So P2 times 1 minus P2. Now, what I want to do, and I think I said this at the beginning of the video, is I want to figure out if there's a meaningful difference between the way that the men will vote and the women will vote. I want to figure out is this meaningful? So is there a meaningful difference here? And what we're going to do in this video is try to come up with a 95% confidence interval for this parameter. This difference of parameters is still a parameter."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "I want to figure out is this meaningful? So is there a meaningful difference here? And what we're going to do in this video is try to come up with a 95% confidence interval for this parameter. This difference of parameters is still a parameter. We don't know what the true difference of these two population parameters are, or these two population proportions, but we're going to try to come up with a 95% confidence interval for that difference. And the way we do that, we go out and we find 1,000 likely men to vote, or men likely to vote, and 1,000 women likely to vote. So let's write this down."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This difference of parameters is still a parameter. We don't know what the true difference of these two population parameters are, or these two population proportions, but we're going to try to come up with a 95% confidence interval for that difference. And the way we do that, we go out and we find 1,000 likely men to vote, or men likely to vote, and 1,000 women likely to vote. So let's write this down. So we get 1,000 men. When we survey the 1,000 men, let's say 642 say that they will vote for the candidate, so they are 1s. And then the remainder, we could even, what would it be, 358?"}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's write this down. So we get 1,000 men. When we survey the 1,000 men, let's say 642 say that they will vote for the candidate, so they are 1s. And then the remainder, we could even, what would it be, 358? Well, I'll just say the remainder. So the rest are 0s. And we do the same thing with women."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the remainder, we could even, what would it be, 358? Well, I'll just say the remainder. So the rest are 0s. And we do the same thing with women. We survey 1,000 women who are likely to vote, but we survey them randomly. And let's say 591 say that they will vote for the candidate, and the rest say that they will not vote for the candidate. So just here, based on our sample proportions or our sample means, it looks like there is a difference, but we still have to come up with our confidence interval."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we do the same thing with women. We survey 1,000 women who are likely to vote, but we survey them randomly. And let's say 591 say that they will vote for the candidate, and the rest say that they will not vote for the candidate. So just here, based on our sample proportions or our sample means, it looks like there is a difference, but we still have to come up with our confidence interval. And let's just make sure we understand what we just did. We just found, so we could figure out a sample proportion over here for the men, so the sample proportion here for the men, which is really just the sample mean of this sample right over here. We have 642 1s, the rest are 0, so we have 642 in the numerator."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just here, based on our sample proportions or our sample means, it looks like there is a difference, but we still have to come up with our confidence interval. And let's just make sure we understand what we just did. We just found, so we could figure out a sample proportion over here for the men, so the sample proportion here for the men, which is really just the sample mean of this sample right over here. We have 642 1s, the rest are 0, so we have 642 in the numerator. We have 1,000 samples. 642 divided by 1,000 is 0.642. So you could view this as a sample mean or as a sample proportion."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 642 1s, the rest are 0, so we have 642 in the numerator. We have 1,000 samples. 642 divided by 1,000 is 0.642. So you could view this as a sample mean or as a sample proportion. And if you do the same thing for the women, the sample proportion is going to be 0.591, or you could even just view this as the sample mean of the sample of 1,000 women. We're the ones voting for it, there's 1, the rest are 0. And just to visualize it properly, let me draw the sampling distribution for the sample proportion."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could view this as a sample mean or as a sample proportion. And if you do the same thing for the women, the sample proportion is going to be 0.591, or you could even just view this as the sample mean of the sample of 1,000 women. We're the ones voting for it, there's 1, the rest are 0. And just to visualize it properly, let me draw the sampling distribution for the sample proportion. So the sampling distribution, we have a large sample size. We have a large sample size here, and especially because the proportions that we're dealing with aren't close to 1 or 0, and we have a large sample size, the sampling distribution will be approximately normal. So this sampling distribution, let me write this, the sampling distribution of the sample proportion, so it's going to have some mean over here, so the mean of the sampling distribution of the sample proportion, and we've seen it multiple times."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just to visualize it properly, let me draw the sampling distribution for the sample proportion. So the sampling distribution, we have a large sample size. We have a large sample size here, and especially because the proportions that we're dealing with aren't close to 1 or 0, and we have a large sample size, the sampling distribution will be approximately normal. So this sampling distribution, let me write this, the sampling distribution of the sample proportion, so it's going to have some mean over here, so the mean of the sampling distribution of the sample proportion, and we've seen it multiple times. It's going to be the same thing as the mean of the population, and the mean of the population is actually the true population proportion. So this is going to be equal to P1. This is something that we don't know about."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this sampling distribution, let me write this, the sampling distribution of the sample proportion, so it's going to have some mean over here, so the mean of the sampling distribution of the sample proportion, and we've seen it multiple times. It's going to be the same thing as the mean of the population, and the mean of the population is actually the true population proportion. So this is going to be equal to P1. This is something that we don't know about. And then the variance of this, and we've seen this several times already, the variance of this distribution, I have to put a 1 here, we're dealing with the men, the variance of this distribution by the central limit theorem is going to be the variance of this distribution up here, which is P1 times 1 minus P1, over our sample size, over 1,000. And we can do the exact same thing for the women. The exact same thing for the women."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is something that we don't know about. And then the variance of this, and we've seen this several times already, the variance of this distribution, I have to put a 1 here, we're dealing with the men, the variance of this distribution by the central limit theorem is going to be the variance of this distribution up here, which is P1 times 1 minus P1, over our sample size, over 1,000. And we can do the exact same thing for the women. The exact same thing for the women. So this is the sampling distribution. Let me write it over here. This is the sampling distribution."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "The exact same thing for the women. So this is the sampling distribution. Let me write it over here. This is the sampling distribution. This is for P2 bar, or this sample mean over here, let me put a 1 over here. Remember, this is for all four of the men, and then this over here is all for the women. Can't forget those 2's over there."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the sampling distribution. This is for P2 bar, or this sample mean over here, let me put a 1 over here. Remember, this is for all four of the men, and then this over here is all for the women. Can't forget those 2's over there. So this distribution is going to have some mean. Let me draw it right over here. So mu sub P2 with a bar over it, so the mean of the sampling distribution for this sample proportion for the women, which is going to be the same thing as the mean of the population, which we already saw is going to be equal to P2."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Can't forget those 2's over there. So this distribution is going to have some mean. Let me draw it right over here. So mu sub P2 with a bar over it, so the mean of the sampling distribution for this sample proportion for the women, which is going to be the same thing as the mean of the population, which we already saw is going to be equal to P2. And then the variance for this distribution, for this sampling distribution over here, is going to be P2, is going to be this variance, this variance over here, divided by our sample size. So P2 times 1 minus P2, all of that over N. Now, our whole goal is to get a 95% confidence interval for that. And so what we're going to do is we're going to think about the sampling distribution not for this, and not the sampling distribution for this, but we're going to think about the sampling distribution for the difference of this sample, this sample proportion, and this sample proportion."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So mu sub P2 with a bar over it, so the mean of the sampling distribution for this sample proportion for the women, which is going to be the same thing as the mean of the population, which we already saw is going to be equal to P2. And then the variance for this distribution, for this sampling distribution over here, is going to be P2, is going to be this variance, this variance over here, divided by our sample size. So P2 times 1 minus P2, all of that over N. Now, our whole goal is to get a 95% confidence interval for that. And so what we're going to do is we're going to think about the sampling distribution not for this, and not the sampling distribution for this, but we're going to think about the sampling distribution for the difference of this sample, this sample proportion, and this sample proportion. We've seen it already. This is really, you know, we're talking about proportions, but it's really the same exact ideas that we did when we just compared sample means generally. So let's look at that."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so what we're going to do is we're going to think about the sampling distribution not for this, and not the sampling distribution for this, but we're going to think about the sampling distribution for the difference of this sample, this sample proportion, and this sample proportion. We've seen it already. This is really, you know, we're talking about proportions, but it's really the same exact ideas that we did when we just compared sample means generally. So let's look at that. Let's look at that distribution. Let's look at this distribution. And just to be clear, when we got this sample mean here, this sample proportion, we just sampled it."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look at that. Let's look at that distribution. Let's look at this distribution. And just to be clear, when we got this sample mean here, this sample proportion, we just sampled it. You could view it as taking a sample from this distribution over here. When we got this sample proportion, it was like taking a sample from this over here. We took 591, or we took 1,000 samples from this when we took their mean, where it's equivalent to taking a sample from the sampling distribution."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just to be clear, when we got this sample mean here, this sample proportion, we just sampled it. You could view it as taking a sample from this distribution over here. When we got this sample proportion, it was like taking a sample from this over here. We took 591, or we took 1,000 samples from this when we took their mean, where it's equivalent to taking a sample from the sampling distribution. Now, this distribution over here is going to be the distribution of all of the differences of the sampling proportions, or of the sample proportions. And so it will look like this. It will have some mean value."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "We took 591, or we took 1,000 samples from this when we took their mean, where it's equivalent to taking a sample from the sampling distribution. Now, this distribution over here is going to be the distribution of all of the differences of the sampling proportions, or of the sample proportions. And so it will look like this. It will have some mean value. I should do this in a different color. I'll do it in green. Yellow and blue make green."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "It will have some mean value. I should do this in a different color. I'll do it in green. Yellow and blue make green. Let me do this one in green. So I'll call this the sampling distribution. This is the sampling distribution of this statistic, of P1 minus P2."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yellow and blue make green. Let me do this one in green. So I'll call this the sampling distribution. This is the sampling distribution of this statistic, of P1 minus P2. And so it has some mean over here. It has some mean, the mean of the sample of P1 minus the sample mean, or the sample proportion of P2. And we know from things that we've done in the last several videos, that this is going to be the exact same thing as this mean minus this mean, which is the exact same thing as P1 minus P2."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the sampling distribution of this statistic, of P1 minus P2. And so it has some mean over here. It has some mean, the mean of the sample of P1 minus the sample mean, or the sample proportion of P2. And we know from things that we've done in the last several videos, that this is going to be the exact same thing as this mean minus this mean, which is the exact same thing as P1 minus P2. So this is going to be equal to P1 minus P2. And the variance of this distribution, so the variance of this distribution, P1 minus P2, just like this, is going to be the sum of the variances of these two distributions. So it's going to be this thing over here."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we know from things that we've done in the last several videos, that this is going to be the exact same thing as this mean minus this mean, which is the exact same thing as P1 minus P2. So this is going to be equal to P1 minus P2. And the variance of this distribution, so the variance of this distribution, P1 minus P2, just like this, is going to be the sum of the variances of these two distributions. So it's going to be this thing over here. I'll just copy and paste it. It's going to be that thing over there, plus this variance over here. There's no radical sign because we're not taking the standard deviation."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be this thing over here. I'll just copy and paste it. It's going to be that thing over there, plus this variance over here. There's no radical sign because we're not taking the standard deviation. We're focused on the variance right now. So plus this thing right over here. So let me copy and let me paste it."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's no radical sign because we're not taking the standard deviation. We're focused on the variance right now. So plus this thing right over here. So let me copy and let me paste it. So plus this thing right over here. So that's going to be the variance. This is going to be the variance."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me copy and let me paste it. So plus this thing right over here. So that's going to be the variance. This is going to be the variance. And if you want the standard deviation, you can literally just get rid of this. You're taking the square root of both sides. So you take the square root of the variance, you get the standard deviation."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be the variance. And if you want the standard deviation, you can literally just get rid of this. You're taking the square root of both sides. So you take the square root of the variance, you get the standard deviation. That's why I got rid of that to the second power. And you want to take a square root of the right-hand side, just like that. Now, all I did right now is just to kind of conceptually set things up in our brain."}, {"video_title": "Comparing population proportions 1   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you take the square root of the variance, you get the standard deviation. That's why I got rid of that to the second power. And you want to take a square root of the right-hand side, just like that. Now, all I did right now is just to kind of conceptually set things up in our brain. What we now need to do is actually tackle the confidence interval. We actually need to come up with a 95% confidence interval for P1 minus P2, or a 95% confidence interval for this mean right over here. And because I'm trying to make my best effort not to make videos too long, I'll do part two in the next video where we actually solve the confidence interval."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "It's a little unusual looking. It looks more like a triangle than our standard density curves, but it's valid. Which of the following statements are true? Choose all answers that apply. The mean of the density curve is less than the median. Pause this video and see if you can figure out whether that's true. Well, we don't know exactly where the mean and median are just by looking at this, but remember, the median is going to be the value for which the area to the right and the left are going to be equal."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Choose all answers that apply. The mean of the density curve is less than the median. Pause this video and see if you can figure out whether that's true. Well, we don't know exactly where the mean and median are just by looking at this, but remember, the median is going to be the value for which the area to the right and the left are going to be equal. So I would guess the median is going to be someplace like that. So that's my guess, my approximation. That is the median."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we don't know exactly where the mean and median are just by looking at this, but remember, the median is going to be the value for which the area to the right and the left are going to be equal. So I would guess the median is going to be someplace like that. So that's my guess, my approximation. That is the median. And because our distribution goes off further to the left than it does to the right, you could view this as something of a tail, it's reasonable to say that this is left skewed, left skewed. And generally speaking, if a distribution is left skewed, the mean is to the left of the median. So because it is left skewed, the mean might be someplace like right over there."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "That is the median. And because our distribution goes off further to the left than it does to the right, you could view this as something of a tail, it's reasonable to say that this is left skewed, left skewed. And generally speaking, if a distribution is left skewed, the mean is to the left of the median. So because it is left skewed, the mean might be someplace like right over there. Another way to even think about the mean is that the mean would be the balance point where you'd place a fulcrum if this were a mass. And you might say, why doesn't that happen at the median? Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer into the fulcrum."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So because it is left skewed, the mean might be someplace like right over there. Another way to even think about the mean is that the mean would be the balance point where you'd place a fulcrum if this were a mass. And you might say, why doesn't that happen at the median? Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer into the fulcrum. So in terms of this first one, the mean of the density curve is less than the median, in this case, or you could say to the left of the median, we can consider this to be true. Now, what about the median of the density curve is three? Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer into the fulcrum. So in terms of this first one, the mean of the density curve is less than the median, in this case, or you could say to the left of the median, we can consider this to be true. Now, what about the median of the density curve is three? Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area. The median definitely, I might not be right there, but the median is definitely not going to be three. This area right over here is for sure smaller than this area right over here. So we can rule that out."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area. The median definitely, I might not be right there, but the median is definitely not going to be three. This area right over here is for sure smaller than this area right over here. So we can rule that out. The area underneath the density curve is one. Pause this video. Is that true?"}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So we can rule that out. The area underneath the density curve is one. Pause this video. Is that true? Yes, this is true. The area underneath any density curve is going to be one. If we look at the total area under the curve, it's always going to be one."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Is that true? Yes, this is true. The area underneath any density curve is going to be one. If we look at the total area under the curve, it's always going to be one. So we answered this question. I'll leave you with one extra question that we can actually figure out from the information they've given us. What is the height of this point of this density curve right over here?"}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "If we look at the total area under the curve, it's always going to be one. So we answered this question. I'll leave you with one extra question that we can actually figure out from the information they've given us. What is the height of this point of this density curve right over here? What is this value? What is this height going to be? See if you can pause this video and figure it out."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "What is the height of this point of this density curve right over here? What is this value? What is this height going to be? See if you can pause this video and figure it out. And I'll give you a hint. The hint is this third statement. The area under the density curve is one."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "See if you can pause this video and figure it out. And I'll give you a hint. The hint is this third statement. The area under the density curve is one. All right, now let's try to work through it together. If we call this height h, we know how to find the area of a triangle. It's 1 1\u20442 base times height."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "The area under the density curve is one. All right, now let's try to work through it together. If we call this height h, we know how to find the area of a triangle. It's 1 1\u20442 base times height. Area is equal to 1 1\u20442 base times height. We know that the area is one. This is a density curve."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "It's 1 1\u20442 base times height. Area is equal to 1 1\u20442 base times height. We know that the area is one. This is a density curve. So one is going to be equal to, what's the length of the base? We go from one to six. So from one to six, this base, the length of this base is five."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "This is a density curve. So one is going to be equal to, what's the length of the base? We go from one to six. So from one to six, this base, the length of this base is five. 1 1\u20442 times five times height. Or we could say one is equal to 5 1\u20442 times height. Or multiply both sides by 2\u2075 to solve for the height."}, {"video_title": "Density curve worked example   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So from one to six, this base, the length of this base is five. 1 1\u20442 times five times height. Or we could say one is equal to 5 1\u20442 times height. Or multiply both sides by 2\u2075 to solve for the height. And what are we gonna get? We're gonna get the height is equal to 2\u2075. So if you have a very clean triangular density curve like this, you can actually figure out the height with even if it was not directly specified."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say I want to figure out the probability. I'm going to flip a coin eight times, and it's a fair coin. And I want to figure out the probability of getting exactly three out of eight heads. So I say three eight heads. But three of my flips are going to be heads, and the rest are going to be tails. So how do I think about that? Well, let's go back to one of the early definitions we use for probability."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I say three eight heads. But three of my flips are going to be heads, and the rest are going to be tails. So how do I think about that? Well, let's go back to one of the early definitions we use for probability. And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true. So the number of events, I guess trials or situations, in which we get three heads. And exactly three heads."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's go back to one of the early definitions we use for probability. And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true. So the number of events, I guess trials or situations, in which we get three heads. And exactly three heads. We're not saying greater than three heads. So four heads won't count, and two heads won't count, five heads won't. Only three heads."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And exactly three heads. We're not saying greater than three heads. So four heads won't count, and two heads won't count, five heads won't. Only three heads. And then over the total number of equally probable trials. So total number of equally possible outcomes. I should be using the word outcomes."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Only three heads. And then over the total number of equally probable trials. So total number of equally possible outcomes. I should be using the word outcomes. So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes. So let's do the bottom part first. What are the total possible outcomes if I'm flipping a fair coin eight times?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I should be using the word outcomes. So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes. So let's do the bottom part first. What are the total possible outcomes if I'm flipping a fair coin eight times? Well, the first time I flip it, I either get heads or tails, so I get two outcomes. And then when I flip it again, I get two more outcomes for the second one. And then how many total outcomes?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "What are the total possible outcomes if I'm flipping a fair coin eight times? Well, the first time I flip it, I either get heads or tails, so I get two outcomes. And then when I flip it again, I get two more outcomes for the second one. And then how many total outcomes? Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip. And then essentially, we multiply 2 times the number of flips. So that's 5, 6, 7, 8."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then how many total outcomes? Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip. And then essentially, we multiply 2 times the number of flips. So that's 5, 6, 7, 8. And that equals 2 to the 8th. So the number of outcomes is just going to be 2 to the total number of flips. And hopefully that makes sense to you."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's 5, 6, 7, 8. And that equals 2 to the 8th. So the number of outcomes is just going to be 2 to the total number of flips. And hopefully that makes sense to you. If not, you might want to re-watch some of the earlier videos. But that's the easy part. So there's 2 to the 8th possible outcomes when you flip a fair coin eight times."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And hopefully that makes sense to you. If not, you might want to re-watch some of the earlier videos. But that's the easy part. So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. So how many of those outcomes are going to result in exactly three heads? Let's think of it this way. Let's give a name to each of our flips."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. So how many of those outcomes are going to result in exactly three heads? Let's think of it this way. Let's give a name to each of our flips. Let's give a name to them. So let me make a little column. Call these the flips."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's give a name to each of our flips. Let's give a name to them. So let me make a little column. Call these the flips. This is my flips column. And I don't know, I could name them anything. I could name them Larry, Curly Moe."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Call these the flips. This is my flips column. And I don't know, I could name them anything. I could name them Larry, Curly Moe. I could name them the, well, I would need five more names for them. But I could name them the seven dwarves, or the eight dwarves, really, because I have eight flips. But I could, you know, I'll just name, I'll number the flips."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could name them Larry, Curly Moe. I could name them the, well, I would need five more names for them. But I could name them the seven dwarves, or the eight dwarves, really, because I have eight flips. But I could, you know, I'll just name, I'll number the flips. One, two, three, four, five, six, seven, eight. And I'm the god of probability. And essentially, I need to just pick three of these flips that are going to result in heads."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I could, you know, I'll just name, I'll number the flips. One, two, three, four, five, six, seven, eight. And I'm the god of probability. And essentially, I need to just pick three of these flips that are going to result in heads. So another way to think about it is, these could be eight people. And I could pick which of these, you know, how many ways can I pick three of these people to put into the car? Or how many ways can I pick three of these people to sit in chairs?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And essentially, I need to just pick three of these flips that are going to result in heads. So another way to think about it is, these could be eight people. And I could pick which of these, you know, how many ways can I pick three of these people to put into the car? Or how many ways can I pick three of these people to sit in chairs? And it doesn't matter the order that I pick them in, right? It doesn't matter if I say the people that are going to get in the car are going to be people one, two, and three. Or if I say three, two, and one."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or how many ways can I pick three of these people to sit in chairs? And it doesn't matter the order that I pick them in, right? It doesn't matter if I say the people that are going to get in the car are going to be people one, two, and three. Or if I say three, two, and one. Or if I say two, three, and one. Those are all the same combination, right? So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if I say three, two, and one. Or if I say two, three, and one. Those are all the same combination, right? So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. I don't want to confuse you too much. But essentially, I'm just going to choose three things out of the eight. So I'm essentially just saying, how many combinations can I get where I pick three out of these eight?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. I don't want to confuse you too much. But essentially, I'm just going to choose three things out of the eight. So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three. And that's, you know, how many combinations of three can we pick of eight? And that's, we went over in the last video."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three. And that's, you know, how many combinations of three can we pick of eight? And that's, we went over in the last video. And let's do it with the formula first. So let me write the formula up here, just so you remember it. But I also want to give you the intuition again for the formula."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's, we went over in the last video. And let's do it with the formula first. So let me write the formula up here, just so you remember it. But I also want to give you the intuition again for the formula. So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial. So in this situation, it would be, that would equal 8 factorial over 3 factorial times what? 8 minus k times 5 factorial."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I also want to give you the intuition again for the formula. So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial. So in this situation, it would be, that would equal 8 factorial over 3 factorial times what? 8 minus k times 5 factorial. Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1. And of course, that and that cancel out. And all you're left with is 8 times 7 times 6 over 3 factorial."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "8 minus k times 5 factorial. Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1. And of course, that and that cancel out. And all you're left with is 8 times 7 times 6 over 3 factorial. And I did this for a reason. Because I want you to re-get the intuition for, at least for this part of the formula. That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And all you're left with is 8 times 7 times 6 over 3 factorial. And I did this for a reason. Because I want you to re-get the intuition for, at least for this part of the formula. That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? And that's essentially saying, well, before I pick anything, I can pick one of eight. Then I have seven left to pick from for the second spot. And then I have six left to pick for the third spot."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? And that's essentially saying, well, before I pick anything, I can pick one of eight. Then I have seven left to pick from for the second spot. And then I have six left to pick for the third spot. And so that's essentially the number of permutations. But since we don't care, you know, if we picked it, what order we pick them in, we need to divide by the number of ways we can rearrange three things. And that's where the 3 factorial comes from."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then I have six left to pick for the third spot. And so that's essentially the number of permutations. But since we don't care, you know, if we picked it, what order we pick them in, we need to divide by the number of ways we can rearrange three things. And that's where the 3 factorial comes from. And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient. But it's good to have the intuition. And then, well, once we're at this point, we can just calculate this."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's where the 3 factorial comes from. And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient. But it's good to have the intuition. And then, well, once we're at this point, we can just calculate this. Well, what's this? This is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1, so that's 6. So 6 cancels out, so it's 8 times 7."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then, well, once we're at this point, we can just calculate this. Well, what's this? This is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1, so that's 6. So 6 cancels out, so it's 8 times 7. So there's 8 times 7, or what is that, 56. Right, 56. Yeah, that's equal to 56."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 6 cancels out, so it's 8 times 7. So there's 8 times 7, or what is that, 56. Right, 56. Yeah, that's equal to 56. So there's 56 different ways to pick three things out of eight. Or if I have eight people, there's 56 ways of picking three people to sit in the car, or however you want to view it. But if I have eight flips, there's 56 ways of picking three of those flips to be heads."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yeah, that's equal to 56. So there's 56 different ways to pick three things out of eight. Or if I have eight people, there's 56 ways of picking three people to sit in the car, or however you want to view it. But if I have eight flips, there's 56 ways of picking three of those flips to be heads. So let's go to our original probability problem. What is the probability that I get three out of eight heads? Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "But if I have eight flips, there's 56 ways of picking three of those flips to be heads. So let's go to our original probability problem. What is the probability that I get three out of eight heads? Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? The total number of outcomes is 2 to the 8th. Another way I could write that, 56, let me unseparate, that's 8 times 7 over 2 to the 8th. 8 is 2 to the 3rd, right?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? The total number of outcomes is 2 to the 8th. Another way I could write that, 56, let me unseparate, that's 8 times 7 over 2 to the 8th. 8 is 2 to the 3rd, right? Let me erase some of this. Not with that color. Let me erase all of this so I have space."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "8 is 2 to the 3rd, right? Let me erase some of this. Not with that color. Let me erase all of this so I have space. I will switch colors for variety. OK, so I'm back. All right, so 8 is the same thing as 2 to the 3rd times 7."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me erase all of this so I have space. I will switch colors for variety. OK, so I'm back. All right, so 8 is the same thing as 2 to the 3rd times 7. This is all just mathematical simplification, but it's useful, over 2 to the 8th. And so if we just divide both sides, the numerator and the denominator by 2 to the 3rd, this becomes 1, this becomes 2 to the 5th, and so it becomes 7 over 32. Is that right?"}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, so 8 is the same thing as 2 to the 3rd times 7. This is all just mathematical simplification, but it's useful, over 2 to the 8th. And so if we just divide both sides, the numerator and the denominator by 2 to the 3rd, this becomes 1, this becomes 2 to the 5th, and so it becomes 7 over 32. Is that right? So if I were to pick 3 out of 8, yep, I think that is right. And so what does that turn out to be? Let me get my calculator."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is that right? So if I were to pick 3 out of 8, yep, I think that is right. And so what does that turn out to be? Let me get my calculator. I often make careless mistakes. Let me see, my calculator seems to have disappeared. Let me get it back."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me get my calculator. I often make careless mistakes. Let me see, my calculator seems to have disappeared. Let me get it back. There it is. OK, 7 divided by 32 is equal to 0.21875, which is equal to 21, if I were to round, roughly, 21.9% chance. So there's a little bit better than 1 in 5 chance that I get exactly 3 out of the 8 flips as heads."}, {"video_title": "Probability using combinations   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me get it back. There it is. OK, 7 divided by 32 is equal to 0.21875, which is equal to 21, if I were to round, roughly, 21.9% chance. So there's a little bit better than 1 in 5 chance that I get exactly 3 out of the 8 flips as heads. Hopefully I didn't confuse you, and now you can apply that to pretty much anything. You could say, well, what is the probability of getting, if I flip a fair coin, of getting exactly 7 out of 8 heads? Or you could say, what's the probability of getting 2 out of 100 heads?"}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "What we're going to do in this video is dig a little bit deeper into confidence intervals. In other videos, we compute them, we even interpret them, but here we're gonna make sure that we are making the right assumptions so that we can have confidence in our confidence intervals or that we are even calculating them in the right way or in the right context. So just as a bit of review, a lot of what we do in confidence intervals is we are trying to estimate some population parameter. Let's say it's the proportion. Maybe it's the proportion that will vote for a candidate. We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say it's the proportion. Maybe it's the proportion that will vote for a candidate. We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. If you're trying to estimate the proportion of people that are gonna vote for a certain candidate, but you are only surveying people at a senior community, well, that would not be a truly random sample if we were only to survey people on a college campus. So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. If you're trying to estimate the proportion of people that are gonna vote for a certain candidate, but you are only surveying people at a senior community, well, that would not be a truly random sample if we were only to survey people on a college campus. So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume that the distribution of the sample proportions the sampling distribution of the sample proportions has roughly a normal shape like that. But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume that the distribution of the sample proportions the sampling distribution of the sample proportions has roughly a normal shape like that. But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, or 0.5, then you wouldn't meet that normal condition because you would expect five successes and five failures for each sample. Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, or 0.5, then you wouldn't meet that normal condition because you would expect five successes and five failures for each sample. Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem. So you wanna expect, you wanna have at least greater than or equal to 10 successes or failures on each. And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And if we have less than 10 on either one of those, then we are going to have a problem. So you wanna expect, you wanna have at least greater than or equal to 10 successes or failures on each. And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last thing we wanna really make sure is known as the independence condition. Independence condition. And this is the 10% rule."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last thing we wanna really make sure is known as the independence condition. Independence condition. And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's say your population were 100,000 people. And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's say your population were 100,000 people. And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition breaks down. So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition breaks down. So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval that we calculated for that sample. And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval that we calculated for that sample. And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples. This is truly at a 95% confidence level."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples. This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes. And you can see that when I do, once again, I did a bunch of samples here."}, {"video_title": "Conditions for valid confidence intervals   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes. And you can see that when I do, once again, I did a bunch of samples here. I did over 2,000 samples. Even though I'm trying to set up these confidence intervals that every time I compute it, that over time, that there's kind of a 95% hit rate, so to speak, here there's only a 94% hit rate. And I've done a lot of samples here."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "100 randomly assigned people are assigned to group 1 and put on the low-fat diet. Another 100 randomly assigned obese people are assigned to group 2 and put on a diet of approximately the same amount of food, but not as low in fat. So group 2 is the control, just the no diet. Group 1 is the low-fat group, so see if it actually works. After 4 months, the mean weight loss was 9.31 pounds for group 1. Let me write this down. The mean weight loss for group 1, let me make it very clear."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Group 1 is the low-fat group, so see if it actually works. After 4 months, the mean weight loss was 9.31 pounds for group 1. Let me write this down. The mean weight loss for group 1, let me make it very clear. The low-fat group, the mean weight loss was 9.31 pounds. Our sample mean for group 1 is 9.31 pounds, with a sample standard deviation of 4.67. Both of these are obviously very easy to calculate from the actual data."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean weight loss for group 1, let me make it very clear. The low-fat group, the mean weight loss was 9.31 pounds. Our sample mean for group 1 is 9.31 pounds, with a sample standard deviation of 4.67. Both of these are obviously very easy to calculate from the actual data. For our control group, the sample mean is 7.40 pounds for group 2. Our sample mean here for the control is 7.40, with a sample standard deviation of 4.04 pounds. If we look at it superficially, it looks like the low-fat group lost more weight, just based on our samples, than the control group."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Both of these are obviously very easy to calculate from the actual data. For our control group, the sample mean is 7.40 pounds for group 2. Our sample mean here for the control is 7.40, with a sample standard deviation of 4.04 pounds. If we look at it superficially, it looks like the low-fat group lost more weight, just based on our samples, than the control group. If we take the difference of the low-fat group, or between the low-fat group and the control group, we get 9.31 minus 7.40 is equal to 1.91. The difference of our samples is 1.91. So just based on what we see, it says maybe you lose an incremental 1.91 pounds every 4 months if you are on this diet."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we look at it superficially, it looks like the low-fat group lost more weight, just based on our samples, than the control group. If we take the difference of the low-fat group, or between the low-fat group and the control group, we get 9.31 minus 7.40 is equal to 1.91. The difference of our samples is 1.91. So just based on what we see, it says maybe you lose an incremental 1.91 pounds every 4 months if you are on this diet. What we want to do in this video is to get a 95% confidence interval around this number to see that in that 95% confidence interval, maybe do we always lose weight? Or is there a chance that we can actually go the other way with the low-fat diet? So really, just this video, 95% confidence interval."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just based on what we see, it says maybe you lose an incremental 1.91 pounds every 4 months if you are on this diet. What we want to do in this video is to get a 95% confidence interval around this number to see that in that 95% confidence interval, maybe do we always lose weight? Or is there a chance that we can actually go the other way with the low-fat diet? So really, just this video, 95% confidence interval. In the next video, we'll actually do a hypothesis test using this same data. Now to do a 95% confidence interval, let's think about the distribution that we're thinking about. So let's look at the distribution."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So really, just this video, 95% confidence interval. In the next video, we'll actually do a hypothesis test using this same data. Now to do a 95% confidence interval, let's think about the distribution that we're thinking about. So let's look at the distribution. Of course, we're going to think about the distribution that we're thinking about. We want to think about the distribution of the difference of the means. So it's going to have some true mean here."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look at the distribution. Of course, we're going to think about the distribution that we're thinking about. We want to think about the distribution of the difference of the means. So it's going to have some true mean here. It's going to have some true mean over here, which is the mean of the difference of the sample means. Actually, let me write that. It's not a y."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to have some true mean here. It's going to have some true mean over here, which is the mean of the difference of the sample means. Actually, let me write that. It's not a y. It's an x1 and x2. So it's the sample mean of x1 minus the sample mean of x2. And then this distribution right here is going to have some standard deviation."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's not a y. It's an x1 and x2. So it's the sample mean of x1 minus the sample mean of x2. And then this distribution right here is going to have some standard deviation. It's going to have some standard deviation. So it's the standard deviation of the distribution of x, of the mean of 1 of x1 minus the sample mean of x2. It's going to have some standard deviation here."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then this distribution right here is going to have some standard deviation. It's going to have some standard deviation. So it's the standard deviation of the distribution of x, of the mean of 1 of x1 minus the sample mean of x2. It's going to have some standard deviation here. And we want to make an inference about this. I guess this is the best way to think about it. We want to get a 95% confidence interval based on our sample."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to have some standard deviation here. And we want to make an inference about this. I guess this is the best way to think about it. We want to get a 95% confidence interval based on our sample. We want to create an interval around this where we are confident that there's a 95% chance that this true mean, the true mean of the differences lies within that interval. And to do that, let's just think of it the other way. How can we create a 95% interval around this, around the mean, where we are 95% sure, or construct an interval around this, where we're 95% sure that any sample from this distribution, and this is one of those samples, that there's a 95% chance that we will select from this region right over here."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to get a 95% confidence interval based on our sample. We want to create an interval around this where we are confident that there's a 95% chance that this true mean, the true mean of the differences lies within that interval. And to do that, let's just think of it the other way. How can we create a 95% interval around this, around the mean, where we are 95% sure, or construct an interval around this, where we're 95% sure that any sample from this distribution, and this is one of those samples, that there's a 95% chance that we will select from this region right over here. So we care about a 95% region right over here. So how many standard deviations do we have to go in each direction? And to do that, we just have to look at a z-table."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "How can we create a 95% interval around this, around the mean, where we are 95% sure, or construct an interval around this, where we're 95% sure that any sample from this distribution, and this is one of those samples, that there's a 95% chance that we will select from this region right over here. So we care about a 95% region right over here. So how many standard deviations do we have to go in each direction? And to do that, we just have to look at a z-table. And just remember, if we have 95% in the middle right over here, we're going to have 2.5% over here, and we're going to have 2.5% over here. We have to have 5% split between these two symmetric tails. So when we look at a z-table, we want the critical z-value that they give right over here."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to do that, we just have to look at a z-table. And just remember, if we have 95% in the middle right over here, we're going to have 2.5% over here, and we're going to have 2.5% over here. We have to have 5% split between these two symmetric tails. So when we look at a z-table, we want the critical z-value that they give right over here. And we have to be careful here. We're not going to look up 95%, because the z-table gives us a cumulative probability up to that critical z-value. So the z-table is going to be interpreted like this."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when we look at a z-table, we want the critical z-value that they give right over here. And we have to be careful here. We're not going to look up 95%, because the z-table gives us a cumulative probability up to that critical z-value. So the z-table is going to be interpreted like this. So there's going to be some z-value right over here, where we have 2.5% above it. The probability of getting a more extreme result, or a z-score above that, is 2.5%. And the probability of getting one below that is going to be 97.5%."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the z-table is going to be interpreted like this. So there's going to be some z-value right over here, where we have 2.5% above it. The probability of getting a more extreme result, or a z-score above that, is 2.5%. And the probability of getting one below that is going to be 97.5%. But if we can find whatever z-value this is right over here, it's going to be the same z-value as that. Instead of thinking about it in terms of a one-tailed scenario, we're going to think of it in a two-tailed scenario. So let's look it up for 97.5% on our z-table."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the probability of getting one below that is going to be 97.5%. But if we can find whatever z-value this is right over here, it's going to be the same z-value as that. Instead of thinking about it in terms of a one-tailed scenario, we're going to think of it in a two-tailed scenario. So let's look it up for 97.5% on our z-table. Let's see, we have 97 right here. This is 0.975, or 97.5. And this gives us a z-value of 1.96."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look it up for 97.5% on our z-table. Let's see, we have 97 right here. This is 0.975, or 97.5. And this gives us a z-value of 1.96. So this is z is equal to 1.96. Or, only 2.5% of the results, or of the samples from this population, are going to be more than 1.96 standard deviations away from the mean. So this critical z-value right here is 1.96 standard deviations."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this gives us a z-value of 1.96. So this is z is equal to 1.96. Or, only 2.5% of the results, or of the samples from this population, are going to be more than 1.96 standard deviations away from the mean. So this critical z-value right here is 1.96 standard deviations. This is 1.96 times the standard deviation of x1 minus x2. And then this right here is going to be negative 1.96 times the same thing. So let me write that."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this critical z-value right here is 1.96 standard deviations. This is 1.96 times the standard deviation of x1 minus x2. And then this right here is going to be negative 1.96 times the same thing. So let me write that. So this right here, it's symmetric. This distance is going to be the same as that distance. So this is negative 1.96 times the standard deviation of this distribution."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me write that. So this right here, it's symmetric. This distance is going to be the same as that distance. So this is negative 1.96 times the standard deviation of this distribution. And if there's a 95% chance, so let's put it this way. There's a 95% chance that our mean, or I guess we could say that our sample that we got from our distribution, this sample is a difference of these other samples. There's a 95% chance that 1.91 lies within, or let me just write, is within this distance is within 1.96 times the standard deviation of that distribution."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is negative 1.96 times the standard deviation of this distribution. And if there's a 95% chance, so let's put it this way. There's a 95% chance that our mean, or I guess we could say that our sample that we got from our distribution, this sample is a difference of these other samples. There's a 95% chance that 1.91 lies within, or let me just write, is within this distance is within 1.96 times the standard deviation of that distribution. So you could view it as a standard error of this statistic. So x1 minus x2. Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got, is within 1.96 times the standard deviation of this distribution of the mean of the distribution, of the true mean of the distribution."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a 95% chance that 1.91 lies within, or let me just write, is within this distance is within 1.96 times the standard deviation of that distribution. So you could view it as a standard error of this statistic. So x1 minus x2. Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got, is within 1.96 times the standard deviation of this distribution of the mean of the distribution, of the true mean of the distribution. Or we could say it the other way around. There's a 95% chance that the true mean of the distribution is within 1.96 times the standard deviation of the distribution of 1.91. These are equivalent statements."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got, is within 1.96 times the standard deviation of this distribution of the mean of the distribution, of the true mean of the distribution. Or we could say it the other way around. There's a 95% chance that the true mean of the distribution is within 1.96 times the standard deviation of the distribution of 1.91. These are equivalent statements. If I say I'm within 3 feet of you, that's equivalent to saying you're within 3 feet of me. That's all that's saying. But when we construct it this way, it becomes pretty clear how do we actually construct the confidence interval."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are equivalent statements. If I say I'm within 3 feet of you, that's equivalent to saying you're within 3 feet of me. That's all that's saying. But when we construct it this way, it becomes pretty clear how do we actually construct the confidence interval. We just have to figure out what this distance right over here is. And to figure out what that distance is, we're going to have to figure out what the standard deviation of this distribution is. Well, the standard deviation of the differences of the sample means is going to be equal to, and we saw this in the last video."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "But when we construct it this way, it becomes pretty clear how do we actually construct the confidence interval. We just have to figure out what this distance right over here is. And to figure out what that distance is, we're going to have to figure out what the standard deviation of this distribution is. Well, the standard deviation of the differences of the sample means is going to be equal to, and we saw this in the last video. In fact, I think I have it right at the bottom here. It's going to be equal to the square root of the variances of each of those distributions. Or the variance of this distribution is going to be equal to the sum."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the standard deviation of the differences of the sample means is going to be equal to, and we saw this in the last video. In fact, I think I have it right at the bottom here. It's going to be equal to the square root of the variances of each of those distributions. Or the variance of this distribution is going to be equal to the sum. Let me write it this way, right over here. So the variance, I'll re-kind of prove it. The variance of the means, or the variance of our distribution, is going to be equal to the sum of the variances of each of these sampling distributions."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or the variance of this distribution is going to be equal to the sum. Let me write it this way, right over here. So the variance, I'll re-kind of prove it. The variance of the means, or the variance of our distribution, is going to be equal to the sum of the variances of each of these sampling distributions. And we know that the variance of each of these sampling distributions is equal to the variance of this sampling distribution, is equal to the variance of the population distribution, divided by our sample size, and our sample size in this case is 100. And the variance of this sampling distribution, for our control, let me do this in a new color, for our control is going to be equal to the variance of the population distribution for the control, divided by its sample size. And since we don't know what these are, we can approximate them, especially because our n is greater than 30 for both circumstances, we can approximate these with our sample variances for each of these distributions."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "The variance of the means, or the variance of our distribution, is going to be equal to the sum of the variances of each of these sampling distributions. And we know that the variance of each of these sampling distributions is equal to the variance of this sampling distribution, is equal to the variance of the population distribution, divided by our sample size, and our sample size in this case is 100. And the variance of this sampling distribution, for our control, let me do this in a new color, for our control is going to be equal to the variance of the population distribution for the control, divided by its sample size. And since we don't know what these are, we can approximate them, especially because our n is greater than 30 for both circumstances, we can approximate these with our sample variances for each of these distributions. So let me make this clear. Our sample variances for each of these distributions. So this is going to be our sample variance 1, or actually our sample standard deviation 1 squared, which is the sample variance for that distribution, over 100, plus my sample standard deviation for the control squared, which is the sample variance, standard deviation squared is just the variance, divided by 100."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "And since we don't know what these are, we can approximate them, especially because our n is greater than 30 for both circumstances, we can approximate these with our sample variances for each of these distributions. So let me make this clear. Our sample variances for each of these distributions. So this is going to be our sample variance 1, or actually our sample standard deviation 1 squared, which is the sample variance for that distribution, over 100, plus my sample standard deviation for the control squared, which is the sample variance, standard deviation squared is just the variance, divided by 100. And this will give us the variance for this distribution. And if we want the standard deviation, we just take the square roots of both sides. If we want the standard deviation of this distribution right here, this is the variance right now, so we just need to take the square root."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be our sample variance 1, or actually our sample standard deviation 1 squared, which is the sample variance for that distribution, over 100, plus my sample standard deviation for the control squared, which is the sample variance, standard deviation squared is just the variance, divided by 100. And this will give us the variance for this distribution. And if we want the standard deviation, we just take the square roots of both sides. If we want the standard deviation of this distribution right here, this is the variance right now, so we just need to take the square root. So let's calculate this. We actually know these values. S1, our sample standard deviation for group 1 is 4.67."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we want the standard deviation of this distribution right here, this is the variance right now, so we just need to take the square root. So let's calculate this. We actually know these values. S1, our sample standard deviation for group 1 is 4.67. We wrote it right here as well. It's 4.67 and 4.04. So this is 4.67, and this number right here is 4.04."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "S1, our sample standard deviation for group 1 is 4.67. We wrote it right here as well. It's 4.67 and 4.04. So this is 4.67, and this number right here is 4.04. The s is 4.67, we're going to have to square it, and the s2 is 4.04, we're going to have to square it. So let's calculate that. So we get, we're going to take the square root of 4.67 squared, divided by 100, plus 4.04 squared, divided by 100, and then close the parentheses, and we get.617."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is 4.67, and this number right here is 4.04. The s is 4.67, we're going to have to square it, and the s2 is 4.04, we're going to have to square it. So let's calculate that. So we get, we're going to take the square root of 4.67 squared, divided by 100, plus 4.04 squared, divided by 100, and then close the parentheses, and we get.617. So this is equal to, let me write it right here, this is going to be equal to 0.617. So if we go back up over here, we calculated the standard deviation of this distribution to be 0.617. So now we can actually calculate our interval, because this is going to be 0.617."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we get, we're going to take the square root of 4.67 squared, divided by 100, plus 4.04 squared, divided by 100, and then close the parentheses, and we get.617. So this is equal to, let me write it right here, this is going to be equal to 0.617. So if we go back up over here, we calculated the standard deviation of this distribution to be 0.617. So now we can actually calculate our interval, because this is going to be 0.617. So if you want 1.96 times that, so we get 1.96 times that.617, I'll just write the answer we just got, so we get 1.21. So this is, this number right here, this number right here is 1.21. So the 95% confidence interval is going to be, is going to be the difference of our means, 1.91 plus or minus this number, 1.21."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So now we can actually calculate our interval, because this is going to be 0.617. So if you want 1.96 times that, so we get 1.96 times that.617, I'll just write the answer we just got, so we get 1.21. So this is, this number right here, this number right here is 1.21. So the 95% confidence interval is going to be, is going to be the difference of our means, 1.91 plus or minus this number, 1.21. So what's our confidence interval? If we subtract it, so the low end of our confidence interval, and I'm running out of space, the low end, 1.91 minus 1.21 is just, what is that? That's just.7."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the 95% confidence interval is going to be, is going to be the difference of our means, 1.91 plus or minus this number, 1.21. So what's our confidence interval? If we subtract it, so the low end of our confidence interval, and I'm running out of space, the low end, 1.91 minus 1.21 is just, what is that? That's just.7. So the low end is.7, and then the high end, 1.91 plus 1.21, what is that? That's 2.12. Let me just make sure that my brain sometimes doesn't work properly when I'm making these videos."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's just.7. So the low end is.7, and then the high end, 1.91 plus 1.21, what is that? That's 2.12. Let me just make sure that my brain sometimes doesn't work properly when I'm making these videos. 3.12, good thing I read it, 3.12, of course. Yeah, 3.12. So let me, so it is 3.12."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me just make sure that my brain sometimes doesn't work properly when I'm making these videos. 3.12, good thing I read it, 3.12, of course. Yeah, 3.12. So let me, so it is 3.12. So, and just be clear, there's not a pure 95% chance that the true difference of the true means lies in this. We are just confident that there's a 95% chance, and we always have to put that little confidence there because remember, we didn't actually know the population standard deviations or the population variances. We estimated them with our sample, and because of that, we don't know that it's an exact probability."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me, so it is 3.12. So, and just be clear, there's not a pure 95% chance that the true difference of the true means lies in this. We are just confident that there's a 95% chance, and we always have to put that little confidence there because remember, we didn't actually know the population standard deviations or the population variances. We estimated them with our sample, and because of that, we don't know that it's an exact probability. We just have to say we're confident that it's a 95% probability, and that's why it's really just, we just say it's a confidence interval. It's a pure probability. But it's a pretty neat result."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "We estimated them with our sample, and because of that, we don't know that it's an exact probability. We just have to say we're confident that it's a 95% probability, and that's why it's really just, we just say it's a confidence interval. It's a pure probability. But it's a pretty neat result. We are now, we have this 95% confidence interval, so we're confident that there's a 95% chance that the true difference of these two samples, and remember, the sample means, the means of the sample, the difference between the, let me make it very clear, the difference between the means of the sample is, or let me put it, the sample means, the expected value of the sample means is actually the same thing as the expected value of the populations. And so, what this is giving us is actually a confidence interval for the true difference between the populations. If you were to give everyone, every possible person, diet one, and every possible person, diet two, this is giving us a confidence interval for the true population means."}, {"video_title": "Confidence interval of difference of means   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it's a pretty neat result. We are now, we have this 95% confidence interval, so we're confident that there's a 95% chance that the true difference of these two samples, and remember, the sample means, the means of the sample, the difference between the, let me make it very clear, the difference between the means of the sample is, or let me put it, the sample means, the expected value of the sample means is actually the same thing as the expected value of the populations. And so, what this is giving us is actually a confidence interval for the true difference between the populations. If you were to give everyone, every possible person, diet one, and every possible person, diet two, this is giving us a confidence interval for the true population means. And so when you look at this, it looks like diet one actually does do something because in any case, even at the low end of the confidence interval, you still have a greater weight loss than diet two. Hopefully that doesn't confuse you too much. In the next video, we're actually going to do a hypothesis test with the same data."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You could go on an island beach vacation. Island. Island beach vacation. You could go skiing on a ski vacation. Or you could go camping. Now those aren't the only possibilities because for each of those vacations, there's different amount of time that you could go on them. So you could go for one day."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You could go skiing on a ski vacation. Or you could go camping. Now those aren't the only possibilities because for each of those vacations, there's different amount of time that you could go on them. So you could go for one day. You could go for two days. Two days. Or you could go for three days."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So you could go for one day. You could go for two days. Two days. Or you could go for three days. Three, it's in a different color. You could go for three days. You could go for three days."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Or you could go for three days. Three, it's in a different color. You could go for three days. You could go for three days. So the first question I'd want to know is, well what is the, and they're gonna randomly pick either a one day ski vacation or a two day island vacation. But the first question I want to know is what are all of the possible outcomes here? What is the sample space?"}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You could go for three days. So the first question I'd want to know is, well what is the, and they're gonna randomly pick either a one day ski vacation or a two day island vacation. But the first question I want to know is what are all of the possible outcomes here? What is the sample space? What is the space from which we are going to pick your particular vacation package? Well for the sample space, we can construct a grid which you can see that I've essentially been constructing while I wrote down all of the possibilities. So let me draw out the sample space with these uneven looking grid lines."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "What is the sample space? What is the space from which we are going to pick your particular vacation package? Well for the sample space, we can construct a grid which you can see that I've essentially been constructing while I wrote down all of the possibilities. So let me draw out the sample space with these uneven looking grid lines. Alright. I think you get the picture. Alright, so you could go, and I'll just abbreviate it."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So let me draw out the sample space with these uneven looking grid lines. Alright. I think you get the picture. Alright, so you could go, and I'll just abbreviate it. You could go on a one day island. A one day island trip. This one I, this is a one day island trip."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Alright, so you could go, and I'll just abbreviate it. You could go on a one day island. A one day island trip. This one I, this is a one day island trip. You could go on a two day. Two day. Actually let me just write it this way."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "This one I, this is a one day island trip. You could go on a two day. Two day. Actually let me just write it this way. All of these are gonna be one day. Right, cause on the one day column. All of these are going to be two days."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Actually let me just write it this way. All of these are gonna be one day. Right, cause on the one day column. All of these are going to be two days. Two days. And all of these are going to be three days cause it's on the three day column. And all of the ones in this row are gonna be island trips."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "All of these are going to be two days. Two days. And all of these are going to be three days cause it's on the three day column. And all of the ones in this row are gonna be island trips. So it's a one day island trip, two day island trip, three day island trip. This second row, it's all ski trips. One day ski trip, two day ski trip, three day ski trip."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And all of the ones in this row are gonna be island trips. So it's a one day island trip, two day island trip, three day island trip. This second row, it's all ski trips. One day ski trip, two day ski trip, three day ski trip. And then finally everything in this third row, they're camping trips. One day camping trip, two day camping trip, three day camping trip. So just like that, we have constructed the sample space right over here."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "One day ski trip, two day ski trip, three day ski trip. And then finally everything in this third row, they're camping trips. One day camping trip, two day camping trip, three day camping trip. So just like that, we have constructed the sample space right over here. You see that there's one, two, three, four, five, six, seven, eight, nine outcomes. And let's say that each of these outcomes are a little piece of paper and they put it in a barrel and they roll it up. And for our purposes, we can assume that they are all equally likely outcomes."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So just like that, we have constructed the sample space right over here. You see that there's one, two, three, four, five, six, seven, eight, nine outcomes. And let's say that each of these outcomes are a little piece of paper and they put it in a barrel and they roll it up. And for our purposes, we can assume that they are all equally likely outcomes. So we're gonna assume equally likely outcomes. So if we do assume equally likely outcomes, we can figure out a probability. Maybe you live in some place that's cold and you're really not in the mood to go skiing."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And for our purposes, we can assume that they are all equally likely outcomes. So we're gonna assume equally likely outcomes. So if we do assume equally likely outcomes, we can figure out a probability. Maybe you live in some place that's cold and you're really not in the mood to go skiing. In fact, you'd like to spend several days away from the snow. So let's ask ourselves a question. What is the probability that you're going to win something at least two days on a vacation without snow?"}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Maybe you live in some place that's cold and you're really not in the mood to go skiing. In fact, you'd like to spend several days away from the snow. So let's ask ourselves a question. What is the probability that you're going to win something at least two days on a vacation without snow? Two days on vacation without snow. You're going to randomly pick one of these nine outcomes. What's the probability that it's going to be at least, it's going to give you a vacation that gives you at least a vacation with two days without snow?"}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "What is the probability that you're going to win something at least two days on a vacation without snow? Two days on vacation without snow. You're going to randomly pick one of these nine outcomes. What's the probability that it's going to be at least, it's going to give you a vacation that gives you at least a vacation with two days without snow? Well, let's just think a little bit about it. We know the sample space and we know each of the outcomes are equally likely. There are nine equal outcomes here."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "What's the probability that it's going to be at least, it's going to give you a vacation that gives you at least a vacation with two days without snow? Well, let's just think a little bit about it. We know the sample space and we know each of the outcomes are equally likely. There are nine equal outcomes here. So let's write that down. We got nine equal outcomes. Now how many of the outcomes satisfy this event, this constraint, at least two days of vacation, let me write this, two days without snow, whether it falls or touching it or whatever."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "There are nine equal outcomes here. So let's write that down. We got nine equal outcomes. Now how many of the outcomes satisfy this event, this constraint, at least two days of vacation, let me write this, two days without snow, whether it falls or touching it or whatever. So this is, you're essentially avoiding skiing. You want at least two days on something other than skiing. We're assuming you're not going to go camping in some type of alpine."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now how many of the outcomes satisfy this event, this constraint, at least two days of vacation, let me write this, two days without snow, whether it falls or touching it or whatever. So this is, you're essentially avoiding skiing. You want at least two days on something other than skiing. We're assuming you're not going to go camping in some type of alpine. You're camping in some place that's warm. Let's think about these outcomes. So this one is no snow, but it's only one day."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "We're assuming you're not going to go camping in some type of alpine. You're camping in some place that's warm. Let's think about these outcomes. So this one is no snow, but it's only one day. This is two days without snow. So we can circle that one. This is three days without snow."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So this one is no snow, but it's only one day. This is two days without snow. So we can circle that one. This is three days without snow. So we can circle that one. All of these have snow. This is one day without snow, so we're not going to do this one."}, {"video_title": "Probability from compound sample space   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "This is three days without snow. So we can circle that one. All of these have snow. This is one day without snow, so we're not going to do this one. This is two days without snow. And this is three days without snow. And so four of the equally likely outcomes satisfy this constraint."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Find the range and the mid-range of the following sets of numbers. So what the range tells us is essentially how spread apart these numbers are. And the way that you calculate it is you just take the difference between the largest of these numbers and the smallest of these numbers. And so if we look at the largest of these numbers, I'll circle it in magenta, it looks like it is 94. 94 is larger than every other number here. So that's the largest of the numbers. And from that we want to subtract the smallest of the numbers."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if we look at the largest of these numbers, I'll circle it in magenta, it looks like it is 94. 94 is larger than every other number here. So that's the largest of the numbers. And from that we want to subtract the smallest of the numbers. And the smallest of the numbers in our set right over here is 65. So you want to subtract 65 from 94. And this is equal to, let's see, if this was 95 minus 65 would be 30."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And from that we want to subtract the smallest of the numbers. And the smallest of the numbers in our set right over here is 65. So you want to subtract 65 from 94. And this is equal to, let's see, if this was 95 minus 65 would be 30. 94 is one less than that, so it is 29. So the larger this number is, that means the more spread out, the larger the difference between the largest and the smallest number. The smaller this is, that means the tighter the range, just to use the word itself, of the numbers actually are."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is equal to, let's see, if this was 95 minus 65 would be 30. 94 is one less than that, so it is 29. So the larger this number is, that means the more spread out, the larger the difference between the largest and the smallest number. The smaller this is, that means the tighter the range, just to use the word itself, of the numbers actually are. So that's the range. The mid-range is one way of thinking, to some degree, of kind of central tendency. So mid-range."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The smaller this is, that means the tighter the range, just to use the word itself, of the numbers actually are. So that's the range. The mid-range is one way of thinking, to some degree, of kind of central tendency. So mid-range. And what you do with the mid-range is you take the average of the largest number and the smallest number. So here we took the difference, that's the range. The mid-range would be the average of these two numbers."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So mid-range. And what you do with the mid-range is you take the average of the largest number and the smallest number. So here we took the difference, that's the range. The mid-range would be the average of these two numbers. So it would be 94 plus 65. When we talk about average, I'm talking about the arithmetic mean, over 2. So this is going to be what?"}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mid-range would be the average of these two numbers. So it would be 94 plus 65. When we talk about average, I'm talking about the arithmetic mean, over 2. So this is going to be what? 90 plus 60 is 150. 150 plus 4 plus 5 is 159. 159 divided by 2 is equal to, 150 divided by 2 is 75."}, {"video_title": "Finding the range and mid-range   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be what? 90 plus 60 is 150. 150 plus 4 plus 5 is 159. 159 divided by 2 is equal to, 150 divided by 2 is 75. 9 divided by 2 is 4 and a half, so this would be 79.5. So it's one kind of way of thinking about the middle of these numbers. Another way is obviously the arithmetic mean, where you actually take the arithmetic mean of everything here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the last several videos, we did some fairly hairy mathematics, and you might have even skipped them. But we got to a pretty neat result. We got to a formula for the slope and y-intercept of the best-fitting regression line when you measure the error by the squared distance to that line. And our formula is, and I'll just rewrite it here just so we have something neat to look at. So the slope of that line is going to be the mean of x's times the mean of the y's minus the mean of the xy's. And don't worry, this seems really confusing. We're going to actually do an example of this in a few seconds."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And our formula is, and I'll just rewrite it here just so we have something neat to look at. So the slope of that line is going to be the mean of x's times the mean of the y's minus the mean of the xy's. And don't worry, this seems really confusing. We're going to actually do an example of this in a few seconds. Divided by the mean of x squared minus the mean of the x squared's. And if this looks a little different than what you see in your statistics class or your textbook, you might see this swapped around. If you multiply both the numerator and the denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's, all of that over the mean of the x squared's minus the mean of the x's squared."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to actually do an example of this in a few seconds. Divided by the mean of x squared minus the mean of the x squared's. And if this looks a little different than what you see in your statistics class or your textbook, you might see this swapped around. If you multiply both the numerator and the denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's, all of that over the mean of the x squared's minus the mean of the x's squared. These are obviously the same thing. They're just multiplying the numerator and the denominator by negative 1, which is the same thing as multiplying the whole thing by 1. And of course, whatever you get for y, oh sorry, whatever you get for m, you can then just substitute back in this, back over here to get your b."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you multiply both the numerator and the denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's, all of that over the mean of the x squared's minus the mean of the x's squared. These are obviously the same thing. They're just multiplying the numerator and the denominator by negative 1, which is the same thing as multiplying the whole thing by 1. And of course, whatever you get for y, oh sorry, whatever you get for m, you can then just substitute back in this, back over here to get your b. Your b is going to be equal to the mean of the y's minus your m, whatever m value you got over here. Let me write that in yellow so it's very clear. You solved for the m value here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And of course, whatever you get for y, oh sorry, whatever you get for m, you can then just substitute back in this, back over here to get your b. Your b is going to be equal to the mean of the y's minus your m, whatever m value you got over here. Let me write that in yellow so it's very clear. You solved for the m value here. Minus m times the mean of the x's. And this is all you need. So let's actually put that into practice."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "You solved for the m value here. Minus m times the mean of the x's. And this is all you need. So let's actually put that into practice. So let's say I have 3 points and I'm going to make sure that these points aren't collinear. Because otherwise it wouldn't be interesting. So let me draw 3 points over here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's actually put that into practice. So let's say I have 3 points and I'm going to make sure that these points aren't collinear. Because otherwise it wouldn't be interesting. So let me draw 3 points over here. Let's say that one point is the point 1, 1, 2. So this is 1, 2. So we have the point right over here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me draw 3 points over here. Let's say that one point is the point 1, 1, 2. So this is 1, 2. So we have the point right over here. We have the point 1, 2. And then we also have the point, let's say we also have the point, oh I don't know, let's say we also have the point 2, 1. Let's say we have the point 2, 1."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have the point right over here. We have the point 1, 2. And then we also have the point, let's say we also have the point, oh I don't know, let's say we also have the point 2, 1. Let's say we have the point 2, 1. And then let's say we also have the point 3, I don't know, let's do something a little bit crazy. Let's do 3, 4. So 3, well let's do it over here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say we have the point 2, 1. And then let's say we also have the point 3, I don't know, let's do something a little bit crazy. Let's do 3, 4. So 3, well let's do it over here. Let's do 4, 3 just so we can actually fit it on the page. So 4, 3 is going to be something right over here. So this is 4, 3."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 3, well let's do it over here. Let's do 4, 3 just so we can actually fit it on the page. So 4, 3 is going to be something right over here. So this is 4, 3. So those are our 3 points and what we want to do is find the best fitting regression line, which we suspect is going to look something, we'll see what it looks like, but I suspect it's going to look something like that. We'll see what it actually looks like using our formulas, which we have proven. So a good place to start is just to calculate these things ahead of time and then just substitute them back into the equation."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is 4, 3. So those are our 3 points and what we want to do is find the best fitting regression line, which we suspect is going to look something, we'll see what it looks like, but I suspect it's going to look something like that. We'll see what it actually looks like using our formulas, which we have proven. So a good place to start is just to calculate these things ahead of time and then just substitute them back into the equation. So what's the mean of our x's? The mean of our x's is going to be 1 plus, I'll do the same colors, 1 plus 2 plus 4 divided by, I'll do this in a neutral color, the mean of our x's divided by 3. And what's this going to be?"}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So a good place to start is just to calculate these things ahead of time and then just substitute them back into the equation. So what's the mean of our x's? The mean of our x's is going to be 1 plus, I'll do the same colors, 1 plus 2 plus 4 divided by, I'll do this in a neutral color, the mean of our x's divided by 3. And what's this going to be? 1 plus 2 is 3 plus 4 is 7 divided by 3. It is equal to 7 over 3. Now, what is the mean of our y's?"}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what's this going to be? 1 plus 2 is 3 plus 4 is 7 divided by 3. It is equal to 7 over 3. Now, what is the mean of our y's? The mean of our y's, once again I'm going to do this in a neutral color, the mean of our y's is equal to 2 plus 1 plus 3, all of that over 3. So this is 2 plus 1 is 3 plus 3 is 6 divided by 3 is equal to 2. This is 6 divided by 3 is equal to 2."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what is the mean of our y's? The mean of our y's, once again I'm going to do this in a neutral color, the mean of our y's is equal to 2 plus 1 plus 3, all of that over 3. So this is 2 plus 1 is 3 plus 3 is 6 divided by 3 is equal to 2. This is 6 divided by 3 is equal to 2. Now, what is the mean of our xy's? Well, over here, it's going to be, so our first xy over here is 1 times 2 plus 2 times 1 plus 4 times 3. And we have 3 of these xy's, so divided by 3."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 6 divided by 3 is equal to 2. Now, what is the mean of our xy's? Well, over here, it's going to be, so our first xy over here is 1 times 2 plus 2 times 1 plus 4 times 3. And we have 3 of these xy's, so divided by 3. So what's this going to be equal to? 2 plus 2, which is 4, 4 plus 12, which is 16. So it's going to be 16 over 3."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we have 3 of these xy's, so divided by 3. So what's this going to be equal to? 2 plus 2, which is 4, 4 plus 12, which is 16. So it's going to be 16 over 3. Did I get that right? 4 plus 12, yep, 16 over 3. And then the last one we have to calculate is the mean of the x squareds."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be 16 over 3. Did I get that right? 4 plus 12, yep, 16 over 3. And then the last one we have to calculate is the mean of the x squareds. So what's the mean of the x squareds? The first x squared is just going to be 1 squared, this 1 squared right over here, plus this 2 squared, plus 2 squared right over here, plus this 4 squared, plus this 4 squared. And we have 3 data points again."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the last one we have to calculate is the mean of the x squareds. So what's the mean of the x squareds? The first x squared is just going to be 1 squared, this 1 squared right over here, plus this 2 squared, plus 2 squared right over here, plus this 4 squared, plus this 4 squared. And we have 3 data points again. So this is 1 plus 4, which is 5, plus 16. 5 plus 16 is equal to 21 over 3, which is equal to 7. So that worked out to a pretty neat number."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we have 3 data points again. So this is 1 plus 4, which is 5, plus 16. 5 plus 16 is equal to 21 over 3, which is equal to 7. So that worked out to a pretty neat number. So let's actually find our m's and our b's. So our slope, our optimal slope for our regression line, the mean of the x's, it's going to be 7 thirds, 7 over 3, times the mean of the y's, the mean of the y's is 2, minus the mean of the xy's, well that's 16 over 3, 16 over 3, and then all of that, all of that over the mean of the x's, the mean of the x's is 7 thirds squared. So 7 over 3 squared, minus the mean of the x squared, so it's going to be minus this 7 right over here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that worked out to a pretty neat number. So let's actually find our m's and our b's. So our slope, our optimal slope for our regression line, the mean of the x's, it's going to be 7 thirds, 7 over 3, times the mean of the y's, the mean of the y's is 2, minus the mean of the xy's, well that's 16 over 3, 16 over 3, and then all of that, all of that over the mean of the x's, the mean of the x's is 7 thirds squared. So 7 over 3 squared, minus the mean of the x squared, so it's going to be minus this 7 right over here. And now we just have to do a little bit of mathematics here. I'm tempted to get out my calculator, but I'll resist the temptation. It's nice to keep things as fractions."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 7 over 3 squared, minus the mean of the x squared, so it's going to be minus this 7 right over here. And now we just have to do a little bit of mathematics here. I'm tempted to get out my calculator, but I'll resist the temptation. It's nice to keep things as fractions. So let's see if I can calculate this. So this is going to be equal to, this is 14 over 3, 14 over 3 minus 16 over 3, all of that over, this is 49, 49 over 9, right? 7 thirds squared is 49 over 9."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's nice to keep things as fractions. So let's see if I can calculate this. So this is going to be equal to, this is 14 over 3, 14 over 3 minus 16 over 3, all of that over, this is 49, 49 over 9, right? 7 thirds squared is 49 over 9. And then minus 7, if I wanted to express that as something over 9, that's the same thing as 63, that's the same thing as 63 over 9. And so in our numerator, we get negative 2 thirds, negative 2 over 3, and then in our denominator, what's 49 minus 63? That's negative, let's see, that's negative 14, that's negative 14 over 9, and this is the same thing as negative 2 thirds times 9 over 14, negative 9 over 14, and then if we divide the numerator and denominator by 3, well the negatives are going to cancel out first of all, we divide by 3, that becomes a 1, that becomes a 3, divide by 2, becomes a 1, that becomes a 7."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "7 thirds squared is 49 over 9. And then minus 7, if I wanted to express that as something over 9, that's the same thing as 63, that's the same thing as 63 over 9. And so in our numerator, we get negative 2 thirds, negative 2 over 3, and then in our denominator, what's 49 minus 63? That's negative, let's see, that's negative 14, that's negative 14 over 9, and this is the same thing as negative 2 thirds times 9 over 14, negative 9 over 14, and then if we divide the numerator and denominator by 3, well the negatives are going to cancel out first of all, we divide by 3, that becomes a 1, that becomes a 3, divide by 2, becomes a 1, that becomes a 7. So our slope is 3 sevenths, not too bad. Now we can go back and figure out our y intercept. So let's figure out our y intercept using this right over here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's negative, let's see, that's negative 14, that's negative 14 over 9, and this is the same thing as negative 2 thirds times 9 over 14, negative 9 over 14, and then if we divide the numerator and denominator by 3, well the negatives are going to cancel out first of all, we divide by 3, that becomes a 1, that becomes a 3, divide by 2, becomes a 1, that becomes a 7. So our slope is 3 sevenths, not too bad. Now we can go back and figure out our y intercept. So let's figure out our y intercept using this right over here. So our y intercept, b, is going to be equal to the mean of the y's, the mean of the y's is 2, minus our slope, we just figured out our slope to be 3 sevenths, so minus 3 sevenths, times the mean of the x's, which is 7 thirds, times 7 thirds. Well these just are the reciprocal of each other, so they cancel out, that just becomes 1. So our y intercept is literally just 2 minus 1, so it equals 1."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's figure out our y intercept using this right over here. So our y intercept, b, is going to be equal to the mean of the y's, the mean of the y's is 2, minus our slope, we just figured out our slope to be 3 sevenths, so minus 3 sevenths, times the mean of the x's, which is 7 thirds, times 7 thirds. Well these just are the reciprocal of each other, so they cancel out, that just becomes 1. So our y intercept is literally just 2 minus 1, so it equals 1. So we have the equation for our line. Our regression line is going to be y is equal to, we figured out m, m is 3 sevenths, y is equal to 3 sevenths x, plus our y intercept is 1, plus 1. And we are done."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our y intercept is literally just 2 minus 1, so it equals 1. So we have the equation for our line. Our regression line is going to be y is equal to, we figured out m, m is 3 sevenths, y is equal to 3 sevenths x, plus our y intercept is 1, plus 1. And we are done. We are done. So let's actually try to graph this. So our y intercept is going to be 1, it's going to be right over there, and the slope of our line is 3 sevenths."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we are done. We are done. So let's actually try to graph this. So our y intercept is going to be 1, it's going to be right over there, and the slope of our line is 3 sevenths. So for every 7 we run, we rise 3. Another way to think of it, for every 3 and a half we run, we rise 1 and a half. So we are going to go 1 and a half right over here."}, {"video_title": "Regression line example   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our y intercept is going to be 1, it's going to be right over there, and the slope of our line is 3 sevenths. So for every 7 we run, we rise 3. Another way to think of it, for every 3 and a half we run, we rise 1 and a half. So we are going to go 1 and a half right over here. So this line, if you were to graph it, and obviously I'm hand drawing it, so it's not going to be that exact, is going to look like that right over there. It actually won't go directly through that line, so I don't want to give you that impression. So it might look something like this."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So we've defined two random variables here. The first random variable, X, is the weight of the cereal in a random box of our favorite cereal, Mattheys, a random closed box of our favorite cereal, Mattheys. And we know a few other things about it. We know what the expected value of X is. It is equal to 16 ounces. In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We know what the expected value of X is. It is equal to 16 ounces. In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certain weight, then the company that produces it just throws out that box. And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certain weight, then the company that produces it just throws out that box. And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. But once again, there's going to be some variation. Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. But once again, there's going to be some variation. Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So given all of this information, what I wanna do is, let's just say I take a random box of mathies, and I take a random filled bowl, and I wanna think about the combined weight in the closed box and the filled bowl. So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So given all of this information, what I wanna do is, let's just say I take a random box of mathies, and I take a random filled bowl, and I wanna think about the combined weight in the closed box and the filled bowl. So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value of Y, and so it would be 16 plus four ounces. In this case, this would be equal to 20 ounces. But what about the variation?"}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value of Y, and so it would be 16 plus four ounces. In this case, this would be equal to 20 ounces. But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum?"}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario?"}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer here and a three-ouncer here, and it is 18 ounces. And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer here and a three-ouncer here, and it is 18 ounces. And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were from their respective means. So hopefully this gives you an intuition for why this makes sense. But let me ask you another question."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were from their respective means. So hopefully this gives you an intuition for why this makes sense. But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be? Would you subtract the variances of each of the random variables here?"}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be? Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on?"}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here. That would be the lowest value that you could take on."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here. That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know this. Let me just write it using this sigma notation."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know this. Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X?"}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64. The standard deviation of Y is 0.6."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64. The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. You add these two up, and you are going to get one."}, {"video_title": "Variance of sum and difference of random variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. You add these two up, and you are going to get one. So the variance of the sum is one. And then if you take the square root of both of these, you get the standard deviation of the sum is also going to be one. And that just happened to work out because we're dealing with the scenario where the variance, where the square root of one is, well, one."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I'm trying to judge how many years of experience we have at the Khan Academy, or on average, how many years of experience we have. And in particular, the particular type of average we'll focus on is the arithmetic mean. So I go and I survey the folks there. And let's say this was when Khan Academy was a smaller organization, when there were only five people in the organization. And I find, and I'm surveying the entire population, so years of experience, the entire population of Khan Academy, because that's what I care about, years of experience at our organization, at Khan Academy. This is when we had five people. And I were to go, we're now 36 people."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's say this was when Khan Academy was a smaller organization, when there were only five people in the organization. And I find, and I'm surveying the entire population, so years of experience, the entire population of Khan Academy, because that's what I care about, years of experience at our organization, at Khan Academy. This is when we had five people. And I were to go, we're now 36 people. I don't want to date this video too much. But let's say I go and I say, OK, there's one person straight out of college. They have one year of experience, or recently out of college, somebody with three years of experience, someone with five years of experience, someone with seven years of experience, and someone very experienced, or reasonably experienced, with 14 years of experience."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I were to go, we're now 36 people. I don't want to date this video too much. But let's say I go and I say, OK, there's one person straight out of college. They have one year of experience, or recently out of college, somebody with three years of experience, someone with five years of experience, someone with seven years of experience, and someone very experienced, or reasonably experienced, with 14 years of experience. So based on this data point, and this is our population for years of experience. I'm assuming that we only have five people in the organization at this point. What would be the population mean for the years of experience?"}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "They have one year of experience, or recently out of college, somebody with three years of experience, someone with five years of experience, someone with seven years of experience, and someone very experienced, or reasonably experienced, with 14 years of experience. So based on this data point, and this is our population for years of experience. I'm assuming that we only have five people in the organization at this point. What would be the population mean for the years of experience? What is the mean years of experience for my population? Well, we can just calculate that. Our mean experience, and I'm going to denote it with mu, because we're talking about the population now."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What would be the population mean for the years of experience? What is the mean years of experience for my population? Well, we can just calculate that. Our mean experience, and I'm going to denote it with mu, because we're talking about the population now. This is a parameter for the population. It's going to be equal to the sum from our first data point, so data point one, all the way to data point, in this case, data point five. We have five data points."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our mean experience, and I'm going to denote it with mu, because we're talking about the population now. This is a parameter for the population. It's going to be equal to the sum from our first data point, so data point one, all the way to data point, in this case, data point five. We have five data points. So we're going to take all from the first data point, the second data point, the third data point, all the way to the fifth. So this is going to be equal to x1 plus x, and I'm going to divide it all by the number of data points I have, plus x2, plus x3, plus x4, plus x sub 5, subscript 5, all of that over 5. And as we said, this is a very fancy way of saying, I'm going to sum up all of these things, I'm going to sum up all of these things, and then divide by the number of things we have."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have five data points. So we're going to take all from the first data point, the second data point, the third data point, all the way to the fifth. So this is going to be equal to x1 plus x, and I'm going to divide it all by the number of data points I have, plus x2, plus x3, plus x4, plus x sub 5, subscript 5, all of that over 5. And as we said, this is a very fancy way of saying, I'm going to sum up all of these things, I'm going to sum up all of these things, and then divide by the number of things we have. So let's do that. Get the calculator out. So I'm going to add them all up."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And as we said, this is a very fancy way of saying, I'm going to sum up all of these things, I'm going to sum up all of these things, and then divide by the number of things we have. So let's do that. Get the calculator out. So I'm going to add them all up. So I'm going to add them all up. 1 plus 3 plus 5. I really don't need a calculator for this."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm going to add them all up. So I'm going to add them all up. 1 plus 3 plus 5. I really don't need a calculator for this. Plus 7 plus 14. So that's five data points, and I'm going to divide by 5. And I get 6."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I really don't need a calculator for this. Plus 7 plus 14. So that's five data points, and I'm going to divide by 5. And I get 6. So the population mean for years of experience at my organization is 6. 6 years of experience. Well, that's, I guess, interesting."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I get 6. So the population mean for years of experience at my organization is 6. 6 years of experience. Well, that's, I guess, interesting. But now I want to ask another question. I want to get some measure of how much spread there is around that mean, or how much do the data points vary around that mean? And obviously, I can give someone all the data points."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that's, I guess, interesting. But now I want to ask another question. I want to get some measure of how much spread there is around that mean, or how much do the data points vary around that mean? And obviously, I can give someone all the data points. But instead, I actually want to come up with a parameter that somehow represents how much all of these things, on average, are varying from this number right here. Or maybe I will call that thing the variance. And so what I do, so the variance, and I will do, and this is a population variance that I'm talking about, just to be clear."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And obviously, I can give someone all the data points. But instead, I actually want to come up with a parameter that somehow represents how much all of these things, on average, are varying from this number right here. Or maybe I will call that thing the variance. And so what I do, so the variance, and I will do, and this is a population variance that I'm talking about, just to be clear. It's a parameter. The population variance I'm going to denote with the Greek letter sigma, lowercase sigma. This is capital sigma."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so what I do, so the variance, and I will do, and this is a population variance that I'm talking about, just to be clear. It's a parameter. The population variance I'm going to denote with the Greek letter sigma, lowercase sigma. This is capital sigma. Lowercase sigma squared. And I'm going to say, well, I'm going to take the distance from each of these points to the mean. And just so I get a positive value, I'm going to square it."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is capital sigma. Lowercase sigma squared. And I'm going to say, well, I'm going to take the distance from each of these points to the mean. And just so I get a positive value, I'm going to square it. And then I'm going to divide by the number of data points that I have. So essentially, I'm going to find the average squared distance. Now, that might sound very complicated, but let's actually work it out."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just so I get a positive value, I'm going to square it. And then I'm going to divide by the number of data points that I have. So essentially, I'm going to find the average squared distance. Now, that might sound very complicated, but let's actually work it out. So I'll take my first data point. I'll take that data point, and I will subtract our mean from it. So this is going to give me a negative number, but if I square it, it's going to be positive."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, that might sound very complicated, but let's actually work it out. So I'll take my first data point. I'll take that data point, and I will subtract our mean from it. So this is going to give me a negative number, but if I square it, it's going to be positive. So it's essentially going to be the squared distance between 1 and my mean. And then to that, I'm going to add the squared distance between 3 and my mean. And to that, I'm going to add the squared distance between 5 and my mean."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to give me a negative number, but if I square it, it's going to be positive. So it's essentially going to be the squared distance between 1 and my mean. And then to that, I'm going to add the squared distance between 3 and my mean. And to that, I'm going to add the squared distance between 5 and my mean. And since I'm squaring, it doesn't matter if I do 5 minus 6 or 6 minus 5. When I square it, I'm going to get a positive result regardless. And then to that, I'm going to add the squared distance between 7 and my mean."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to that, I'm going to add the squared distance between 5 and my mean. And since I'm squaring, it doesn't matter if I do 5 minus 6 or 6 minus 5. When I square it, I'm going to get a positive result regardless. And then to that, I'm going to add the squared distance between 7 and my mean. So 7 minus 6 squared. All of this, this is my population mean that I'm finding the difference between. And then finally, the squared difference between 14 and my mean."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then to that, I'm going to add the squared distance between 7 and my mean. So 7 minus 6 squared. All of this, this is my population mean that I'm finding the difference between. And then finally, the squared difference between 14 and my mean. And then I'm going to find essentially the mean of these squared distances. So I have 5 squared distances right over here. So let me divide by 5."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, the squared difference between 14 and my mean. And then I'm going to find essentially the mean of these squared distances. So I have 5 squared distances right over here. So let me divide by 5. So what will I get when I make this calculation right over here? Well, let's figure this out. This is going to be equal to 1 minus 6 is negative 5."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me divide by 5. So what will I get when I make this calculation right over here? Well, let's figure this out. This is going to be equal to 1 minus 6 is negative 5. Negative 5 squared is 25. 3 minus 6 is negative 3. If I square that, I get 9."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be equal to 1 minus 6 is negative 5. Negative 5 squared is 25. 3 minus 6 is negative 3. If I square that, I get 9. 5 minus 6 is negative 1. If I square it, I get positive 1. 7 minus 6 is 1."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I square that, I get 9. 5 minus 6 is negative 1. If I square it, I get positive 1. 7 minus 6 is 1. If I square it, I get positive 1. And 14 minus 6 is 8. If I square it, I get 64."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "7 minus 6 is 1. If I square it, I get positive 1. And 14 minus 6 is 8. If I square it, I get 64. And then I'm going to divide all of that by 5. And I don't need to use a calculator, but I tend to make a lot of careless mistakes when I do things while I'm making a video. So let me get 25 plus 9 plus 1 plus 1 plus 64 divided by 5."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I square it, I get 64. And then I'm going to divide all of that by 5. And I don't need to use a calculator, but I tend to make a lot of careless mistakes when I do things while I'm making a video. So let me get 25 plus 9 plus 1 plus 1 plus 64 divided by 5. So I get 20. So the average squared distance, or the mean squared distance from our population mean is equal to 20. And you might say, wait, these things aren't 20."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me get 25 plus 9 plus 1 plus 1 plus 64 divided by 5. So I get 20. So the average squared distance, or the mean squared distance from our population mean is equal to 20. And you might say, wait, these things aren't 20. Remember, it's the squared distance away from my population mean. So I squared each of these things. I liked it because it made it positive."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you might say, wait, these things aren't 20. Remember, it's the squared distance away from my population mean. So I squared each of these things. I liked it because it made it positive. And we'll see later it has other nice properties about it. Now, the last thing is how can we represent this mathematically? We already saw that we know how to represent a population mean and a sample mean mathematically like this."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I liked it because it made it positive. And we'll see later it has other nice properties about it. Now, the last thing is how can we represent this mathematically? We already saw that we know how to represent a population mean and a sample mean mathematically like this. And hopefully, we don't find it that daunting anymore. But how would we do the exact same thing? How would we denote what we did right over here?"}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We already saw that we know how to represent a population mean and a sample mean mathematically like this. And hopefully, we don't find it that daunting anymore. But how would we do the exact same thing? How would we denote what we did right over here? Well, let's just think it through. We're just saying that the population variance, we're taking the sum of each. So we're going to take each item."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "How would we denote what we did right over here? Well, let's just think it through. We're just saying that the population variance, we're taking the sum of each. So we're going to take each item. We'll start with the first item. And we're going to go to the nth item in our population. We're talking about a population here."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're going to take each item. We'll start with the first item. And we're going to go to the nth item in our population. We're talking about a population here. And we're not going to just take the item. This would just be the item. But we're going to take the item."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're talking about a population here. And we're not going to just take the item. This would just be the item. But we're going to take the item. And from that, we're going to subtract the population mean. We're going to subtract this thing. We're going to subtract this thing."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we're going to take the item. And from that, we're going to subtract the population mean. We're going to subtract this thing. We're going to subtract this thing. We're going to square it. So the way I've written it right now, this would just be the numerator. I've just taken the sum of each of these things, the sum of the difference between each data point and the population mean and squared it."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to subtract this thing. We're going to square it. So the way I've written it right now, this would just be the numerator. I've just taken the sum of each of these things, the sum of the difference between each data point and the population mean and squared it. If I really want to get the way I figured out this variance right over here, I have to divide the whole thing by the number of data points we have. So this might seem very daunting and very intimidating. But all it says is, take each of your data points."}, {"video_title": "Variance of a population   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I've just taken the sum of each of these things, the sum of the difference between each data point and the population mean and squared it. If I really want to get the way I figured out this variance right over here, I have to divide the whole thing by the number of data points we have. So this might seem very daunting and very intimidating. But all it says is, take each of your data points. Well, one, it says, figure out your population mean. Figure that out first. And then from each data point in your population, subtract out that population mean, square it, take the sum of all of those things, and then just divide by the number of data points you have."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So every day after school, you decide to go to the frozen yogurt store at exactly four o'clock, four o'clock p.m. Now because you like frozen yogurt so much, you are not a big fan of having to wait in line when you get there. You're impatient. You want your frozen yogurt immediately. And so you decide to conduct a study. You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "And so you decide to conduct a study. You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of the line. So let me make two columns here. Line size is the left column."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of the line. So let me make two columns here. Line size is the left column. And on the right column, let's say this is the number of times observed. So times, times observed. Observed."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Line size is the left column. And on the right column, let's say this is the number of times observed. So times, times observed. Observed. Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Observed. Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times. And you see two people in line, exactly eight times. And in your 50 visits, you don't see more than, you never see more than two people in line."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times. And you see two people in line, exactly eight times. And in your 50 visits, you don't see more than, you never see more than two people in line. I guess this is a very efficient cashier at this frozen yogurt store. So based on this, based on what you have observed, what would be your estimate of the probability of finding no people in line, one people in line, or two people in line, at 4 p.m. on the days after school that you visit the frozen yogurt store? And you'll say you only visited on weekdays where there are school days."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "And in your 50 visits, you don't see more than, you never see more than two people in line. I guess this is a very efficient cashier at this frozen yogurt store. So based on this, based on what you have observed, what would be your estimate of the probability of finding no people in line, one people in line, or two people in line, at 4 p.m. on the days after school that you visit the frozen yogurt store? And you'll say you only visited on weekdays where there are school days. So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day? Well, all you can do is estimate the true probability, the true theoretical probability. We don't know what that is, but you've done 50 observations here, right?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "And you'll say you only visited on weekdays where there are school days. So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day? Well, all you can do is estimate the true probability, the true theoretical probability. We don't know what that is, but you've done 50 observations here, right? This is, and notice this adds up to 50. 18 plus eight is 26, 26 plus 24 is 50. So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "We don't know what that is, but you've done 50 observations here, right? This is, and notice this adds up to 50. 18 plus eight is 26, 26 plus 24 is 50. So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? What is the relative frequency of one person, or the relative frequency of two people in line? And then we can use that as the estimates for the probability. So let's do that."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? What is the relative frequency of one person, or the relative frequency of two people in line? And then we can use that as the estimates for the probability. So let's do that. So probability estimate. I'll do it in the next column. So probability, probability estimate."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So let's do that. So probability estimate. I'll do it in the next column. So probability, probability estimate. And once again, we can do that by looking at the relative frequency. The relative frequency of zero, well, we observe that 24 times out of 50, and so 24 out of 50 is the same thing as 0.48, or you could even say that this is 48%. Now, what's the relative frequency of seeing one person in line?"}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So probability, probability estimate. And once again, we can do that by looking at the relative frequency. The relative frequency of zero, well, we observe that 24 times out of 50, and so 24 out of 50 is the same thing as 0.48, or you could even say that this is 48%. Now, what's the relative frequency of seeing one person in line? Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits. And then finally, the relative frequency of seeing a two-person line, that was eight out of the 50 visits, and so that is 0.16, and that is equal to 16% of the visits. And so there's interesting things here."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Now, what's the relative frequency of seeing one person in line? Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits. And then finally, the relative frequency of seeing a two-person line, that was eight out of the 50 visits, and so that is 0.16, and that is equal to 16% of the visits. And so there's interesting things here. Remember, these are estimates of the probability. You're doing this by essentially sampling what the line on 50 different days. You don't know, it's not gonna always be exactly this, but it's a good estimate."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "And so there's interesting things here. Remember, these are estimates of the probability. You're doing this by essentially sampling what the line on 50 different days. You don't know, it's not gonna always be exactly this, but it's a good estimate. You did it 50 times. And so based on this, you'd say, well, I'd estimate the probability of having a zero-person line is 48%. I'd estimate that the probability of having a one-person line is 36%."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "You don't know, it's not gonna always be exactly this, but it's a good estimate. You did it 50 times. And so based on this, you'd say, well, I'd estimate the probability of having a zero-person line is 48%. I'd estimate that the probability of having a one-person line is 36%. I'd estimate that the probability of having a two-person line is 16%, or 0.16. And it's important to realize that these are legitimate probabilities. Remember, to be a probability, it has to be between zero and one."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "I'd estimate that the probability of having a one-person line is 36%. I'd estimate that the probability of having a two-person line is 16%, or 0.16. And it's important to realize that these are legitimate probabilities. Remember, to be a probability, it has to be between zero and one. It has to be zero and one. And if you look at all of the possible events, it should add up to one, because at least based on your observations, these are the possibilities. Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Remember, to be a probability, it has to be between zero and one. It has to be zero and one. And if you look at all of the possible events, it should add up to one, because at least based on your observations, these are the possibilities. Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. Let's see, 36 plus 16 is 52. 52 plus 48, they add up to one. Now, once you do this, you might do something interesting."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. Let's see, 36 plus 16 is 52. 52 plus 48, they add up to one. Now, once you do this, you might do something interesting. You might say, okay, you know what? Over the next two years, you plan on visiting 500 times. So visiting 500 times."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Now, once you do this, you might do something interesting. You might say, okay, you know what? Over the next two years, you plan on visiting 500 times. So visiting 500 times. So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far? Well, it's reasonable to say, well, a good estimate of the number of times you'll see a two-person line when you visit 500 times. Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So visiting 500 times. So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far? Well, it's reasonable to say, well, a good estimate of the number of times you'll see a two-person line when you visit 500 times. Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. And so what is this going to be? Let's see, 500 divided by 50 is just 10. So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. And so what is this going to be? Let's see, 500 divided by 50 is just 10. So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. Now, to be clear, I would be shocked if it's exactly 80 ends up being the case, but this is actually a very good expectation, based on your observations. It is completely possible, first of all, that your observations were off, that this is just a random chance that you happened to observe this many, or this few times that there were two people in line. So that could be off, but even if these are very good estimates, it's possible that something, that you see a two-person line 85 out of the 500 times, or 65 out of the 500 times."}, {"video_title": "Constructing probability model from observations   7th grade   Khan Academy (2).mp3", "Sentence": "So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. Now, to be clear, I would be shocked if it's exactly 80 ends up being the case, but this is actually a very good expectation, based on your observations. It is completely possible, first of all, that your observations were off, that this is just a random chance that you happened to observe this many, or this few times that there were two people in line. So that could be off, but even if these are very good estimates, it's possible that something, that you see a two-person line 85 out of the 500 times, or 65 out of the 500 times. All of those things are possible. Whenever you think, and it's always very important to keep in mind, you're estimating the true probability here, which it's very hard to know for sure what the true probability is, but you can make estimates based on sampling the line on different days, by making these observations, by having these experiments, so to speak, each of these observations you can use an experiment, and then you can use those to set an expectation. But none of these things do you know for sure, that they're definitely gonna be exactly 80 out of the next 500 times."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment?"}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average?"}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy.mp3", "Sentence": "Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get?"}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "A statistician for a basketball team tracked the number of points that each of the 12 players on the team had in one game, and then made a stem and the leaf plot to show the data. Sometimes it's called a stem plot. How many points did the team score? And when you first look at this plot right over here, it seems a little hard to understand. Under stem you have zero, one, two. Under leaf you have all of these digits here. How does this relate to the number of points each student or each player actually scored?"}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And when you first look at this plot right over here, it seems a little hard to understand. Under stem you have zero, one, two. Under leaf you have all of these digits here. How does this relate to the number of points each student or each player actually scored? And the way to interpret a stem and leaf plot is the leaf contain, at least the way that this statistician used it, the leaf contains the smallest digit, or the ones digit, in the number of points that each player scored, and the stem contains the tens digits. And usually the leaf will contain the rightmost digit, or the ones digit, and then the stem will contain all of the other digits. And what's useful about this is it gives kind of a distribution of where the players were."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "How does this relate to the number of points each student or each player actually scored? And the way to interpret a stem and leaf plot is the leaf contain, at least the way that this statistician used it, the leaf contains the smallest digit, or the ones digit, in the number of points that each player scored, and the stem contains the tens digits. And usually the leaf will contain the rightmost digit, or the ones digit, and then the stem will contain all of the other digits. And what's useful about this is it gives kind of a distribution of where the players were. You see that most of the players scored points that started with a zero, then a few more scored points that started with a one, and then only one score scored points that started with a two, and it was actually 20 points. So let me actually write down all of this data in a way that maybe you're a little bit more used to understanding it. So I'm gonna write the zeros in purple."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And what's useful about this is it gives kind of a distribution of where the players were. You see that most of the players scored points that started with a zero, then a few more scored points that started with a one, and then only one score scored points that started with a two, and it was actually 20 points. So let me actually write down all of this data in a way that maybe you're a little bit more used to understanding it. So I'm gonna write the zeros in purple. So there's, let's see, one, two, three, four, five, six, seven players had zero as the first digit. So one, two, three, four, five, six, seven. I wrote seven zeros, and then this player also had a zero in his ones digit."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So I'm gonna write the zeros in purple. So there's, let's see, one, two, three, four, five, six, seven players had zero as the first digit. So one, two, three, four, five, six, seven. I wrote seven zeros, and then this player also had a zero in his ones digit. This player, I'm gonna try to do all the colors, this player also had a zero in his ones digit. This player right here had a two in his ones digit, so he scored a total of two points. This player, let me do orange, this player had four for his ones digit."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "I wrote seven zeros, and then this player also had a zero in his ones digit. This player, I'm gonna try to do all the colors, this player also had a zero in his ones digit. This player right here had a two in his ones digit, so he scored a total of two points. This player, let me do orange, this player had four for his ones digit. This player had seven for his ones digit. Then this player had seven for his ones digit. And then, let me see, I'm almost using all the colors, this player had nine for his ones digit."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "This player, let me do orange, this player had four for his ones digit. This player had seven for his ones digit. Then this player had seven for his ones digit. And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. So the way to read this is you had one player with zero points, zero, two, four, seven, nine, and nine. But you can see, I'm just kind of silly saying the zero was the tens digit. You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. So the way to read this is you had one player with zero points, zero, two, four, seven, nine, and nine. But you can see, I'm just kind of silly saying the zero was the tens digit. You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. But these are the actual scores for those seven players. Now let's go to the next row in the stem and leaf plot. So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. But these are the actual scores for those seven players. Now let's go to the next row in the stem and leaf plot. So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. So one, one, one, and one. And then we have this player over here, it's one, his ones digit, or her ones digit is a one. So this player, this represents 11."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. So one, one, one, and one. And then we have this player over here, it's one, his ones digit, or her ones digit is a one. So this player, this represents 11. One in the tens place, one in the ones place. This player over here also got 11. One in the tens place, one in the ones place."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So this player, this represents 11. One in the tens place, one in the ones place. This player over here also got 11. One in the tens place, one in the ones place. This player, let me do orange. This player has three in the ones place. So he or she scored 13 points."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "One in the tens place, one in the ones place. This player, let me do orange. This player has three in the ones place. So he or she scored 13 points. One in the tens place, three in the ones place, 13 points. And then, I will do this in purple, this player has eight in their ones place. So he or she scored 18 points."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So he or she scored 13 points. One in the tens place, three in the ones place, 13 points. And then, I will do this in purple, this player has eight in their ones place. So he or she scored 18 points. One in the tens place, eight in the ones place, 18 points. And then finally, you have this player that has two, the tens digit is a two, and then the ones digit is a zero. Is a zero, I'll circle that in yellow."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So he or she scored 18 points. One in the tens place, eight in the ones place, 18 points. And then finally, you have this player that has two, the tens digit is a two, and then the ones digit is a zero. Is a zero, I'll circle that in yellow. It is a zero. So he or she scored 20 points. So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Is a zero, I'll circle that in yellow. It is a zero. So he or she scored 20 points. So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. And once again, what was useful about this is you see how many players scored between zero and nine points, including nine points. How many scored between 10 and 19 points, and then how many scored 20 points or over. And you see the distribution right over here."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. And once again, what was useful about this is you see how many players scored between zero and nine points, including nine points. How many scored between 10 and 19 points, and then how many scored 20 points or over. And you see the distribution right over here. But let's actually answer the question that they're asking us to answer. How many points did the team score? So here, we just have to add up all of these numbers right over here."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And you see the distribution right over here. But let's actually answer the question that they're asking us to answer. How many points did the team score? So here, we just have to add up all of these numbers right over here. So we're going to add up, I'll start with the largest. So 20 plus 18 plus 13 plus 11 plus 11, 13, 11, 11, plus nine plus seven, plus seven again, plus four, plus two. Did I do that right?"}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So here, we just have to add up all of these numbers right over here. So we're going to add up, I'll start with the largest. So 20 plus 18 plus 13 plus 11 plus 11, 13, 11, 11, plus nine plus seven, plus seven again, plus four, plus two. Did I do that right? We have two 11s, then a nine, then two sevens, then a four, then a two, and then these two characters didn't score anything. So let's add up all of these together. Let's add them all up."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Did I do that right? We have two 11s, then a nine, then two sevens, then a four, then a two, and then these two characters didn't score anything. So let's add up all of these together. Let's add them all up. So zero plus eight is eight, plus three is 11, plus one is 12, plus one is 13, plus nine is 22, plus seven is 27, 34, 38, 30, or 40. So that gets us to 40. Let me do that one more time."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let's add them all up. So zero plus eight is eight, plus three is 11, plus one is 12, plus one is 13, plus nine is 22, plus seven is 27, 34, 38, 30, or 40. So that gets us to 40. Let me do that one more time. Eight, 11, 11, 12, 13, 22, 29, 29, and then, 29, 36, 40, and 42. Looks like I actually might have messed, let me do it one more time. This is the hardest part, adding these up."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let me do that one more time. Eight, 11, 11, 12, 13, 22, 29, 29, and then, 29, 36, 40, and 42. Looks like I actually might have messed, let me do it one more time. This is the hardest part, adding these up. So let me try that one last time. I'm just going to state where my sum is. So zero, eight, add three, 11, 12, 13, 22, 29, 36, 40, 42."}, {"video_title": "Stem-and-leaf plots   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "This is the hardest part, adding these up. So let me try that one last time. I'm just going to state where my sum is. So zero, eight, add three, 11, 12, 13, 22, 29, 36, 40, 42. So it's a good thing that I double-checked that, I made a mistake the first time. Four plus two is six, seven, eight, nine, 10. So we get to 102 points."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we got that to be 30. That's our total sum of squares. And we asked ourselves, how much of that variation is due to variation within each of these groups versus variation between the groups themselves. So for the variation within the groups, we had our sum of squares within. And there we got 6. And then the balance of this 30, there's no units here, the balance of this variation came from variation between the groups. And we actually calculated it and we got 24."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for the variation within the groups, we had our sum of squares within. And there we got 6. And then the balance of this 30, there's no units here, the balance of this variation came from variation between the groups. And we actually calculated it and we got 24. What I want to do in this video is actually use this type of information, essentially these statistics we've calculated, to do some inferential statistics, to come to some type of conclusion, or maybe not to come to some type of conclusion. And what I want to do is just to put some context around these groups. We've just been dealing with them in the abstract right now."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we actually calculated it and we got 24. What I want to do in this video is actually use this type of information, essentially these statistics we've calculated, to do some inferential statistics, to come to some type of conclusion, or maybe not to come to some type of conclusion. And what I want to do is just to put some context around these groups. We've just been dealing with them in the abstract right now. But you can imagine that these are kind of the results of some type of experiment. Let's say that I gave 3 different types of pills, or 3 different types of food, to people taking a test. And these are the scores on the test."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "We've just been dealing with them in the abstract right now. But you can imagine that these are kind of the results of some type of experiment. Let's say that I gave 3 different types of pills, or 3 different types of food, to people taking a test. And these are the scores on the test. So this is food 1, this is food 1, this is food 2, this is food 2, and then this right over here is food 3. And I want to figure out if the type of food that people take going into the test, does it really affect their scores? Are the differences in these scores, if you look at these means, it looks like they perform best in group 3, then in group 2, or then in group 1."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And these are the scores on the test. So this is food 1, this is food 1, this is food 2, this is food 2, and then this right over here is food 3. And I want to figure out if the type of food that people take going into the test, does it really affect their scores? Are the differences in these scores, if you look at these means, it looks like they perform best in group 3, then in group 2, or then in group 1. But is that difference purely random, random chance? Or can I be pretty confident that it's due to actual differences in the population means of all of the people who would ever take food 3 versus food 2 versus food 1? What I want to do is say, so my question here is, are the true population means the same?"}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "Are the differences in these scores, if you look at these means, it looks like they perform best in group 3, then in group 2, or then in group 1. But is that difference purely random, random chance? Or can I be pretty confident that it's due to actual differences in the population means of all of the people who would ever take food 3 versus food 2 versus food 1? What I want to do is say, so my question here is, are the true population means the same? So if the true population means, this is a sample mean just based on 3 samples, but if I knew the true population means, so my question is, is the mean of the population of people taking food 1 equal to the mean of food 2? Obviously I'll never be able to give that food to every human being that could ever live and then make them all take an exam, but we're trying to get a sense of, there is some true mean there, it's just not really measurable. So my question is, this equal to this equal to the mean 3, the true population mean 3."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I want to do is say, so my question here is, are the true population means the same? So if the true population means, this is a sample mean just based on 3 samples, but if I knew the true population means, so my question is, is the mean of the population of people taking food 1 equal to the mean of food 2? Obviously I'll never be able to give that food to every human being that could ever live and then make them all take an exam, but we're trying to get a sense of, there is some true mean there, it's just not really measurable. So my question is, this equal to this equal to the mean 3, the true population mean 3. And my question is, are these equal? It's because if they're not equal, then that means that the food, that the food actually, the different foods that you get, actually do have some type of impact on how people perform on a test. So let's do a little bit of a hypothesis test here."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "So my question is, this equal to this equal to the mean 3, the true population mean 3. And my question is, are these equal? It's because if they're not equal, then that means that the food, that the food actually, the different foods that you get, actually do have some type of impact on how people perform on a test. So let's do a little bit of a hypothesis test here. So let's say that my null hypothesis, let's say that my null hypothesis is that the means are the same, or food doesn't make a difference. Food doesn't make a difference. The food doesn't make a difference, but my alternate hypothesis is that it does."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's do a little bit of a hypothesis test here. So let's say that my null hypothesis, let's say that my null hypothesis is that the means are the same, or food doesn't make a difference. Food doesn't make a difference. The food doesn't make a difference, but my alternate hypothesis is that it does. And a way of thinking about this a little quantitatively is that if it doesn't make a difference, the true population means of the groups will be the same. So that means the true population mean of the group that took food 1 will be the same as the group that took food 2, which will be the same as the group that took food 3. If our alternate hypothesis is correct, then these means will not all be the same."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "The food doesn't make a difference, but my alternate hypothesis is that it does. And a way of thinking about this a little quantitatively is that if it doesn't make a difference, the true population means of the groups will be the same. So that means the true population mean of the group that took food 1 will be the same as the group that took food 2, which will be the same as the group that took food 3. If our alternate hypothesis is correct, then these means will not all be the same. So how can we test this hypothesis? So what we're going to do, we're going to assume the null hypothesis. This is what we always do in our hypothesis testing."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "If our alternate hypothesis is correct, then these means will not all be the same. So how can we test this hypothesis? So what we're going to do, we're going to assume the null hypothesis. This is what we always do in our hypothesis testing. We're going to assume our null hypothesis, and then essentially figure out what are the chances of getting a certain statistic this extreme. And I haven't even defined what that statistic are. So we're going to define, we're going to assume our null hypothesis, and then we're going to come up with a statistic called the F statistic."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is what we always do in our hypothesis testing. We're going to assume our null hypothesis, and then essentially figure out what are the chances of getting a certain statistic this extreme. And I haven't even defined what that statistic are. So we're going to define, we're going to assume our null hypothesis, and then we're going to come up with a statistic called the F statistic. So our F statistic, which has an F distribution, and we won't go in real deep into the details of the F distribution, but you can already start to think of it as a ratio of 2 chi-squared distributions that may or may not have different degrees of freedom. Our F statistic is going to be the ratio of our sum of squares between the samples. So it's going to be our sum of squares between, divided by our degrees of freedom between, and sometimes this is called the mean squares between MSB, either way, it's going to be that, divided by the sum of squares within."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're going to define, we're going to assume our null hypothesis, and then we're going to come up with a statistic called the F statistic. So our F statistic, which has an F distribution, and we won't go in real deep into the details of the F distribution, but you can already start to think of it as a ratio of 2 chi-squared distributions that may or may not have different degrees of freedom. Our F statistic is going to be the ratio of our sum of squares between the samples. So it's going to be our sum of squares between, divided by our degrees of freedom between, and sometimes this is called the mean squares between MSB, either way, it's going to be that, divided by the sum of squares within. So that's what I had done up here. The sum of squares within in blue, divided by the sum of squares within, divided by the degrees of freedom of the sum of squares within. And that was M times N minus 1."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be our sum of squares between, divided by our degrees of freedom between, and sometimes this is called the mean squares between MSB, either way, it's going to be that, divided by the sum of squares within. So that's what I had done up here. The sum of squares within in blue, divided by the sum of squares within, divided by the degrees of freedom of the sum of squares within. And that was M times N minus 1. Now let's just think about what this is doing right here. If this number, if the numerator is much larger than the denominator, then what that tells us is that the variation in this data is due mostly to the differences between the actual means, and is due less to the actual variation within the means. That's if this numerator is much bigger than this denominator over here."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that was M times N minus 1. Now let's just think about what this is doing right here. If this number, if the numerator is much larger than the denominator, then what that tells us is that the variation in this data is due mostly to the differences between the actual means, and is due less to the actual variation within the means. That's if this numerator is much bigger than this denominator over here. So that would make us believe, that should make us believe, that there is a difference in the true population mean. So if this number is really big, it should tell us that there's a lower probability that our null hypothesis is correct. If this number is really small, let's say that this is larger, let's say that our denominator is larger, that means that our variation within each sample makes up more of the total variation than our variation between the samples."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's if this numerator is much bigger than this denominator over here. So that would make us believe, that should make us believe, that there is a difference in the true population mean. So if this number is really big, it should tell us that there's a lower probability that our null hypothesis is correct. If this number is really small, let's say that this is larger, let's say that our denominator is larger, that means that our variation within each sample makes up more of the total variation than our variation between the samples. So that means that our variation within each of these samples is a bigger percentage of the total variation versus the variation between the samples. So that would make us believe that, hey, any difference that we actually see in the means is probably just random. And that would make it maybe a little harder to reject our null hypothesis."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "If this number is really small, let's say that this is larger, let's say that our denominator is larger, that means that our variation within each sample makes up more of the total variation than our variation between the samples. So that means that our variation within each of these samples is a bigger percentage of the total variation versus the variation between the samples. So that would make us believe that, hey, any difference that we actually see in the means is probably just random. And that would make it maybe a little harder to reject our null hypothesis. So let's just actually calculate it for this. So in this case, our sum of squares between, we calculated over here was 24, and we had 2 degrees of freedom, and our sum of squares within was 6, and we had how many degrees of freedom? Our degrees of freedom there were also 6, 6 degrees of freedom right over there."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that would make it maybe a little harder to reject our null hypothesis. So let's just actually calculate it for this. So in this case, our sum of squares between, we calculated over here was 24, and we had 2 degrees of freedom, and our sum of squares within was 6, and we had how many degrees of freedom? Our degrees of freedom there were also 6, 6 degrees of freedom right over there. So this is going to be 24 divided by 2, which is 12 divided by 1. So our F statistic that we've calculated is going to be equal to 12. And this stands for Fisher, who was the biologist and statistician who came up with this."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our degrees of freedom there were also 6, 6 degrees of freedom right over there. So this is going to be 24 divided by 2, which is 12 divided by 1. So our F statistic that we've calculated is going to be equal to 12. And this stands for Fisher, who was the biologist and statistician who came up with this. So our F statistic is going to be 12. And what we're going to see is this is a pretty high number. Now one thing I forgot to mention, any hypothesis test, we need to have some type of significance level."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this stands for Fisher, who was the biologist and statistician who came up with this. So our F statistic is going to be 12. And what we're going to see is this is a pretty high number. Now one thing I forgot to mention, any hypothesis test, we need to have some type of significance level. And so let's say the significance level that we care about for our hypothesis test is 10%, is 0.10, which means that if we assume, if assuming the null hypothesis, there is less than a 10% chance of getting the result that we got, of getting this F statistic, then we will reject the null hypothesis. So what we want to do is figure out a critical F statistic value that getting that extreme of a value or greater is 10%, and if this is bigger than our critical F statistic value, then we're going to reject the null hypothesis. If it's less, we can't reject the null hypothesis."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now one thing I forgot to mention, any hypothesis test, we need to have some type of significance level. And so let's say the significance level that we care about for our hypothesis test is 10%, is 0.10, which means that if we assume, if assuming the null hypothesis, there is less than a 10% chance of getting the result that we got, of getting this F statistic, then we will reject the null hypothesis. So what we want to do is figure out a critical F statistic value that getting that extreme of a value or greater is 10%, and if this is bigger than our critical F statistic value, then we're going to reject the null hypothesis. If it's less, we can't reject the null hypothesis. And so I'm not going to go a lot into the guts of the F statistic, but you can already appreciate that each of these sum of squares has a chi-square distribution. This has a chi-square distribution, and this has a different chi-square distribution. This is a chi-square distribution with 2 degrees of freedom."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's less, we can't reject the null hypothesis. And so I'm not going to go a lot into the guts of the F statistic, but you can already appreciate that each of these sum of squares has a chi-square distribution. This has a chi-square distribution, and this has a different chi-square distribution. This is a chi-square distribution with 2 degrees of freedom. It's a chi-square distribution with, and we haven't normalized it and all of that, but roughly a chi-square distribution with 6 degrees of freedom. So the F statistic is actually, the ratio or the F distribution is the ratio of 2 chi-square distributions. And I got this, this is a screenshot from a professor's course at UCLA, hope they don't mind, I actually needed, I had to find an F table for us to look into."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is a chi-square distribution with 2 degrees of freedom. It's a chi-square distribution with, and we haven't normalized it and all of that, but roughly a chi-square distribution with 6 degrees of freedom. So the F statistic is actually, the ratio or the F distribution is the ratio of 2 chi-square distributions. And I got this, this is a screenshot from a professor's course at UCLA, hope they don't mind, I actually needed, I had to find an F table for us to look into. But this is what an F distribution looks like. And obviously it's going to look different depending on the degrees of freedom of the numerator and the denominator. There's kind of 2 degrees of freedom to think about, the numerator's degree of freedom and the denominator's degree of freedom."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I got this, this is a screenshot from a professor's course at UCLA, hope they don't mind, I actually needed, I had to find an F table for us to look into. But this is what an F distribution looks like. And obviously it's going to look different depending on the degrees of freedom of the numerator and the denominator. There's kind of 2 degrees of freedom to think about, the numerator's degree of freedom and the denominator's degree of freedom. But with that said, let's calculate the critical F statistic. The critical F statistic for alpha is equal to 0.10, and you're actually going to see different F tables for each different alpha, where our numerator degree of freedom is 2 and our denominator degree of freedom is 6. So this table that I got, this whole table is for an alpha, our significance level of 10% or 0.10, and our numerator's degree of freedom was 2 and our denominator's degree of freedom is 6."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's kind of 2 degrees of freedom to think about, the numerator's degree of freedom and the denominator's degree of freedom. But with that said, let's calculate the critical F statistic. The critical F statistic for alpha is equal to 0.10, and you're actually going to see different F tables for each different alpha, where our numerator degree of freedom is 2 and our denominator degree of freedom is 6. So this table that I got, this whole table is for an alpha, our significance level of 10% or 0.10, and our numerator's degree of freedom was 2 and our denominator's degree of freedom is 6. So our critical F value is 3.46. So our critical F value is 3.46. So this right over here, this value right here is 3.46."}, {"video_title": "ANOVA 3 Hypothesis test with F-statistic   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this table that I got, this whole table is for an alpha, our significance level of 10% or 0.10, and our numerator's degree of freedom was 2 and our denominator's degree of freedom is 6. So our critical F value is 3.46. So our critical F value is 3.46. So this right over here, this value right here is 3.46. The value that we got based on our data is much larger than that, way above it. It's going to have a very, very small p-value. The probability of getting something this extreme just by chance, assuming the null hypothesis, is very low."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "We already know a little bit about random variables. What we're going to see in this video is that random variables come in two varieties. You have discrete random variables and you have continuous random variables. Continuous. And discrete random variables, these are essentially random variables that can take on distinct or separate values. And we'll give examples of that in a second. That comes straight from the meaning of the word discrete in the English language."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Continuous. And discrete random variables, these are essentially random variables that can take on distinct or separate values. And we'll give examples of that in a second. That comes straight from the meaning of the word discrete in the English language. Distinct or separate values. While continuous, and I guess there's another definition for the word discrete in the English language, which would be polite or not obnoxious or kind of subtle, that is not what we're talking about. We are not talking about random variables that are polite."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "That comes straight from the meaning of the word discrete in the English language. Distinct or separate values. While continuous, and I guess there's another definition for the word discrete in the English language, which would be polite or not obnoxious or kind of subtle, that is not what we're talking about. We are not talking about random variables that are polite. We're talking about ones that can take on distinct values. And continuous random variables, they can take on any value in a range. And that range could even be infinite."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "We are not talking about random variables that are polite. We're talking about ones that can take on distinct values. And continuous random variables, they can take on any value in a range. And that range could even be infinite. So any value in an interval. So with those two definitions out of the way, let's look at some actual random variable definitions. And I want to think together about whether you would classify them as discrete or continuous random variables."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that range could even be infinite. So any value in an interval. So with those two definitions out of the way, let's look at some actual random variable definitions. And I want to think together about whether you would classify them as discrete or continuous random variables. So let's say that I have a random variable capital X. And it is equal to, well this is one that we covered in the last video. It's one if my fair coin is heads."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I want to think together about whether you would classify them as discrete or continuous random variables. So let's say that I have a random variable capital X. And it is equal to, well this is one that we covered in the last video. It's one if my fair coin is heads. It's zero if my fair coin is tails. So is this a discrete or a continuous random variable? Well this random variable right over here can take on distinct values."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's one if my fair coin is heads. It's zero if my fair coin is tails. So is this a discrete or a continuous random variable? Well this random variable right over here can take on distinct values. It can take on either a one or it could take on a zero. Another way to think about it is you can count the number of different values it can take on. This is the first value it can take on."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well this random variable right over here can take on distinct values. It can take on either a one or it could take on a zero. Another way to think about it is you can count the number of different values it can take on. This is the first value it can take on. This is the second value that it can take on. So this is clearly a discrete random variable. Let's think about another one."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the first value it can take on. This is the second value that it can take on. So this is clearly a discrete random variable. Let's think about another one. Let's define random variable Y as equal to the mass of a random animal selected at the New Orleans Zoo. New Orleans Zoo where I grew up. The Ottoman Zoo."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's think about another one. Let's define random variable Y as equal to the mass of a random animal selected at the New Orleans Zoo. New Orleans Zoo where I grew up. The Ottoman Zoo. At the New Orleans Zoo. The Ottoman Zoo. Y is the mass of a random animal selected at the New Orleans Zoo."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The Ottoman Zoo. At the New Orleans Zoo. The Ottoman Zoo. Y is the mass of a random animal selected at the New Orleans Zoo. Is this a discrete random variable or a continuous random variable? Well the exact mass, and I should probably put that qualifier here. I'll even add it here just to make it really clear."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Y is the mass of a random animal selected at the New Orleans Zoo. Is this a discrete random variable or a continuous random variable? Well the exact mass, and I should probably put that qualifier here. I'll even add it here just to make it really clear. The exact mass of a random animal or a random object in our universe, it can take on any of a whole set of values. I mean who knows exactly the exact number of electrons that are a part of that object right at that moment. Who knows the neutrons, the protons, the exact number of molecules in that object or part of that animal exactly that moment."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll even add it here just to make it really clear. The exact mass of a random animal or a random object in our universe, it can take on any of a whole set of values. I mean who knows exactly the exact number of electrons that are a part of that object right at that moment. Who knows the neutrons, the protons, the exact number of molecules in that object or part of that animal exactly that moment. So that mass, for example at the zoo, it might take on a value anywhere between, well maybe close to zero. Close to zero. There's no animal that has zero mass."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Who knows the neutrons, the protons, the exact number of molecules in that object or part of that animal exactly that moment. So that mass, for example at the zoo, it might take on a value anywhere between, well maybe close to zero. Close to zero. There's no animal that has zero mass. But it could be close to zero if we're thinking about an ant or we're thinking about a dust mite or maybe if you consider even a bacterium an animal. I believe bacterium is a singular of bacteria. And it could go all the way, maybe the most massive animal in the zoo is the elephant of some kind."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's no animal that has zero mass. But it could be close to zero if we're thinking about an ant or we're thinking about a dust mite or maybe if you consider even a bacterium an animal. I believe bacterium is a singular of bacteria. And it could go all the way, maybe the most massive animal in the zoo is the elephant of some kind. I don't know what it would be in kilograms but it would be fairly large. So maybe you can get up all the way to 3,000 kilograms or probably larger. Say 5,000 kilograms."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it could go all the way, maybe the most massive animal in the zoo is the elephant of some kind. I don't know what it would be in kilograms but it would be fairly large. So maybe you can get up all the way to 3,000 kilograms or probably larger. Say 5,000 kilograms. I don't know what the mass of a very heavy elephant or a very massive elephant I should say actually is. Maybe something fun for you to look at. But any animal could have a mass anywhere in between here."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Say 5,000 kilograms. I don't know what the mass of a very heavy elephant or a very massive elephant I should say actually is. Maybe something fun for you to look at. But any animal could have a mass anywhere in between here. It does not take on discrete values. You could have an animal that is exactly maybe 123.75921 kilograms. And even there that actually might not be the exact mass."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "But any animal could have a mass anywhere in between here. It does not take on discrete values. You could have an animal that is exactly maybe 123.75921 kilograms. And even there that actually might not be the exact mass. You might have to get even more precise. 10732. I think you get the picture."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And even there that actually might not be the exact mass. You might have to get even more precise. 10732. I think you get the picture. Even though this is the way I've defined it now, a finite interval, you can take on any value in between here. There are not discrete values. So this one here is clearly a continuous random variable."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "I think you get the picture. Even though this is the way I've defined it now, a finite interval, you can take on any value in between here. There are not discrete values. So this one here is clearly a continuous random variable. Let's think about another one. Let's say that random variable Y, instead of it being this, let's say it's the year that a random student in the class was born. Random student in a class was born."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this one here is clearly a continuous random variable. Let's think about another one. Let's say that random variable Y, instead of it being this, let's say it's the year that a random student in the class was born. Random student in a class was born. Is this a discrete or a continuous random variable? Well that year, you literally can define it as a specific discrete year. It could be 1992 or it could be 1985 or it could be 2000 and 2001."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Random student in a class was born. Is this a discrete or a continuous random variable? Well that year, you literally can define it as a specific discrete year. It could be 1992 or it could be 1985 or it could be 2000 and 2001. There are discrete values that this random variable can actually take on. It won't be able to take on any value between 2000 and 2001. It will either be 2000 or it will be 2001 or 2002."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "It could be 1992 or it could be 1985 or it could be 2000 and 2001. There are discrete values that this random variable can actually take on. It won't be able to take on any value between 2000 and 2001. It will either be 2000 or it will be 2001 or 2002. Once again, you can count the values it can take on. Most of the times that you're dealing with, as in the case right here, a discrete random variable. Let me make it clear."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "It will either be 2000 or it will be 2001 or 2002. Once again, you can count the values it can take on. Most of the times that you're dealing with, as in the case right here, a discrete random variable. Let me make it clear. This one over here is also a discrete random variable. Most of the time that you're dealing with a discrete random variable, you're probably going to be dealing with a finite number of values. But it does not have to be a finite number of values."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me make it clear. This one over here is also a discrete random variable. Most of the time that you're dealing with a discrete random variable, you're probably going to be dealing with a finite number of values. But it does not have to be a finite number of values. You can actually have an infinite potential number of values that it can take on as long as the values are countable. As long as you can literally say, okay, this is the first value it can take on, the second, the third, and you might be counting forever. But as long as you can literally list, and it could be an infinite list, but if you could list the values that it could take on, then you're dealing with a discrete random variable."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it does not have to be a finite number of values. You can actually have an infinite potential number of values that it can take on as long as the values are countable. As long as you can literally say, okay, this is the first value it can take on, the second, the third, and you might be counting forever. But as long as you can literally list, and it could be an infinite list, but if you could list the values that it could take on, then you're dealing with a discrete random variable. Notice, in this scenario with the zoo, you could not list all of the possible masses. You could not even count them. You might attempt to, and it's a fun exercise to try at least once, to try to list all of the values this might take on."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "But as long as you can literally list, and it could be an infinite list, but if you could list the values that it could take on, then you're dealing with a discrete random variable. Notice, in this scenario with the zoo, you could not list all of the possible masses. You could not even count them. You might attempt to, and it's a fun exercise to try at least once, to try to list all of the values this might take on. You might say, okay, maybe you could take on 0.01 and maybe 0.02. But wait, you just skipped an infinite number of values that it could take on because it could have taken on 0.011, 0.012. Even between those, there's an infinite number of values it could take on."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "You might attempt to, and it's a fun exercise to try at least once, to try to list all of the values this might take on. You might say, okay, maybe you could take on 0.01 and maybe 0.02. But wait, you just skipped an infinite number of values that it could take on because it could have taken on 0.011, 0.012. Even between those, there's an infinite number of values it could take on. There's no way for you to count the number of values that a continuous random variable can take on. There's no way for you to list them. With a discrete random variable, you can count the values."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Even between those, there's an infinite number of values it could take on. There's no way for you to count the number of values that a continuous random variable can take on. There's no way for you to list them. With a discrete random variable, you can count the values. You can list the values. Let's do another example. Let's let random variable Z be the number of ants born tomorrow in the universe."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "With a discrete random variable, you can count the values. You can list the values. Let's do another example. Let's let random variable Z be the number of ants born tomorrow in the universe. Now, you're probably arguing that there aren't ants on other planets, or maybe they're ant-like creatures, but they're not going to be ants as we define them, but how do we know? The number of ants born in the universe, maybe some ants have figured out interstellar travel of some kind. The number of ants born tomorrow in the universe, that's my random variable Z."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's let random variable Z be the number of ants born tomorrow in the universe. Now, you're probably arguing that there aren't ants on other planets, or maybe they're ant-like creatures, but they're not going to be ants as we define them, but how do we know? The number of ants born in the universe, maybe some ants have figured out interstellar travel of some kind. The number of ants born tomorrow in the universe, that's my random variable Z. Is this a discrete random variable or a continuous random variable? Once again, we can count the number of values this could take on. This could be 1, it could be 2, it could be 3, it could be 4, it could be 5 quadrillion ants, it could be 5 quadrillion and 1."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The number of ants born tomorrow in the universe, that's my random variable Z. Is this a discrete random variable or a continuous random variable? Once again, we can count the number of values this could take on. This could be 1, it could be 2, it could be 3, it could be 4, it could be 5 quadrillion ants, it could be 5 quadrillion and 1. We can actually count the values. Those values are discrete. Once again, this right over here is a discrete random variable."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "This could be 1, it could be 2, it could be 3, it could be 4, it could be 5 quadrillion ants, it could be 5 quadrillion and 1. We can actually count the values. Those values are discrete. Once again, this right over here is a discrete random variable. This is fun, so let's keep doing more of these. Let's say that I have random variable X. We're not using this definition anymore."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, this right over here is a discrete random variable. This is fun, so let's keep doing more of these. Let's say that I have random variable X. We're not using this definition anymore. Now, I'm going to define random variable X to be the winning time. Let me write it this way, the exact winning time for the men's 100 meter dash at the 2016 Olympics. The exact time that it took for the winner, who's probably going to be Usain Bolt, but it might not be."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not using this definition anymore. Now, I'm going to define random variable X to be the winning time. Let me write it this way, the exact winning time for the men's 100 meter dash at the 2016 Olympics. The exact time that it took for the winner, who's probably going to be Usain Bolt, but it might not be. Actually, he's aging a little bit. Remember the exact winning time for the men's 100 meter dash at the 2016 Olympics. Not the one that you necessarily see on the clock."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The exact time that it took for the winner, who's probably going to be Usain Bolt, but it might not be. Actually, he's aging a little bit. Remember the exact winning time for the men's 100 meter dash at the 2016 Olympics. Not the one that you necessarily see on the clock. The exact, the precise time that you would see at the men's 100 meter dash. Is this a discrete or continuous random variable? The way I've defined it, and this one's a little bit tricky, because you might say it's countable."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Not the one that you necessarily see on the clock. The exact, the precise time that you would see at the men's 100 meter dash. Is this a discrete or continuous random variable? The way I've defined it, and this one's a little bit tricky, because you might say it's countable. You might say it could be 9.56 seconds, or 9.57 seconds, or 9.58 seconds. You might be tempted to believe that, because when you watch the 100 meter dash at the Olympics, they measure it to the nearest hundredths. They round to the nearest hundredths."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The way I've defined it, and this one's a little bit tricky, because you might say it's countable. You might say it could be 9.56 seconds, or 9.57 seconds, or 9.58 seconds. You might be tempted to believe that, because when you watch the 100 meter dash at the Olympics, they measure it to the nearest hundredths. They round to the nearest hundredths. That's how precise their timing is. I'm talking about the exact winning time, the exact number of seconds it takes for that person from the starting gun to cross the finish line. There, it can take on any value."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "They round to the nearest hundredths. That's how precise their timing is. I'm talking about the exact winning time, the exact number of seconds it takes for that person from the starting gun to cross the finish line. There, it can take on any value. It can take on any value between, well, I guess they're limited by the speed of light. It could take on any value you could imagine. It might be anywhere between 5 seconds and maybe 12 seconds."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "There, it can take on any value. It can take on any value between, well, I guess they're limited by the speed of light. It could take on any value you could imagine. It might be anywhere between 5 seconds and maybe 12 seconds. It could be anywhere in between there. It might not be 9.57. That might be what the clock says, but in reality, the exact winning time could be 9.571."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "It might be anywhere between 5 seconds and maybe 12 seconds. It could be anywhere in between there. It might not be 9.57. That might be what the clock says, but in reality, the exact winning time could be 9.571. Or it could be 9.572359. I think you see what I'm saying. The exact precise time could be any value in an interval."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "That might be what the clock says, but in reality, the exact winning time could be 9.571. Or it could be 9.572359. I think you see what I'm saying. The exact precise time could be any value in an interval. This right over here is a continuous random variable. Now, what would be the case, instead of saying the exact winning time, if instead I defined x to be the winning time of the men's 100-meter dash at the 2016 Olympics, rounded to the nearest hundredth? Is this a discrete or a continuous random variable?"}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The exact precise time could be any value in an interval. This right over here is a continuous random variable. Now, what would be the case, instead of saying the exact winning time, if instead I defined x to be the winning time of the men's 100-meter dash at the 2016 Olympics, rounded to the nearest hundredth? Is this a discrete or a continuous random variable? Let me delete this. I've changed the random variable now. Is this going to be a discrete or a continuous random variable?"}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is this a discrete or a continuous random variable? Let me delete this. I've changed the random variable now. Is this going to be a discrete or a continuous random variable? Now we can actually count the actual values that this random variable can take on. It might be 9.56. It could be 9.57."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is this going to be a discrete or a continuous random variable? Now we can actually count the actual values that this random variable can take on. It might be 9.56. It could be 9.57. It could be 9.58. We can actually list them. In this case, when we round it to the nearest hundredth, we can actually list the values."}, {"video_title": "Discrete and continuous random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "It could be 9.57. It could be 9.58. We can actually list them. In this case, when we round it to the nearest hundredth, we can actually list the values. We are now dealing with a discrete random variable. Anyway, I'll let you go there. Hopefully this gives you a sense of the distinction between discrete and continuous random variables."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then in the other data set, I have a one, I'm gonna do this on the right side of the screen, a one, a one, a six, and a four. Now the first thing I wanna think about is, well how do I, is there a number that can give me a measure of center of each of these data sets? And one of the ways that we know how to do that is by finding the mean. So let's figure out the mean of each of these data sets. So this first data set, the mean, well we just need to sum up all of the numbers. So it's gonna be two plus two plus four plus four, and then we're gonna divide by the number of numbers that we have. So we have one, two, three, four numbers, so that's that four right over there."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's figure out the mean of each of these data sets. So this first data set, the mean, well we just need to sum up all of the numbers. So it's gonna be two plus two plus four plus four, and then we're gonna divide by the number of numbers that we have. So we have one, two, three, four numbers, so that's that four right over there. And this is going to be two plus two is four, plus four is eight, plus four is 12, so it's gonna be 12 over four, which is equal to three. So actually let's just, let's see if we can visualize this a little bit on a number line. So, and actually I'll do kind of a, I'll do a little bit of a dot plot here, so we can see all of the values."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we have one, two, three, four numbers, so that's that four right over there. And this is going to be two plus two is four, plus four is eight, plus four is 12, so it's gonna be 12 over four, which is equal to three. So actually let's just, let's see if we can visualize this a little bit on a number line. So, and actually I'll do kind of a, I'll do a little bit of a dot plot here, so we can see all of the values. So if this is zero, one, two, three, four, and five. And so we have two twos, and so why don't I just do, so for each of these twos, actually I'll just do it in yellow, so I have one two, and then I'll have another two, I'm just gonna do a dot plot here, and then I have two fours. So one four and another four, right over there."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So, and actually I'll do kind of a, I'll do a little bit of a dot plot here, so we can see all of the values. So if this is zero, one, two, three, four, and five. And so we have two twos, and so why don't I just do, so for each of these twos, actually I'll just do it in yellow, so I have one two, and then I'll have another two, I'm just gonna do a dot plot here, and then I have two fours. So one four and another four, right over there. And we calculated that the mean is three. The mean is three. A measure of central tendency, it is three."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So one four and another four, right over there. And we calculated that the mean is three. The mean is three. A measure of central tendency, it is three. So I'll just put three right over here, I'll just mark it with that dotted line. That's where the mean is. All right, well we've visualized that a little bit, and that does look like it's the center."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "A measure of central tendency, it is three. So I'll just put three right over here, I'll just mark it with that dotted line. That's where the mean is. All right, well we've visualized that a little bit, and that does look like it's the center. It's a pretty, it makes sense. So now let's look at this other data set right over here. So the mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still have four data points."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "All right, well we've visualized that a little bit, and that does look like it's the center. It's a pretty, it makes sense. So now let's look at this other data set right over here. So the mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still have four data points. And this is two plus six is eight plus four is 12. 12 divided by four, this is also three. So this also has the same mean."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still have four data points. And this is two plus six is eight plus four is 12. 12 divided by four, this is also three. So this also has the same mean. We have different numbers, but we have the same mean. But there's something about this data set that feels a little bit different about this. And let's visualize it to see if we can see a difference."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this also has the same mean. We have different numbers, but we have the same mean. But there's something about this data set that feels a little bit different about this. And let's visualize it to see if we can see a difference. Let's see if we can visualize it. So now I have to go all the way up to six. So let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And let's visualize it to see if we can see a difference. Let's see if we can visualize it. So now I have to go all the way up to six. So let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven. So we have a one, we have a one, we have another one, we have a six, and then we have a four. And we calculated that the mean is three. So we calculated that the mean is three."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven. So we have a one, we have a one, we have another one, we have a six, and then we have a four. And we calculated that the mean is three. So we calculated that the mean is three. So the mean is three. So when we measure it by the mean, the central point or measure of that central point, which we use as the mean, well, it looks the same, but the data sets look different. And how do they look different?"}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we calculated that the mean is three. So the mean is three. So when we measure it by the mean, the central point or measure of that central point, which we use as the mean, well, it looks the same, but the data sets look different. And how do they look different? Well, we've talked about notions of variability or variation and it looks like this data set is more spread out. It looks like the data points are on average further away from the mean than these data points are. And so that's an interesting question that we ask ourselves in statistics."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And how do they look different? Well, we've talked about notions of variability or variation and it looks like this data set is more spread out. It looks like the data points are on average further away from the mean than these data points are. And so that's an interesting question that we ask ourselves in statistics. We just don't want a measure of center like the mean. We might also want a measure of variability. And one of the more straightforward ways to think about variability is, well, on average, how far are each of the data points from the mean?"}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And so that's an interesting question that we ask ourselves in statistics. We just don't want a measure of center like the mean. We might also want a measure of variability. And one of the more straightforward ways to think about variability is, well, on average, how far are each of the data points from the mean? And that might sound a little complicated, but we're gonna figure out what that means in a second, and not to overuse the word mean. So we wanna figure out on average how far each of these data points from the mean. And what we're about to calculate, this is called mean absolute deviation."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And one of the more straightforward ways to think about variability is, well, on average, how far are each of the data points from the mean? And that might sound a little complicated, but we're gonna figure out what that means in a second, and not to overuse the word mean. So we wanna figure out on average how far each of these data points from the mean. And what we're about to calculate, this is called mean absolute deviation. Absolute deviation. Mean absolute deviation, or if you just use the acronym MAD, MAD, for mean absolute deviation. And all we're talking about, we're gonna figure out how much do each of these points, their distance, so absolute deviation, how much do they deviate from the mean, but the absolute of it, so each of these points at two, they are one away from the mean."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And what we're about to calculate, this is called mean absolute deviation. Absolute deviation. Mean absolute deviation, or if you just use the acronym MAD, MAD, for mean absolute deviation. And all we're talking about, we're gonna figure out how much do each of these points, their distance, so absolute deviation, how much do they deviate from the mean, but the absolute of it, so each of these points at two, they are one away from the mean. Doesn't matter if they're less or more, they're one away from the mean. And then we find the mean of all of the deviations. So what does that mean?"}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And all we're talking about, we're gonna figure out how much do each of these points, their distance, so absolute deviation, how much do they deviate from the mean, but the absolute of it, so each of these points at two, they are one away from the mean. Doesn't matter if they're less or more, they're one away from the mean. And then we find the mean of all of the deviations. So what does that mean? I'm using the word mean, well, using it a little bit too much. So let's figure out the mean absolute deviation of this first data set. So we've been able to figure out what the mean is."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So what does that mean? I'm using the word mean, well, using it a little bit too much. So let's figure out the mean absolute deviation of this first data set. So we've been able to figure out what the mean is. The mean is three. So we take each of the data points, and we figure out what's its absolute deviation from the mean? So we take the first two."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we've been able to figure out what the mean is. The mean is three. So we take each of the data points, and we figure out what's its absolute deviation from the mean? So we take the first two. So we say two minus the mean. Two minus the mean, and we take the absolute value. So that's its absolute deviation."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we take the first two. So we say two minus the mean. Two minus the mean, and we take the absolute value. So that's its absolute deviation. Then we have another two. So we find that's absolute deviation from three. Remember, if we're just taking two minus three and taking the absolute value, that's just saying it's absolute deviation."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So that's its absolute deviation. Then we have another two. So we find that's absolute deviation from three. Remember, if we're just taking two minus three and taking the absolute value, that's just saying it's absolute deviation. How far is it from three? It's fairly easy to calculate in this case. Then we have a four and another four."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Remember, if we're just taking two minus three and taking the absolute value, that's just saying it's absolute deviation. How far is it from three? It's fairly easy to calculate in this case. Then we have a four and another four. So let me write that. So then we have the mean, or we have the absolute deviation of four from three, from the mean. And then plus, we have another four."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Then we have a four and another four. So let me write that. So then we have the mean, or we have the absolute deviation of four from three, from the mean. And then plus, we have another four. We have this other four right up here. Four minus three. We take the absolute value, because once again, it's absolute deviation."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then plus, we have another four. We have this other four right up here. Four minus three. We take the absolute value, because once again, it's absolute deviation. And then we divide it. And then we divide it by the number of data points we have. So what is this going to be?"}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We take the absolute value, because once again, it's absolute deviation. And then we divide it. And then we divide it by the number of data points we have. So what is this going to be? Two minus three is negative one, but we take the absolute value. It's just going to be one. Two minus three is negative one."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So what is this going to be? Two minus three is negative one, but we take the absolute value. It's just going to be one. Two minus three is negative one. We take the absolute value, and it's just going to be one. And you see that here visually. This point is just one away."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Two minus three is negative one. We take the absolute value, and it's just going to be one. And you see that here visually. This point is just one away. It's just one away from three. This point is just one away from three. Four minus three is one."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This point is just one away. It's just one away from three. This point is just one away from three. Four minus three is one. Absolute value of that is one. This point is just one away from three. Four minus three, absolute value, that's another one."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Four minus three is one. Absolute value of that is one. This point is just one away from three. Four minus three, absolute value, that's another one. So you see in this case, every data point was exactly one away from the mean. And we took the absolute value so that we don't have negative ones here. We just care how far it is in absolute terms."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Four minus three, absolute value, that's another one. So you see in this case, every data point was exactly one away from the mean. And we took the absolute value so that we don't have negative ones here. We just care how far it is in absolute terms. So you have four data points. Each of their absolute deviations is four away, so the mean of the absolute deviations are one plus one plus one plus one, which is four, over four. So it's equal to one."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We just care how far it is in absolute terms. So you have four data points. Each of their absolute deviations is four away, so the mean of the absolute deviations are one plus one plus one plus one, which is four, over four. So it's equal to one. So one way to think about it, it's saying on average, the mean of the distances of these points away from the actual mean is one. And that makes sense because all of these are exactly one away from the mean. Now let's see what results we get for this data set right over here."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it's equal to one. So one way to think about it, it's saying on average, the mean of the distances of these points away from the actual mean is one. And that makes sense because all of these are exactly one away from the mean. Now let's see what results we get for this data set right over here. And I'll do it, let me actually get some space over here. At any point, if you get inspired, I encourage you to calculate the mean absolute deviation on your own. So let's calculate it."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now let's see what results we get for this data set right over here. And I'll do it, let me actually get some space over here. At any point, if you get inspired, I encourage you to calculate the mean absolute deviation on your own. So let's calculate it. So the mean absolute deviation here, I'll write MAD, is going to be equal to, well let's figure out the absolute deviation of each of these points from the mean. So it's the absolute value of one minus three, that's this first one, plus the absolute deviation, so one minus three, that's the second one, and then plus the absolute value of six minus three, that's the sixth, and then we have the four, plus the absolute value of four minus three, and then we have four points. So one minus three is negative two, absolute value is two, and we see that here."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's calculate it. So the mean absolute deviation here, I'll write MAD, is going to be equal to, well let's figure out the absolute deviation of each of these points from the mean. So it's the absolute value of one minus three, that's this first one, plus the absolute deviation, so one minus three, that's the second one, and then plus the absolute value of six minus three, that's the sixth, and then we have the four, plus the absolute value of four minus three, and then we have four points. So one minus three is negative two, absolute value is two, and we see that here. This is two away from three. We just care about absolute deviation. We don't care if it's to the left or to the right."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So one minus three is negative two, absolute value is two, and we see that here. This is two away from three. We just care about absolute deviation. We don't care if it's to the left or to the right. Then we have another one minus three is negative two, but it's absolute value, so this is two, and that's this. This is two away from the mean. Then we have six minus three, absolute value of that's just going to be three, and that's this right over here."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We don't care if it's to the left or to the right. Then we have another one minus three is negative two, but it's absolute value, so this is two, and that's this. This is two away from the mean. Then we have six minus three, absolute value of that's just going to be three, and that's this right over here. We see this six is three to the right of the mean. We don't care whether it's to the right or the left, and then four minus three. Four minus three is one, absolute value is one, and we see that."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Then we have six minus three, absolute value of that's just going to be three, and that's this right over here. We see this six is three to the right of the mean. We don't care whether it's to the right or the left, and then four minus three. Four minus three is one, absolute value is one, and we see that. It is one to the right of three, and so what do we have? We have two plus two is four, plus three is seven, plus one is eight over four, which is equal to two. So the mean absolute deviation, let me write it down, it fell off over here."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Four minus three is one, absolute value is one, and we see that. It is one to the right of three, and so what do we have? We have two plus two is four, plus three is seven, plus one is eight over four, which is equal to two. So the mean absolute deviation, let me write it down, it fell off over here. Here for this data set, the mean absolute deviation is equal to two, while for this data set, the mean absolute deviation is equal to one. That makes sense. They have the exact same means."}, {"video_title": "Mean absolute deviation   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the mean absolute deviation, let me write it down, it fell off over here. Here for this data set, the mean absolute deviation is equal to two, while for this data set, the mean absolute deviation is equal to one. That makes sense. They have the exact same means. They both have a mean of three, but this one is more spread out. The one on the right is more spread out because on average, each of these points are two away from three, while on average, each of these points are one away from three. The means of the absolute deviations on this one is one."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "This right over here is a scratch pad on Khan Academy created by Khan Academy user Charlotte Auen. And what you see here is a simulation that allows us to keep sampling from our gumball machine and start approximating the sampling distribution of the sample proportion. So her simulation focuses on green gumballs, but we talked about yellow before. In the yellow gumballs, we said 60% were yellow, so let's make 60% here green. And then let's take samples of 10, just like we did before. And then let's just start with one sample. So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "In the yellow gumballs, we said 60% were yellow, so let's make 60% here green. And then let's take samples of 10, just like we did before. And then let's just start with one sample. So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%. We have one situation where 50% were green. Now let's do another sample."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%. We have one situation where 50% were green. Now let's do another sample. So this sample, 60% are green, and so let's keep going. Let's draw another sample. And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's do another sample. So this sample, 60% are green, and so let's keep going. Let's draw another sample. And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. And let's just really increase. So we're gonna do 50 samples of 10 at a time. And so here, we can quickly get to a fairly large number of samples."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. And let's just really increase. So we're gonna do 50 samples of 10 at a time. And so here, we can quickly get to a fairly large number of samples. So here, we're over 1,000 samples. And what's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "And so here, we can quickly get to a fairly large number of samples. So here, we're over 1,000 samples. And what's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6. We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. And as we draw more and more samples, we should get even closer and closer to those values. And we see that for the most part, we are getting closer and closer."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "What we calculated a few minutes ago was that it should be 0.6. We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. And as we draw more and more samples, we should get even closer and closer to those values. And we see that for the most part, we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before. Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "And we see that for the most part, we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before. Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. But what if our population proportion is closer to zero? So let's say our population proportion is 10%, 0.1. What do you think the distribution is going to look like then?"}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. But what if our population proportion is closer to zero? So let's say our population proportion is 10%, 0.1. What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed. But let's actually see that. So here we see that our distribution is indeed right skewed."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed. But let's actually see that. So here we see that our distribution is indeed right skewed. And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew. And if your population proportion was close to one, well, you can imagine the opposite's going to happen. You're going to end up with a left skew."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "So here we see that our distribution is indeed right skewed. And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew. And if your population proportion was close to one, well, you can imagine the opposite's going to happen. You're going to end up with a left skew. And we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation. And so let's do a population proportion that is right in between."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "You're going to end up with a left skew. And we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation. And so let's do a population proportion that is right in between. And so here, this is similar to what we saw before. This is looking roughly normal. But now, and that's when we had sample size of 10."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "And so let's do a population proportion that is right in between. And so here, this is similar to what we saw before. This is looking roughly normal. But now, and that's when we had sample size of 10. But what if we have a sample size of 50 every time? Well, notice, now it looks like a much tighter distribution. This isn't even going all the way to one yet, but it is a much tighter distribution."}, {"video_title": "Sampling distribution of sample proportion part 2   AP Statistics   Khan Academy.mp3", "Sentence": "But now, and that's when we had sample size of 10. But what if we have a sample size of 50 every time? Well, notice, now it looks like a much tighter distribution. This isn't even going all the way to one yet, but it is a much tighter distribution. And the reason why that made sense, the standard deviation of your sample proportion, it is inversely proportional to the square root of n. And so that makes sense. So hopefully you have a good intuition now for the sample proportion, its distribution, the sampling distribution of the sample proportion, that you can calculate its mean and its standard deviation. And you feel good about it because we saw it in a simulation."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "The possible colors are blue, yellow, white, red, orange, and green. How many four-color codes can be made if the colors cannot be repeated? To some degree, this whole paragraph in the beginning doesn't even matter. If we're just choosing from how many colors are there? There's one, two, three, four, five, six colors. And we're going to pick four of them. How many four-color codes can be made if the colors cannot be repeated?"}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "If we're just choosing from how many colors are there? There's one, two, three, four, five, six colors. And we're going to pick four of them. How many four-color codes can be made if the colors cannot be repeated? And since these are codes, we're going to assume that blue, red, yellow, and green is different than green, red, yellow, and blue. We're going to assume that these are not the same code. Even though we've picked the same four colors, we're going to assume that these are two different codes."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "How many four-color codes can be made if the colors cannot be repeated? And since these are codes, we're going to assume that blue, red, yellow, and green is different than green, red, yellow, and blue. We're going to assume that these are not the same code. Even though we've picked the same four colors, we're going to assume that these are two different codes. And that makes sense because we're dealing with codes. These are different codes. This would count as two different codes right here, even though we've picked the same actual colors, the same four colors."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Even though we've picked the same four colors, we're going to assume that these are two different codes. And that makes sense because we're dealing with codes. These are different codes. This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. We've picked them in different orders. With that out of the way, let's think about how many different ways we can pick four colors. Let's say we have four slots here."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. We've picked them in different orders. With that out of the way, let's think about how many different ways we can pick four colors. Let's say we have four slots here. One slot, two slot, three slot, and four slots. At first, we care only about how many ways can we pick a color for that slot right there, that first slot. We haven't picked any colors yet."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let's say we have four slots here. One slot, two slot, three slot, and four slots. At first, we care only about how many ways can we pick a color for that slot right there, that first slot. We haven't picked any colors yet. We have six possible colors. One, two, three, four, five, six. There's going to be six different possibilities for this slot right there."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We haven't picked any colors yet. We have six possible colors. One, two, three, four, five, six. There's going to be six different possibilities for this slot right there. Let's put a six right there. Now, they told us that the colors cannot be repeated. Whatever color is in this slot, we're going to take it out of the possible colors."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "There's going to be six different possibilities for this slot right there. Let's put a six right there. Now, they told us that the colors cannot be repeated. Whatever color is in this slot, we're going to take it out of the possible colors. Now that we've taken that color out, how many possibilities are when we go to this slot, when we go to the next slot? How many possibilities when we go to the next slot right here? We took one of the six out for the first slot, so there's only five possibilities here."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Whatever color is in this slot, we're going to take it out of the possible colors. Now that we've taken that color out, how many possibilities are when we go to this slot, when we go to the next slot? How many possibilities when we go to the next slot right here? We took one of the six out for the first slot, so there's only five possibilities here. By the same logic, when we go to the third slot, we've used up two of the colors already. There will only be four possible colors left. Then for the last slot, we would have used up three of the colors, so there's only three possibilities left."}, {"video_title": "Example Ways to arrange colors   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We took one of the six out for the first slot, so there's only five possibilities here. By the same logic, when we go to the third slot, we've used up two of the colors already. There will only be four possible colors left. Then for the last slot, we would have used up three of the colors, so there's only three possibilities left. If we think about all of the possibilities, all of the permutations, and permutations are when you think about all of the possibilities, and you do care about order, where you say that this is different than this. This is a different permutation than this. All of the different permutations here, when you pick four colors out of a possible of six colors, it's going to be six possibilities for the first one, times five for the second bucket, times four for the third bucket of third position, times three."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that you're curious about people's TV watching habits, and in particular, how much TV do people in the country watch? So what you're concerned with, if we imagine the entire country that we've already talked about, especially if we're talking about a country like the United States, but pretty much any country, is a very large population. In the United States, we're talking about on the order of 300 million people. So ideally, if you could somehow magically do it, you would survey or somehow observe all 300 million people and take the mean of how many hours of TV they watch on a given day. And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So ideally, if you could somehow magically do it, you would survey or somehow observe all 300 million people and take the mean of how many hours of TV they watch on a given day. And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance?"}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that?"}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1?"}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is. These both could be way off."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is. These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances."}, {"video_title": "Sample variance   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances. And then we're dividing not by the number of data points we have, but by 1 less than the number of data points we have. So this calculation, where we summed up all of this, and then we divided by 5, not by 6, this is the standard definition of sample variance. So I'll leave you there."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I've taken some exercises from the Khan Academy exercises here, and I'm just gonna solve it on my scratch pad. The following data points represent the number of animal crackers in each kid's lunchbox. Sort the data from least to greatest, and then find the interquartile range of the data set. And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives?"}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines. Let me write those. We have two nines. Then we have three tens."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we have two nines. Let me write those. We have two nines. Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11. We have an 11. We have two 12s. Two 12s."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Then we have an 11. We have an 11. We have two 12s. Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers?"}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here."}, {"video_title": "How to calculate interquartile range IQR   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here. And the middle of the second half, I have one, two, three, four, five numbers and this 12 is right in the middle. You have two to the left and two to the right. So the median of the second half is 12."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I want to do in this video is build up some tools in our toolkit for dealing with sums and differences of random variables. So let's say that we have two random variables, X and Y, and they are completely independent. They are independent random variables. And I'm just going to go over a little bit of notation here. If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm just going to go over a little bit of notation here. If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y. There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's the same thing as sigma squared of Y. There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y. What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The random variable X plus the random variable Y. What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be?"}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is what the mean of our random variable A would be equal to. All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is what the mean of our random variable A would be equal to. All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of random variable Y. So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of random variable Y. So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the difference?"}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements. So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are equivalent statements. So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is. Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's all the variance actually is. Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y?"}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable. So if you look at this, we can rewrite this. I'll give myself a little bit more space."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable. So if you look at this, we can rewrite this. I'll give myself a little bit more space. We can rewrite this as the expected value, the variance of negative Y is the expected value. This is just 1. Negative 1 squared is just 1."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll give myself a little bit more space. We can rewrite this as the expected value, the variance of negative Y is the expected value. This is just 1. Negative 1 squared is just 1. And over here you have Y. And instead of just writing plus the expected value of negative Y, that's the same thing as minus the expected value of Y. So you have that and then all of that squared."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Negative 1 squared is just 1. And over here you have Y. And instead of just writing plus the expected value of negative Y, that's the same thing as minus the expected value of Y. So you have that and then all of that squared. Now notice, this is the exact same thing by definition as the variance of Y. So we just showed you just now, so this is the variance of Y. So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have that and then all of that squared. Now notice, this is the exact same thing by definition as the variance of Y. So we just showed you just now, so this is the variance of Y. So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. You could definitely believe this. It's equal to the sum of the variance of the first one plus the variance of the negative of the second one. And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. You could definitely believe this. It's equal to the sum of the variance of the first one plus the variance of the negative of the second one. And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable. You just care about absolute distance. So it makes complete sense that that quantity and that quantity is going to be the same thing."}, {"video_title": "Variance of differences of random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable. You just care about absolute distance. So it makes complete sense that that quantity and that quantity is going to be the same thing. So the whole reason why I went through this exercise, the important takeaways here, is that the mean of differences right over here, so I could rewrite it as the mean of the differences of the random variable is the same thing as the differences of their means. And then the other important takeaway, and I'm going to build on this in the next few videos, is that the variance of the difference, if I define a new random variable as the difference of two other random variables, the variance of that random variable is actually the sum of the variances of the two random variables. So these are the two important takeaways that we'll use to build on in future videos."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "In other videos, we've done linear regressions by hand, but we mentioned that most regressions are actually done using some type of computer or calculator. And so what we're going to do in this video is look at an example of the output that we might see from a computer and to not be intimidated by it and to see how it gives us the equation for the regression line and some of the other data it gives us. So here it tells us Cheryl Dixon is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake in milligrams and the number of hours spent studying. A scatterplot of the data showed a linear relationship. This is a computer output from a least squares regression analysis on the data. So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "She randomly selects 20 students at her school and records their caffeine intake in milligrams and the number of hours spent studying. A scatterplot of the data showed a linear relationship. This is a computer output from a least squares regression analysis on the data. So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. How do we make sense of this in order to come up with an equation for our linear regression? So let's just get straight on our variables. Let's just say that we say that Y is the thing that we're trying to predict."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. How do we make sense of this in order to come up with an equation for our linear regression? So let's just get straight on our variables. Let's just say that we say that Y is the thing that we're trying to predict. So this is the hours spent studying, hours studying, and then let's say X is what we think explains the hours studying or is one of the things that explains the hours studying, and this is the amount of caffeine ingested. So this is caffeine consumed in milligrams. And so our regression line would have the form Y hat."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "Let's just say that we say that Y is the thing that we're trying to predict. So this is the hours spent studying, hours studying, and then let's say X is what we think explains the hours studying or is one of the things that explains the hours studying, and this is the amount of caffeine ingested. So this is caffeine consumed in milligrams. And so our regression line would have the form Y hat. This tells us that this is a linear regression. It's trying to estimate the actual Y values for given Xs. It's going to be equal to MX plus B."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "And so our regression line would have the form Y hat. This tells us that this is a linear regression. It's trying to estimate the actual Y values for given Xs. It's going to be equal to MX plus B. Now how do we figure out what M and B are based on this computer output? So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine. And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "It's going to be equal to MX plus B. Now how do we figure out what M and B are based on this computer output? So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine. And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. There is the constant value, and there is your variable, in this case caffeine, that you're using to predict the amount that you study. And so this tells you the coefficients on each. The coefficient on a constant is the constant."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. There is the constant value, and there is your variable, in this case caffeine, that you're using to predict the amount that you study. And so this tells you the coefficients on each. The coefficient on a constant is the constant. You could view this as the coefficient on the X to the zeroth term. And so the coefficient on the constant, that is the constant, 2.544. And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "The coefficient on a constant is the constant. You could view this as the coefficient on the X to the zeroth term. And so the coefficient on the constant, that is the constant, 2.544. And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. So just like that, we actually have the equation for the regression line. That is why these computer things are useful. So we can just write it out."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. So just like that, we actually have the equation for the regression line. That is why these computer things are useful. So we can just write it out. Y hat is equal to 0.164X plus 2.544, 2.544. So that's the regression line. What is this other information they give us?"}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "So we can just write it out. Y hat is equal to 0.164X plus 2.544, 2.544. So that's the regression line. What is this other information they give us? Well, I won't give you a very satisfying answer because all of this is actually useful for inferential statistics. To think about things like, well, what is the probability that this is chance that we got something to fit this well? So this right over here is the R squared."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "What is this other information they give us? Well, I won't give you a very satisfying answer because all of this is actually useful for inferential statistics. To think about things like, well, what is the probability that this is chance that we got something to fit this well? So this right over here is the R squared. And if you wanted to figure out the R from this, you would just take the square root here. We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have. But you might say, well, how do we know if R is a positive square root or the negative square root of that?"}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "So this right over here is the R squared. And if you wanted to figure out the R from this, you would just take the square root here. We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have. But you might say, well, how do we know if R is a positive square root or the negative square root of that? R can take on values between negative one and positive one. And the answer is, you would look at the slope here. We have a positive slope, which tells us that R is going to be positive."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "But you might say, well, how do we know if R is a positive square root or the negative square root of that? R can take on values between negative one and positive one. And the answer is, you would look at the slope here. We have a positive slope, which tells us that R is going to be positive. If we had a negative slope, then we would take the negative square root. Now, this right here is the adjusted R squared. And we really don't have to worry about it too much when we're thinking about just bivariate data."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "We have a positive slope, which tells us that R is going to be positive. If we had a negative slope, then we would take the negative square root. Now, this right here is the adjusted R squared. And we really don't have to worry about it too much when we're thinking about just bivariate data. We're talking about caffeine and hour studying in this case. If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet. Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "And we really don't have to worry about it too much when we're thinking about just bivariate data. We're talking about caffeine and hour studying in this case. If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet. Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. And why is that useful? Well, that's a measure of how well the regression line fits the data. It's a measure of, we could say, the typical error."}, {"video_title": "Interpreting computer regression data   AP Statistics   Khan Academy.mp3", "Sentence": "Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. And why is that useful? Well, that's a measure of how well the regression line fits the data. It's a measure of, we could say, the typical error. So big takeaway, computers are useful. They'll give you a lot of data. And the key thing is how do you pick out the things that you actually need?"}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So you go around the restaurant, and you write down everyone's age. And so these are the ages of everyone in the restaurant at that moment. And so you're interested in somehow presenting this, somehow visualizing the distribution of the ages, because you want to just say, well, are there more young people, are there more teenagers, are there more middle-aged people, are there more seniors here? And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins?"}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops. It's the number."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops. It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket?"}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three. Three people."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three. Three people. And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Three people. And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people. Two people are in that bucket. And then 50 to 59. So you have one, two people."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have one, two people. Two people are in that bucket. And then 50 to 59. So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories?"}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19. There are three people. So 10 to 19, there are three people. So I'll do a bar like this."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then we have the 10 to 19. There are three people. So 10 to 19, there are three people. So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. There's five people. Five people there. So it'll look like this."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So 20 to 29, this is gonna be this bar. There's five people. Five people there. So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person. One person."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person. One person. And then we have 40 to 49. We have two people. 40 to 49, two people."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "One person. And then we have 40 to 49. We have two people. 40 to 49, two people. So it looks like this. 40 to 49, two people. Almost there, 50 to 59."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "40 to 49, two people. So it looks like this. 40 to 49, two people. Almost there, 50 to 59. We have two people. 50 to 59, we also have two people. So that's that right over there."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Almost there, 50 to 59. We have two people. 50 to 59, we also have two people. So that's that right over there. That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So that's that right over there. That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there?"}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot."}, {"video_title": "How to create a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. But as a histogram, we're able to put them into buckets. And we're just like, hey, generally between the ages zero and nine, we have six people. And so you see that plotted out just like that."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Hey everybody, LeBron here. Got another quick brain teaser for you. Do I have a better odds of making three free throws in a row or one three pointer? Here's my friend Sal with the answer. Excellent question LeBron. But before I answer it, I want to point out an interesting trend related to your question that I just dug up. This is from the New York Times, October 2009, so it's a couple of years old."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here's my friend Sal with the answer. Excellent question LeBron. But before I answer it, I want to point out an interesting trend related to your question that I just dug up. This is from the New York Times, October 2009, so it's a couple of years old. But it's really interesting. It shows that since three pointers were first introduced, they were first introduced in the 1979-1980 season, that three pointers have become more and more frequent. So what they are showing here is the average attempts per team season by season."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is from the New York Times, October 2009, so it's a couple of years old. But it's really interesting. It shows that since three pointers were first introduced, they were first introduced in the 1979-1980 season, that three pointers have become more and more frequent. So what they are showing here is the average attempts per team season by season. It looks like there's just a steady upward trend here related to our question. There's a couple of anomalies here. The ones that really jump out are these three seasons in the late 90s."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what they are showing here is the average attempts per team season by season. It looks like there's just a steady upward trend here related to our question. There's a couple of anomalies here. The ones that really jump out are these three seasons in the late 90s. That's because the actual three pointer line was pulled in to get higher scoring games, so people attempted more, but then it was put back to where it was originally. This was a shortened season. I'm not really sure what happened in the 2000-2001-2002 season."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "The ones that really jump out are these three seasons in the late 90s. That's because the actual three pointer line was pulled in to get higher scoring games, so people attempted more, but then it was put back to where it was originally. This was a shortened season. I'm not really sure what happened in the 2000-2001-2002 season. But it's something to think about. There is just this trend that more and more people are taking three pointers. With that out of the way, let's think about your actual question."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm not really sure what happened in the 2000-2001-2002 season. But it's something to think about. There is just this trend that more and more people are taking three pointers. With that out of the way, let's think about your actual question. To answer it, I dug up your stats right over here from NBA.com. We'll use your career stats. We want to compare three free throws to a three pointer."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "With that out of the way, let's think about your actual question. To answer it, I dug up your stats right over here from NBA.com. We'll use your career stats. We want to compare three free throws to a three pointer. Right over here we have your three pointer percentage. This is in your career. I'll round it to the nearest hundredths."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to compare three free throws to a three pointer. Right over here we have your three pointer percentage. This is in your career. I'll round it to the nearest hundredths. It looks like it's about 33%. Your three point percentage is 33%. Then your free throw percentage, your career free throw percentage, and this is in your career, we'll round to the nearest hundredths."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll round it to the nearest hundredths. It looks like it's about 33%. Your three point percentage is 33%. Then your free throw percentage, your career free throw percentage, and this is in your career, we'll round to the nearest hundredths. We'll round up right over here. That gets us to right at about 75%. Clearly, looking at these numbers right over here, you're much more likely to make a given free throw than making a given three pointer."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then your free throw percentage, your career free throw percentage, and this is in your career, we'll round to the nearest hundredths. We'll round up right over here. That gets us to right at about 75%. Clearly, looking at these numbers right over here, you're much more likely to make a given free throw than making a given three pointer. You're more than twice as likely to make a free throw. But that's not what you asked. You asked, what about three free throws in a row?"}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Clearly, looking at these numbers right over here, you're much more likely to make a given free throw than making a given three pointer. You're more than twice as likely to make a free throw. But that's not what you asked. You asked, what about three free throws in a row? We'll do an analysis very similar to the last time when we asked about 10 free throws in a row. Let's think about the first free throw. Free throw number one."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "You asked, what about three free throws in a row? We'll do an analysis very similar to the last time when we asked about 10 free throws in a row. Let's think about the first free throw. Free throw number one. If we imagined a gazillion LeBron Jameses, identical LeBron Jameses, all taking that first free throw, we would expect on average that 75% would make that first free throw. 75% is three-fourths. 75% would make that first free throw."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Free throw number one. If we imagined a gazillion LeBron Jameses, identical LeBron Jameses, all taking that first free throw, we would expect on average that 75% would make that first free throw. 75% is three-fourths. 75% would make that first free throw. 25% we would expect on average, won't always be the case, but this is what we would expect, 25% would miss that first free throw. Let's go to the second free throw. Free throw number two."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "75% would make that first free throw. 25% we would expect on average, won't always be the case, but this is what we would expect, 25% would miss that first free throw. Let's go to the second free throw. Free throw number two. We only care about the ones that the LeBron Jameses that keep making their free throws. Let's think about of the 75% that made that first one. Some of the 25% might make that second one and then maybe even the third one, but let's just think about the ones that made the first one."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Free throw number two. We only care about the ones that the LeBron Jameses that keep making their free throws. Let's think about of the 75% that made that first one. Some of the 25% might make that second one and then maybe even the third one, but let's just think about the ones that made the first one. Of the ones that made the first one, we would expect 75% of them to make the second one. 75% of the 75%, that's half of the 75%, that's about 75% of the 75% would make that second free throw and the first free throw. Now we have, this is going to be 75% times 75%."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Some of the 25% might make that second one and then maybe even the third one, but let's just think about the ones that made the first one. Of the ones that made the first one, we would expect 75% of them to make the second one. 75% of the 75%, that's half of the 75%, that's about 75% of the 75% would make that second free throw and the first free throw. Now we have, this is going to be 75% times 75%. Of course, there's other combinations out here where someone's missed at least one of the free throws. Let's go to the third free throw. Free throw number three."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now we have, this is going to be 75% times 75%. Of course, there's other combinations out here where someone's missed at least one of the free throws. Let's go to the third free throw. Free throw number three. What percentage of these LeBron Jameses right here will make the third free throw? 75% of these will make the third free throw. 75% of this number, let me just draw it visually, that's half, that's about 75% of that number."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Free throw number three. What percentage of these LeBron Jameses right here will make the third free throw? 75% of these will make the third free throw. 75% of this number, let me just draw it visually, that's half, that's about 75% of that number. They will make the third one as well. This is 75% of this number, which is 75% of 75%. This is how many LeBron Jameses are going to make all three of the free throws."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "75% of this number, let me just draw it visually, that's half, that's about 75% of that number. They will make the third one as well. This is 75% of this number, which is 75% of 75%. This is how many LeBron Jameses are going to make all three of the free throws. Once again, we can write this as, we can either multiply it out or we can just write this as 75% to the third power, which is the same thing as 75%, literally means 75 per 100, same thing as 75 over 100 to the third power, which is the same thing as 0.75 to the third power. Let's calculate it, get the calculator out. Actually, let me show you, we get the same result."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is how many LeBron Jameses are going to make all three of the free throws. Once again, we can write this as, we can either multiply it out or we can just write this as 75% to the third power, which is the same thing as 75%, literally means 75 per 100, same thing as 75 over 100 to the third power, which is the same thing as 0.75 to the third power. Let's calculate it, get the calculator out. Actually, let me show you, we get the same result. We can write 0.75 times 0.75. On this calculator, that little snowflake looking thing, it means multiplication, times 0.75. Then we get 0.42, I'll round to the nearest hundredths."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me show you, we get the same result. We can write 0.75 times 0.75. On this calculator, that little snowflake looking thing, it means multiplication, times 0.75. Then we get 0.42, I'll round to the nearest hundredths. That's the same thing we would get if we got 0.75 to the third power. Once again, 0.42. Let me write that."}, {"video_title": "Three pointer vs free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we get 0.42, I'll round to the nearest hundredths. That's the same thing we would get if we got 0.75 to the third power. Once again, 0.42. Let me write that. This gets us to approximately 0.42, which is the same thing as 42%. Your probability of making three free throws in a row is 42%, which is still higher than making one three-pointer. I'll leave you there, but I want the people who are watching this video to think about what would happen if the numbers were different, or maybe look up some NBA players or maybe some college players, and figure out and compare the probability of making three free throws to a three-pointer, and see if you can find any players where their probability of making a three-pointer is actually higher than making three free throws in a row."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Accident frequency, and I'm just making this up. And I could just show these data points, maybe for some kind of statistical survey, that when the age is this, whatever number this is, maybe this is 20 years old, this is the accident frequency, and it could be a number of accidents per hundred. And that when the age is 21 years old, this is the frequency. And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes?"}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative?"}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative?"}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective."}, {"video_title": "Bivariate relationship linearity, strength and direction   AP Statistics   Khan Academy.mp3", "Sentence": "These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective. There'll be some cases that are more obvious than others. So for, and oftentimes you wanna make a comparison. That this is a stronger linear, positive linear relationship than this one is, right over here, because you can see most of the data is closer to the line."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Darnell is a middle school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean?"}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's."}, {"video_title": "Standard normal table for proportion below   AP Statistics   Khan Academy.mp3", "Sentence": "And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's. But the answer to this question, what proportion of students' heights are lower than Darnell's height? Well, that would be 0.7157. And they want our answer to four decimal places, which is exactly what we have done."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Well ideally, we would go to the entire population of likely voters right over here. Let's say there's 100,000 likely voters and we would ask every one of them, who do you support? And from that, we would be able to get the population proportion. Which would be, this is the proportion that support candidate A. But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Which would be, this is the proportion that support candidate A. But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions. We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And we know a lot about the sampling distribution of the sample proportions. We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here. Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe 0.54 is right over here. Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean?"}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion. And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "We calculated our sample proportion. And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05. So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our sample proportion that we just happened to calculate. So that would be 0.54 minus two times five hundredths."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to be, this is approximately 0.05. So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our sample proportion that we just happened to calculate. So that would be 0.54 minus two times five hundredths. So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So that would be 0.54 minus two times five hundredths. So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is not going to be fixed every time we take a sample."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is not going to be fixed every time we take a sample. Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use it over and over, it will produce intervals, and the intervals won't always be the same, it's gonna be dependent on our sample proportion, but it will produce intervals which include the true proportion, which we might not know and often don't know. It'll include the true proportion 95% of the time."}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And as I mentioned, this margin of error is not going to be fixed every time we take a sample. Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use it over and over, it will produce intervals, and the intervals won't always be the same, it's gonna be dependent on our sample proportion, but it will produce intervals which include the true proportion, which we might not know and often don't know. It'll include the true proportion 95% of the time. I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interesting question is, well, what if you wanted to tighten up the intervals on average?"}, {"video_title": "Confidence intervals and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "It'll include the true proportion 95% of the time. I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interesting question is, well, what if you wanted to tighten up the intervals on average? How would you do that? Well, if you wanted to lower your margin of error, the best way to lower the margin of error is if you increase this denominator right over here, and increasing that denominator means increasing the sample size. And so one thing that you will often see when people are talking about election coverage is, well, we need to sample more people in order to get a lower margin of error."}, {"video_title": "Estimating the line of best fit exercise   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's see, we have a bunch of data points. And we want to find a line that at least shows the trend in the data. And this one seems a little difficult, because if we ignore these three points down here, maybe we could do a line that looks something like this. It seems like it kind of approximates this trend, although it doesn't seem like a great trend. And if we ignore these two points right over here, we could do something like, maybe something like that. But we can't just ignore points like that, so I would say that there's actually no good line of best fit here. So let me check my answer."}, {"video_title": "Estimating the line of best fit exercise   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It seems like it kind of approximates this trend, although it doesn't seem like a great trend. And if we ignore these two points right over here, we could do something like, maybe something like that. But we can't just ignore points like that, so I would say that there's actually no good line of best fit here. So let me check my answer. Let's try a couple more of these. Find the line of best fit. Well, this feels very similar."}, {"video_title": "Estimating the line of best fit exercise   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me check my answer. Let's try a couple more of these. Find the line of best fit. Well, this feels very similar. It really feels like there's no, I mean, I could do that, but then I'm ignoring these two points. I could do something like that, then I'd be ignoring these points. So I'd also say no good best fit line exists."}, {"video_title": "Estimating the line of best fit exercise   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this feels very similar. It really feels like there's no, I mean, I could do that, but then I'm ignoring these two points. I could do something like that, then I'd be ignoring these points. So I'd also say no good best fit line exists. So let's try one more. So here, it looks like there's very clearly this trend, and I could try to fit it a little bit better than it's fit right now. So it feels like something like that fits this trend line quite well."}, {"video_title": "Estimating the line of best fit exercise   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'd also say no good best fit line exists. So let's try one more. So here, it looks like there's very clearly this trend, and I could try to fit it a little bit better than it's fit right now. So it feels like something like that fits this trend line quite well. I could maybe drop this down a little bit, something like that. Let's check my answer. A good best fit line exists."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's say that you have a cherry pie store, and you've noticed that there is variability in the number of cherries on each pie that you sell. Some pies might have over 100 cherries, while other pies might have fewer than 50 cherries. So what you're curious about is what is the distribution? How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between?"}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries?"}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assuming that they're all accounted for by this histogram? And I encourage you to pause the video and try to figure it out on your own."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assuming that they're all accounted for by this histogram? And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries?"}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pies have 60 or more cherries."}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do we have that have 60 to 89 cherries than 120 to 149 cherries?"}, {"video_title": "How to interpret a histogram   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do we have that have 60 to 89 cherries than 120 to 149 cherries? Well, we say, well, we have eight pies that have 60 to 89 cherries, three that have 120 to 149. So we have five more pies in the 60 to 89 category than we do in the 120 to 149 category. So a lot of questions that we can start to answer, and hopefully this gives you a sense of how you can interpret histograms."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "This one has a population of 383, and then it calculates the parameters for that population directly from it. The mean is 10.9, the variance is 25.5. And then it uses that population and samples from it, and it does samples of size two, three, four, five, all the way up to 10, and it keeps sampling from it, calculates the statistics for those samples, so the sample mean and the sample variance, in particular the biased sample variance, and it starts telling us some things about us that give us some intuition. And you can actually click on each of these and zoom in to really be able to study these graphs in detail. So I've already taken a screenshot of this and put it on my little doodle pad, so it can really delve into some of the math and the intuition of what this is actually showing us. So here I took a screenshot, and you see for this case right over here, the population was 529, population mean was 10.6. And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you can actually click on each of these and zoom in to really be able to study these graphs in detail. So I've already taken a screenshot of this and put it on my little doodle pad, so it can really delve into some of the math and the intuition of what this is actually showing us. So here I took a screenshot, and you see for this case right over here, the population was 529, population mean was 10.6. And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8. So this first chart on the bottom left tells us a couple of interesting things. And just to be clear, this is the biased sample variance that he's calculating. This is the biased sample variance."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8. So this first chart on the bottom left tells us a couple of interesting things. And just to be clear, this is the biased sample variance that he's calculating. This is the biased sample variance. So he's calculating it. That is being calculated for each of our data points, so starting with our first data point in each of our samples, going to our nth data point in the sample. You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the biased sample variance. So he's calculating it. That is being calculated for each of our data points, so starting with our first data point in each of our samples, going to our nth data point in the sample. You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things. The first thing it shows us is that the cases where we are significantly underestimating the sample variance, when we're getting sample variances close to zero, these are also the cases, these are also the cases, or they're disproportionately the cases, where the means for those samples are way far off from the true sample mean. Or you could view that the other way around. The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things. The first thing it shows us is that the cases where we are significantly underestimating the sample variance, when we're getting sample variances close to zero, these are also the cases, these are also the cases, or they're disproportionately the cases, where the means for those samples are way far off from the true sample mean. Or you could view that the other way around. The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations. The other thing that might pop out at you is the realization that the pinker dots are the ones for smaller sample size, while the bluer dots are the ones of a larger sample size. And you see here these two little, the two little, I guess the tails, so to speak, of this hump, that at these ends, you disproportionately, it's more of a reddish color. That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations. The other thing that might pop out at you is the realization that the pinker dots are the ones for smaller sample size, while the bluer dots are the ones of a larger sample size. And you see here these two little, the two little, I guess the tails, so to speak, of this hump, that at these ends, you disproportionately, it's more of a reddish color. That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates. There are some red ones here, and that's why it gives us that purplish color. But out here on these tails, it's almost purely some of these red. Every now and then, by happenstance, you get a little blue one."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates. There are some red ones here, and that's why it gives us that purplish color. But out here on these tails, it's almost purely some of these red. Every now and then, by happenstance, you get a little blue one. But this is disproportionately far more red, which really makes sense. When you have a smaller sample size, you're more likely to get a sample mean that is a bad estimate of the population mean, that's far from the population mean, and you're more likely to significantly underestimate the sample variance. Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population variance and finding the mean over all of those, you see that over many, many, many trials, many, many samples of size two, that that by sample variance over population variance, it's approaching half of the true population variance."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "Every now and then, by happenstance, you get a little blue one. But this is disproportionately far more red, which really makes sense. When you have a smaller sample size, you're more likely to get a sample mean that is a bad estimate of the population mean, that's far from the population mean, and you're more likely to significantly underestimate the sample variance. Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population variance and finding the mean over all of those, you see that over many, many, many trials, many, many samples of size two, that that by sample variance over population variance, it's approaching half of the true population variance. When sample size is three, it's approaching 2 3rds, 66.6% of the true population variance. When sample size is four, it's approaching 3 4ths of the true population variance. And so we can come up with a general theme that's happening."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population variance and finding the mean over all of those, you see that over many, many, many trials, many, many samples of size two, that that by sample variance over population variance, it's approaching half of the true population variance. When sample size is three, it's approaching 2 3rds, 66.6% of the true population variance. When sample size is four, it's approaching 3 4ths of the true population variance. And so we can come up with a general theme that's happening. When we use the biased estimate, when we use the biased estimate, we're not approaching the population variance, we're approaching n minus one, let me write this down, we're approaching n minus one over n times the population variance. When n was two, this approached 1 1.5, 1 1.5. When n is three, this is 2 3rds."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so we can come up with a general theme that's happening. When we use the biased estimate, when we use the biased estimate, we're not approaching the population variance, we're approaching n minus one, let me write this down, we're approaching n minus one over n times the population variance. When n was two, this approached 1 1.5, 1 1.5. When n is three, this is 2 3rds. When n is four, this is 3 4ths. So this is giving us a biased estimate. So how would we unbiased this?"}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "When n is three, this is 2 3rds. When n is four, this is 3 4ths. So this is giving us a biased estimate. So how would we unbiased this? Well, if we really wanna get our best estimate of the true population variance, not n minus one over n times the population variance, we would wanna multiply, let me do this in a color I haven't used yet, we would wanna multiply times n over n minus one. We would wanna multiply n over n minus one to get an unbiased estimate. Here, these cancel out, and you are left just with your population variance."}, {"video_title": "Simulation showing bias in sample variance   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how would we unbiased this? Well, if we really wanna get our best estimate of the true population variance, not n minus one over n times the population variance, we would wanna multiply, let me do this in a color I haven't used yet, we would wanna multiply times n over n minus one. We would wanna multiply n over n minus one to get an unbiased estimate. Here, these cancel out, and you are left just with your population variance. That's what we want to estimate. And over here, you are left with our unbiased estimate, our unbiased estimate of population variance, our unbiased sample variance, which is equal to, and this is what we see, what we saw in the last several videos, what you see in statistics books, and sometimes it's confusing why. Hopefully, Peter's simulation gives you a good idea of why, or at least convinces you that it is the case."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "What proportion of laptop prices are between $624 and $768? So let's think about what they are asking. So we have a normal distribution for the prices. So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624. And $768 would put us right at about there."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624. And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values?"}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624. That will be negative two point something."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624. That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table. And let's see, we want to get 0.30."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table. And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179. So 0.6179."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179. So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be?"}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table. So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table. So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to get us, let's see, negative 2.1, there we go."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768, to get what's in between."}, {"video_title": "Standard normal table for proportion between values   AP Statistics   Khan Academy.mp3", "Sentence": "So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768, to get what's in between. 0.6179, once again, I know I keep repeating it, that's this entire area right over here, and we're gonna subtract out what we have in red. So minus 0.0179, so we're gonna subtract this out, to get 0.6. So if we want to give our answer to four decimal places, it would be 0.6000, or another way to think about it is exactly 60% is between 624 and 768."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Let me throw a few blue ones in there. And what we're going to concern ourselves in this video are the yellow gumballs. And let's say that we know that the proportion of yellow gumballs over here is p. This right over here is a population, population parameter. Parameter. And for the sake of argument, just to make things concrete, let's just say that 60% of the gumballs are yellow, or 0.6 of them. Now let's review some things that we have seen before. I'm gonna define our Bernoulli random variable."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Parameter. And for the sake of argument, just to make things concrete, let's just say that 60% of the gumballs are yellow, or 0.6 of them. Now let's review some things that we have seen before. I'm gonna define our Bernoulli random variable. Let's call this capital Y, which is equal to one if when we take one random gumball out of that machine, we get a, we pick a yellow gumball. And it's equal to zero if when we pick one random gumball out of that machine, we don't pick yellow. So not yellow."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "I'm gonna define our Bernoulli random variable. Let's call this capital Y, which is equal to one if when we take one random gumball out of that machine, we get a, we pick a yellow gumball. And it's equal to zero if when we pick one random gumball out of that machine, we don't pick yellow. So not yellow. From previous videos, we know some interesting things about this Bernoulli random variable. We know its mean. We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "So not yellow. From previous videos, we know some interesting things about this Bernoulli random variable. We know its mean. We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population. So it's going to be equal to P, which in this particular case we know is 0.6. And we know what the standard deviation of this Bernoulli random variable is. It is going to be P times one minus P. Actually, that's the variance."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population. So it's going to be equal to P, which in this particular case we know is 0.6. And we know what the standard deviation of this Bernoulli random variable is. It is going to be P times one minus P. Actually, that's the variance. We want to take the square root of that to get the standard deviation. So in this particular scenario, that's going to be the square root of 0.6 times 0.4. Fair enough."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "It is going to be P times one minus P. Actually, that's the variance. We want to take the square root of that to get the standard deviation. So in this particular scenario, that's going to be the square root of 0.6 times 0.4. Fair enough. This is all review so far. But now let me define, let me define another random variable, X, which is equal to the sum of 10 independent, independent trials, trials, trials of Y. Now, we have seen random variables like this before."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Fair enough. This is all review so far. But now let me define, let me define another random variable, X, which is equal to the sum of 10 independent, independent trials, trials, trials of Y. Now, we have seen random variables like this before. This is a binomial random variable. Now, what do we know about its mean and standard deviation? Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over here."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Now, we have seen random variables like this before. This is a binomial random variable. Now, what do we know about its mean and standard deviation? Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over here. So N times P, which in this particular situation is going to be, N is 10, we're doing 10 trials, and P is 0.6, which is equal to six. And that makes sense. If 60% of the balls here are yellow, and if you were to take a sample, or if you were to take 10 trials, so if you were to take 10 balls one at a time, they have to be independent, so you keep looking at them and then replacing them, but if you took 10, then you would expect that six of them would be yellow."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over here. So N times P, which in this particular situation is going to be, N is 10, we're doing 10 trials, and P is 0.6, which is equal to six. And that makes sense. If 60% of the balls here are yellow, and if you were to take a sample, or if you were to take 10 trials, so if you were to take 10 balls one at a time, they have to be independent, so you keep looking at them and then replacing them, but if you took 10, then you would expect that six of them would be yellow. They're not always going to be six yellow, but that would be maybe what you would expect. All right, now what's the standard deviation here? So our standard deviation is equal to, and we have proved this in other videos, it's equal to the square root of N times P times one minus P. Notice, you just put an N right over here under the radical sign, and so this is going to get us, in this particular situation, to 10 times 0.6 times 0.4, and then the square root of everything."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "If 60% of the balls here are yellow, and if you were to take a sample, or if you were to take 10 trials, so if you were to take 10 balls one at a time, they have to be independent, so you keep looking at them and then replacing them, but if you took 10, then you would expect that six of them would be yellow. They're not always going to be six yellow, but that would be maybe what you would expect. All right, now what's the standard deviation here? So our standard deviation is equal to, and we have proved this in other videos, it's equal to the square root of N times P times one minus P. Notice, you just put an N right over here under the radical sign, and so this is going to get us, in this particular situation, to 10 times 0.6 times 0.4, and then the square root of everything. So all of this is review. If it's unfamiliar, I encourage you to review some of the videos on Bernoulli random variables and on binomial random variables. But what we're going to do in this video is think about a sampling distribution, and it's going to be the sampling distribution for a sample statistic known as the sample proportion, which we actually talked about when we first introduced sampling distributions."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "So our standard deviation is equal to, and we have proved this in other videos, it's equal to the square root of N times P times one minus P. Notice, you just put an N right over here under the radical sign, and so this is going to get us, in this particular situation, to 10 times 0.6 times 0.4, and then the square root of everything. So all of this is review. If it's unfamiliar, I encourage you to review some of the videos on Bernoulli random variables and on binomial random variables. But what we're going to do in this video is think about a sampling distribution, and it's going to be the sampling distribution for a sample statistic known as the sample proportion, which we actually talked about when we first introduced sampling distributions. So let's say, so let's just park all of this. This is background right over here. Let's start taking samples of 10, and I didn't pick that randomly."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "But what we're going to do in this video is think about a sampling distribution, and it's going to be the sampling distribution for a sample statistic known as the sample proportion, which we actually talked about when we first introduced sampling distributions. So let's say, so let's just park all of this. This is background right over here. Let's start taking samples of 10, and I didn't pick that randomly. I want to make it reconcile with what we did with our random variable here. And so let's take a sample of 10 gumballs, and let's calculate the proportion that are yellow. And so we will call that our sample proportion, and I might as well just do that in yellow."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Let's start taking samples of 10, and I didn't pick that randomly. I want to make it reconcile with what we did with our random variable here. And so let's take a sample of 10 gumballs, and let's calculate the proportion that are yellow. And so we will call that our sample proportion, and I might as well just do that in yellow. So we want to calculate the sample proportion that are yellow. And what is this equivalent to? Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, and then I'm gonna divide it by the sample size."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "And so we will call that our sample proportion, and I might as well just do that in yellow. So we want to calculate the sample proportion that are yellow. And what is this equivalent to? Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, and then I'm gonna divide it by the sample size. So I'm gonna divide it by N. And in this case, it would be X divided by 10. I know what some of you are thinking. Wait, wait, wait, hold on for a second."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, and then I'm gonna divide it by the sample size. So I'm gonna divide it by N. And in this case, it would be X divided by 10. I know what some of you are thinking. Wait, wait, wait, hold on for a second. X is sum of 10 independent, independent trials right over here. To be independent, you can't just take 10 gumballs. You have to take them one at a time and then replace them back in order for them to be truly independent."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "Wait, wait, wait, hold on for a second. X is sum of 10 independent, independent trials right over here. To be independent, you can't just take 10 gumballs. You have to take them one at a time and then replace them back in order for them to be truly independent. But remember, we have our 10% rule. We have our 10% rule, which tells us that if a sample is less than or equal to 10% of the population, then you can treat each of the gumballs in this situation as being independent. So let's just say for the sake of argument that there are 10,000 gumballs in here."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "You have to take them one at a time and then replace them back in order for them to be truly independent. But remember, we have our 10% rule. We have our 10% rule, which tells us that if a sample is less than or equal to 10% of the population, then you can treat each of the gumballs in this situation as being independent. So let's just say for the sake of argument that there are 10,000 gumballs in here. And so we can feel pretty good that these samples, that each of the things in the sample are independent of each other by our 10% rule. And so each of these are, each of these 10 gumballs, what we see, they are going to be independent. I'm gonna put them in quotes by the 10% rule."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just say for the sake of argument that there are 10,000 gumballs in here. And so we can feel pretty good that these samples, that each of the things in the sample are independent of each other by our 10% rule. And so each of these are, each of these 10 gumballs, what we see, they are going to be independent. I'm gonna put them in quotes by the 10% rule. And so in that situation, we can make this claim or we can feel good that this claim is roughly true. And so let's say for that first sample that we do, our sample proportion is equal to 0.3. So three of our gumballs just happened, three of our 10 gumballs just happened to be yellow."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "I'm gonna put them in quotes by the 10% rule. And so in that situation, we can make this claim or we can feel good that this claim is roughly true. And so let's say for that first sample that we do, our sample proportion is equal to 0.3. So three of our gumballs just happened, three of our 10 gumballs just happened to be yellow. Then we do it again. So we take another sample. We calculate our sample proportion, the statistic again."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "So three of our gumballs just happened, three of our 10 gumballs just happened to be yellow. Then we do it again. So we take another sample. We calculate our sample proportion, the statistic again. Remember, it's trying to estimate our population parameter. And let's say that time it happens to be seven out of 10. And we just keep doing that."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "We calculate our sample proportion, the statistic again. Remember, it's trying to estimate our population parameter. And let's say that time it happens to be seven out of 10. And we just keep doing that. And if we keep doing that and we plot it on a dot chart or dot distribution, I guess we could say, where our possible outcomes, you could have zero out of 10, one out of 10, two, three, four, five, that's so half of them, six, seven, eight, nine, 10. So that would be all of them. And so you could plot, okay, 0.3, one, two, three."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "And we just keep doing that. And if we keep doing that and we plot it on a dot chart or dot distribution, I guess we could say, where our possible outcomes, you could have zero out of 10, one out of 10, two, three, four, five, that's so half of them, six, seven, eight, nine, 10. So that would be all of them. And so you could plot, okay, 0.3, one, two, three. That's one scenario where I got zero, where my sample proportion is 0.3. Then 0.7, that's one situation where I got a 0.7. And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "And so you could plot, okay, 0.3, one, two, three. That's one scenario where I got zero, where my sample proportion is 0.3. Then 0.7, that's one situation where I got a 0.7. And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here. And if you kept taking samples and kept calculating these sample proportions and you kept plotting it here, you would get a better and better and better approximation for the sampling distribution of the sampling proportion. But how can we actually characterize the true sampling distribution for the sample proportion? What is going to be the mean of this sampling distribution?"}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here. And if you kept taking samples and kept calculating these sample proportions and you kept plotting it here, you would get a better and better and better approximation for the sampling distribution of the sampling proportion. But how can we actually characterize the true sampling distribution for the sample proportion? What is going to be the mean of this sampling distribution? And what is going to be the standard deviation? Well, we can derive that from what we see right over here. The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just going to be the mean of X divided by N, which is equal to what?"}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "What is going to be the mean of this sampling distribution? And what is going to be the standard deviation? Well, we can derive that from what we see right over here. The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just going to be the mean of X divided by N, which is equal to what? Well, the mean of X is N times P. This is N times P. You divided it by N, you're going to get P. And that makes sense. One way to think about it, the expected value for your sample proportion is going to be the proportion of gumballs that you actually see. And so this is also a good indicator that this is going to be a reasonably unbiased estimator."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just going to be the mean of X divided by N, which is equal to what? Well, the mean of X is N times P. This is N times P. You divided it by N, you're going to get P. And that makes sense. One way to think about it, the expected value for your sample proportion is going to be the proportion of gumballs that you actually see. And so this is also a good indicator that this is going to be a reasonably unbiased estimator. Now let's think about the standard deviation for our sample proportion. Well, we can just view that as our standard deviation of our binomial random variable X divided by N. So this is going to be equal to the square root of N times P times one minus P, all of that over N, which is the same thing as if we put this, if we divide by N inside the radical, it'd be the same thing as the square root of N, P times one minus P over N squared. Divide the numerator and denominator by N, you will get the square root of P times one minus P, all of that over N. And so in this particular situation where our parameter is 0.6, where our population parameter is 0.6, so it's going to be 0.6, that's the true proportion for our population."}, {"video_title": "Sampling distribution of sample proportion part 1   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is also a good indicator that this is going to be a reasonably unbiased estimator. Now let's think about the standard deviation for our sample proportion. Well, we can just view that as our standard deviation of our binomial random variable X divided by N. So this is going to be equal to the square root of N times P times one minus P, all of that over N, which is the same thing as if we put this, if we divide by N inside the radical, it'd be the same thing as the square root of N, P times one minus P over N squared. Divide the numerator and denominator by N, you will get the square root of P times one minus P, all of that over N. And so in this particular situation where our parameter is 0.6, where our population parameter is 0.6, so it's going to be 0.6, that's the true proportion for our population. And then what would our standard deviation be for our sample proportion? Well, it's going to be equal to the square root of 0.6 times 0.4, all of that over 10. And we can get a calculator out to calculate that."}, {"video_title": "Using a confidence interval to test slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "Hashim obtained a random sample of students and noticed a positive linear relationship between their ages and their backpack weights. A 95% confidence interval for the slope of the regression line was 0.39 plus or minus 0.23. Hashim wants to use this interval to test the null hypothesis that the true slope of the population regression line, so this is a population parameter right here for the slope of the population regression line, is equal to zero, versus the alternative hypothesis is that the true slope of the population regression line is not equal to zero, at the alpha is equal to 0.05 level of significance. Assume that all conditions for inference have been met. So given the information that we just have about what Hashim is doing, what would be his conclusion? Would he reject the null hypothesis, which would suggest the alternative, or would he be unable to reject the null hypothesis? Well, let's just think about this a little bit."}, {"video_title": "Using a confidence interval to test slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "Assume that all conditions for inference have been met. So given the information that we just have about what Hashim is doing, what would be his conclusion? Would he reject the null hypothesis, which would suggest the alternative, or would he be unable to reject the null hypothesis? Well, let's just think about this a little bit. We have a 95% confidence interval. Let me write this down. So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what?"}, {"video_title": "Using a confidence interval to test slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "Well, let's just think about this a little bit. We have a 95% confidence interval. Let me write this down. So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what? 0.62. Now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 95% confidence interval, that 95% of the time we do this, it should overlap with the true population parameter that we are trying to estimate. But in this hypothesis test, remember, we are assuming that the true population parameter is equal to zero, and that does not overlap with this confidence interval."}, {"video_title": "Using a confidence interval to test slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what? 0.62. Now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 95% confidence interval, that 95% of the time we do this, it should overlap with the true population parameter that we are trying to estimate. But in this hypothesis test, remember, we are assuming that the true population parameter is equal to zero, and that does not overlap with this confidence interval. So assuming, let me write this down, assuming null hypothesis is true, null hypothesis is true, we are in the less than or equal to 5% chance of situations, situations, where, where beta not overlap, overlap, with 95% intervals. And the whole notion of hypothesis testing is you assume the null hypothesis, you take your sample, and then if you get statistics, and if the probability of getting those statistics or something even more extreme than those statistics is less than your significance level, then you reject the null hypothesis. And that's exactly what's happening here."}, {"video_title": "Using a confidence interval to test slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "But in this hypothesis test, remember, we are assuming that the true population parameter is equal to zero, and that does not overlap with this confidence interval. So assuming, let me write this down, assuming null hypothesis is true, null hypothesis is true, we are in the less than or equal to 5% chance of situations, situations, where, where beta not overlap, overlap, with 95% intervals. And the whole notion of hypothesis testing is you assume the null hypothesis, you take your sample, and then if you get statistics, and if the probability of getting those statistics or something even more extreme than those statistics is less than your significance level, then you reject the null hypothesis. And that's exactly what's happening here. And this null hypothesis value is nowhere even close to overlapping, it's over 16 hundredths below the low end of this bound. And so because of that, we would reject the null hypothesis. Reject the null hypothesis, which suggests the alternative, which suggests the alternative hypothesis."}, {"video_title": "Using a confidence interval to test slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "And that's exactly what's happening here. And this null hypothesis value is nowhere even close to overlapping, it's over 16 hundredths below the low end of this bound. And so because of that, we would reject the null hypothesis. Reject the null hypothesis, which suggests the alternative, which suggests the alternative hypothesis. And one way to interpret this alternative hypothesis, that beta is not equal to zero, is that there is, there is a non-zero linear relationship, relationship, between, between ages and backpack weights. Ages and backpack weights. And we are done."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The male and female heights are each normally distributed. We independently, randomly select a man and a woman. What is the probability that the woman is taller than the man? So I encourage you to pause this video and think through it and I'll give you a hint. What if we were to define the random variable m as equal to the height of a randomly selected man, height of random man. What if we defined the random variable w to be equal to the height of a random woman, woman, and we defined a third random variable in terms of these first two? So let me call this d for difference."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So I encourage you to pause this video and think through it and I'll give you a hint. What if we were to define the random variable m as equal to the height of a randomly selected man, height of random man. What if we defined the random variable w to be equal to the height of a random woman, woman, and we defined a third random variable in terms of these first two? So let me call this d for difference. And it is equal to the difference in height between a randomly selected man and a randomly selected woman. So d, the random variable d, is equal to the random variable m minus the random variable w. So the first two are clearly normally distributed. They tell us that right over here."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let me call this d for difference. And it is equal to the difference in height between a randomly selected man and a randomly selected woman. So d, the random variable d, is equal to the random variable m minus the random variable w. So the first two are clearly normally distributed. They tell us that right over here. The male and female heights are each normally distributed. And we also know, or you're about to know, that the difference of random variables that are each normally distributed is also going to be normally distributed. So given this, can you think about how to tackle this question?"}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "They tell us that right over here. The male and female heights are each normally distributed. And we also know, or you're about to know, that the difference of random variables that are each normally distributed is also going to be normally distributed. So given this, can you think about how to tackle this question? The probability that the woman is taller than the man. All right, now let's work through this together and to help us visualize, I'll draw the normal distribution curves for these three random variables. So this first one is for the variable m. And so right here in the middle, that is the mean of m. And we know that this is going to be equal to 178 centimeters."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So given this, can you think about how to tackle this question? The probability that the woman is taller than the man. All right, now let's work through this together and to help us visualize, I'll draw the normal distribution curves for these three random variables. So this first one is for the variable m. And so right here in the middle, that is the mean of m. And we know that this is going to be equal to 178 centimeters. We'll assume everything is in centimeters. We also know that it has a standard deviation of eight centimeters. So for example, if this is one standard deviation above, this is one standard deviation below, this point right over here would be eight centimeters more than 178, so that would be 186."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this first one is for the variable m. And so right here in the middle, that is the mean of m. And we know that this is going to be equal to 178 centimeters. We'll assume everything is in centimeters. We also know that it has a standard deviation of eight centimeters. So for example, if this is one standard deviation above, this is one standard deviation below, this point right over here would be eight centimeters more than 178, so that would be 186. And this would be eight centimeters below that, so this would be 170 centimeters. So this is for the random variable m. Now let's think about the random variable w. The random variable w, the mean of w, they tell us, is 170. And one standard deviation above the mean is going to be six centimeters above the mean."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, if this is one standard deviation above, this is one standard deviation below, this point right over here would be eight centimeters more than 178, so that would be 186. And this would be eight centimeters below that, so this would be 170 centimeters. So this is for the random variable m. Now let's think about the random variable w. The random variable w, the mean of w, they tell us, is 170. And one standard deviation above the mean is going to be six centimeters above the mean. The standard deviation is six, six centimeters. So this would be minus six, is to go to one standard deviation below the mean. Now let's think about the difference between the two, the random variable d. So let me think about this a little bit."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And one standard deviation above the mean is going to be six centimeters above the mean. The standard deviation is six, six centimeters. So this would be minus six, is to go to one standard deviation below the mean. Now let's think about the difference between the two, the random variable d. So let me think about this a little bit. The random variable d, the mean of d is going to be equal to the differences in the means of these random variables. So it's going to be equal to the mean of m, the mean of m, minus the mean of w, minus the mean of w. Well we know both of these, this is gonna be 178 minus 170. So let me write that down."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's think about the difference between the two, the random variable d. So let me think about this a little bit. The random variable d, the mean of d is going to be equal to the differences in the means of these random variables. So it's going to be equal to the mean of m, the mean of m, minus the mean of w, minus the mean of w. Well we know both of these, this is gonna be 178 minus 170. So let me write that down. This is equal to 178 centimeters minus 170 centimeters, which is going to be equal to, I'll do it in this color, this is going to be equal to eight centimeters. So this is eight right over here. Now what about the standard deviation?"}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let me write that down. This is equal to 178 centimeters minus 170 centimeters, which is going to be equal to, I'll do it in this color, this is going to be equal to eight centimeters. So this is eight right over here. Now what about the standard deviation? Assuming these two random variables are independent, and they tell us that we are independently, randomly selecting a man and a woman, the height of the man shouldn't affect the height of the woman or vice versa. Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase. But you won't just add the standard deviations."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now what about the standard deviation? Assuming these two random variables are independent, and they tell us that we are independently, randomly selecting a man and a woman, the height of the man shouldn't affect the height of the woman or vice versa. Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase. But you won't just add the standard deviations. What you would actually do is say, the variance of the difference is going to be the sum of these two variances. So let me write that down. So I could write variance with VAR, or I could write it as the standard deviation squared."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But you won't just add the standard deviations. What you would actually do is say, the variance of the difference is going to be the sum of these two variances. So let me write that down. So I could write variance with VAR, or I could write it as the standard deviation squared. So let me write that. The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive. This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So I could write variance with VAR, or I could write it as the standard deviation squared. So let me write that. The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive. This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. And so we can figure this out. This is going to be equal to, the standard deviation of variable M is eight. So eight squared is going to be 64."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. And so we can figure this out. This is going to be equal to, the standard deviation of variable M is eight. So eight squared is going to be 64. And then we have six squared. This right over here is six. Six squared is going to be 36."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So eight squared is going to be 64. And then we have six squared. This right over here is six. Six squared is going to be 36. You add these two together, this is going to be equal to 100. And so the variance of this distribution right over here is going to be equal to 100. Well what's the standard deviation of that distribution?"}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Six squared is going to be 36. You add these two together, this is going to be equal to 100. And so the variance of this distribution right over here is going to be equal to 100. Well what's the standard deviation of that distribution? Well it's going to be equal to the square root of the variance. So the square root of 100, which is equal to 10. So for example, one standard deviation above the mean is going to be 18."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well what's the standard deviation of that distribution? Well it's going to be equal to the square root of the variance. So the square root of 100, which is equal to 10. So for example, one standard deviation above the mean is going to be 18. One standard deviation below the mean is going to be equal to negative two. And so now using this distribution, we can actually answer this question. What is the probability that the woman is taller than the man?"}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, one standard deviation above the mean is going to be 18. One standard deviation below the mean is going to be equal to negative two. And so now using this distribution, we can actually answer this question. What is the probability that the woman is taller than the man? Well we can rewrite that question as saying, what is the probability that the random variable D is, what conditions would it be? Pause the video and think about it. Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "What is the probability that the woman is taller than the man? Well we can rewrite that question as saying, what is the probability that the random variable D is, what conditions would it be? Pause the video and think about it. Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. Then D is going to be less than zero. So what we really want to do is figure out the probability that D is less than zero. And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that?"}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. Then D is going to be less than zero. So what we really want to do is figure out the probability that D is less than zero. And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? So we want to figure out this entire area. There's a couple of ways you could do this. You could figure out the z-score for D equaling zero, and that's pretty straightforward."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? So we want to figure out this entire area. There's a couple of ways you could do this. You could figure out the z-score for D equaling zero, and that's pretty straightforward. You could just say this z is equal to zero minus our mean of eight divided by our standard deviation of 10. So it's negative eight over 10, which is equal to negative 8 1\u204410. So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8?"}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You could figure out the z-score for D equaling zero, and that's pretty straightforward. You could just say this z is equal to zero minus our mean of eight divided by our standard deviation of 10. So it's negative eight over 10, which is equal to negative 8 1\u204410. So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? Another way you could do this is you could use a graphing calculator. I have a TI-84 here, where you have a normal cumulative distribution function. I'm gonna press second, vars, and that gets me to distribution."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? Another way you could do this is you could use a graphing calculator. I have a TI-84 here, where you have a normal cumulative distribution function. I'm gonna press second, vars, and that gets me to distribution. And so I have these various functions. I want normal cumulative distribution function. So that is choice two."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "I'm gonna press second, vars, and that gets me to distribution. And so I have these various functions. I want normal cumulative distribution function. So that is choice two. And then the lower bound. Well, I want to go to negative infinity. Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So that is choice two. And then the lower bound. Well, I want to go to negative infinity. Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. So we could say negative one times 10 to the 99th power. And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power. So this is a very, very, very negative number."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. So we could say negative one times 10 to the 99th power. And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power. So this is a very, very, very negative number. The upper bound here, we want to go, let me delete this. The upper bound is going to be zero. We're finding the area from negative infinity all the way to zero."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this is a very, very, very negative number. The upper bound here, we want to go, let me delete this. The upper bound is going to be zero. We're finding the area from negative infinity all the way to zero. The mean here, well, we've already figured that out. The mean is eight. And then the standard deviation here, we figured this out too, this is equal to 10."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We're finding the area from negative infinity all the way to zero. The mean here, well, we've already figured that out. The mean is eight. And then the standard deviation here, we figured this out too, this is equal to 10. And so when we pick this, we're gonna go back to the main screen, enter. So this is, we could have just typed this in directly on the main screen. This says, look, we're looking at a normal distribution."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And then the standard deviation here, we figured this out too, this is equal to 10. And so when we pick this, we're gonna go back to the main screen, enter. So this is, we could have just typed this in directly on the main screen. This says, look, we're looking at a normal distribution. We want to find the cumulative area between two bounds. In this case, it's from negative infinity to zero. From negative infinity to zero, where the mean is eight and the standard deviation is 10."}, {"video_title": "Example Analyzing the difference in distributions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This says, look, we're looking at a normal distribution. We want to find the cumulative area between two bounds. In this case, it's from negative infinity to zero. From negative infinity to zero, where the mean is eight and the standard deviation is 10. We press enter, and we get approximately 0.212, is approximately 0.212. Or you could say, what is the probability that the woman is taller than the man? Well, 0.212, or approximately, there's a 21.2% chance of that happening, a little better than one in five."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you need to shake the hand exactly once of every other person in the room so that you all meet. So my question to you is, if each of these people need to shake the hand of every other person exactly once, how many handshakes are going to occur? The number of handshakes that are going to occur. So like always, pause the video and see if you can make sense of this. All right, I'm assuming you've had a go at it. So one way to think about it is, okay, if you say there's a handshake, well, a handshake has two people are party to a handshake. We're not talking about some new three-person handshake or four-person handshake."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So like always, pause the video and see if you can make sense of this. All right, I'm assuming you've had a go at it. So one way to think about it is, okay, if you say there's a handshake, well, a handshake has two people are party to a handshake. We're not talking about some new three-person handshake or four-person handshake. We're just talking about the traditional two people shake their right hands. And so there's one person, and there's another person that's party to it. And so you say, okay, there's four possibilities of one party."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not talking about some new three-person handshake or four-person handshake. We're just talking about the traditional two people shake their right hands. And so there's one person, and there's another person that's party to it. And so you say, okay, there's four possibilities of one party. And if we assume people aren't shaking their own hands, which we are assuming, or they're always going to shake someone else's hand, then for the other party, there's only three other, for each of these four possibilities, who's this party, there's three possibilities, who's the other party. And so you might say that there's four times three handshakes. Since there's four times three, I guess you could say possible handshakes."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so you say, okay, there's four possibilities of one party. And if we assume people aren't shaking their own hands, which we are assuming, or they're always going to shake someone else's hand, then for the other party, there's only three other, for each of these four possibilities, who's this party, there's three possibilities, who's the other party. And so you might say that there's four times three handshakes. Since there's four times three, I guess you could say possible handshakes. And what I'd like you to do is think a little bit about whether this is right, whether there would actually be 12 handshakes. Well, you might have thought about it, and you might say, well, you know, this four times three, this would count, this is actually counting the permutations. This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Since there's four times three, I guess you could say possible handshakes. And what I'd like you to do is think a little bit about whether this is right, whether there would actually be 12 handshakes. Well, you might have thought about it, and you might say, well, you know, this four times three, this would count, this is actually counting the permutations. This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. This would count, this would count A being the number one handshaker and B being the number two handshaker as being different than B being the number one handshaker and A being the number two handshaker. But we don't want both of these things to occur. We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. This would count, this would count A being the number one handshaker and B being the number two handshaker as being different than B being the number one handshaker and A being the number two handshaker. But we don't want both of these things to occur. We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. We only have to do it once. These are actually the same thing, so no reason for both of these to occur. So we are going to be double counting."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. We only have to do it once. These are actually the same thing, so no reason for both of these to occur. So we are going to be double counting. So what we really want to do is think about combinations. One way to think about it is you have four people. In a world of four people or a pool of four people, how many ways can you choose two?"}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we are going to be double counting. So what we really want to do is think about combinations. One way to think about it is you have four people. In a world of four people or a pool of four people, how many ways can you choose two? How many ways to choose two? Because that's what we're doing. Each handshake is just really a selection of two of these people."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "In a world of four people or a pool of four people, how many ways can you choose two? How many ways to choose two? Because that's what we're doing. Each handshake is just really a selection of two of these people. And so we want to say how many ways can we select two people so that we have a different, each combination, each of these ways to select two people should have a different combination of people in it. If two of them had the same, if A, B, and B, A, these are the same combination. And so this is really a combinations problem."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Each handshake is just really a selection of two of these people. And so we want to say how many ways can we select two people so that we have a different, each combination, each of these ways to select two people should have a different combination of people in it. If two of them had the same, if A, B, and B, A, these are the same combination. And so this is really a combinations problem. This is really equivalent to saying how many ways are there to choose two people from a pool of four, or four choose two, or four choose two. And so this is going to be, well, how many ways are there to permute four people into three spots, which is going to be four times three, which we just figured out right over there, which is 12. Actually, why don't I do it in that green color so you see where that came from."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so this is really a combinations problem. This is really equivalent to saying how many ways are there to choose two people from a pool of four, or four choose two, or four choose two. And so this is going to be, well, how many ways are there to permute four people into three spots, which is going to be four times three, which we just figured out right over there, which is 12. Actually, why don't I do it in that green color so you see where that came from. So four times three, and then you're going to divide that by the number of ways you can arrange two people. Well, you can arrange two people in two different ways. One's on the left, one's on the right, or the other one's on the left, and the other one's on the right."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, why don't I do it in that green color so you see where that came from. So four times three, and then you're going to divide that by the number of ways you can arrange two people. Well, you can arrange two people in two different ways. One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. Or you could also view that as two factorial, which is also equal to two. So we could write this down as two. So this is the number of ways to arrange two people."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. Or you could also view that as two factorial, which is also equal to two. So we could write this down as two. So this is the number of ways to arrange two people. To arrange two people. Two people, two people. And this up here, let me just do this in a new color."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the number of ways to arrange two people. To arrange two people. Two people, two people. And this up here, let me just do this in a new color. This up here, that's the permutations. That's the way, number of permutations if you take two people from a pool of four. So here you would care about order."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this up here, let me just do this in a new color. This up here, that's the permutations. That's the way, number of permutations if you take two people from a pool of four. So here you would care about order. And so, one way to think about it, this two is correcting for this double counting here. And if you wanted to apply the formula, you could. I just kind of reasoned through it again."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here you would care about order. And so, one way to think about it, this two is correcting for this double counting here. And if you wanted to apply the formula, you could. I just kind of reasoned through it again. I mean, you could literally say, okay, four times three is 12, we're double counting, because there's two ways to arrange two people, so you just divided it by two. So you just divide by two. And then you are going to be left with six."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I just kind of reasoned through it again. I mean, you could literally say, okay, four times three is 12, we're double counting, because there's two ways to arrange two people, so you just divided it by two. So you just divide by two. And then you are going to be left with six. You could think of it in terms of this. Or you could just apply the formula. You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you are going to be left with six. You could think of it in terms of this. Or you could just apply the formula. You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. This is going to be four factorial over two factorial times four minus two. Four minus two factorial. And I'm going to make this color different just so you can keep track of how I'm at least applying this."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. This is going to be four factorial over two factorial times four minus two. Four minus two factorial. And I'm going to make this color different just so you can keep track of how I'm at least applying this. And so what is this going to be? This is going to be four times three times two times one over two times one times this right over here is two times one. So that will cancel with that."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm going to make this color different just so you can keep track of how I'm at least applying this. And so what is this going to be? This is going to be four times three times two times one over two times one times this right over here is two times one. So that will cancel with that. Four divided by two is two. Two times three divided by one is equal to six. And to just really hit the point home, let's actually draw it out."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that will cancel with that. Four divided by two is two. Two times three divided by one is equal to six. And to just really hit the point home, let's actually draw it out. Let's draw it out. So A could shake B's hand. A could shake C's hand."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to just really hit the point home, let's actually draw it out. Let's draw it out. So A could shake B's hand. A could shake C's hand. A could shake D's hand. Or B could shake, and I may just do what we calculated first, the 12, B could shake A's hand. B could shake C's hand."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "A could shake C's hand. A could shake D's hand. Or B could shake, and I may just do what we calculated first, the 12, B could shake A's hand. B could shake C's hand. B could shake, whoops. B could shake D's hand. And you can say C could shake A's hand, C could shake B's hand, C could shake, C could shake D's hand."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "B could shake C's hand. B could shake, whoops. B could shake D's hand. And you can say C could shake A's hand, C could shake B's hand, C could shake, C could shake D's hand. and you could say D could shake A's hand, D could shake B's hand, D could shake C's hand. And this is 12 right over here, and this is a permutations. If D shaking C's hand was actually different than C shaking D's hand, then we would count 12."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you can say C could shake A's hand, C could shake B's hand, C could shake, C could shake D's hand. and you could say D could shake A's hand, D could shake B's hand, D could shake C's hand. And this is 12 right over here, and this is a permutations. If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. But we just wanted to say, well, how many ways, they just have to meet each other once. And so we're double counting. So, A, B is the same thing as B, A."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. But we just wanted to say, well, how many ways, they just have to meet each other once. And so we're double counting. So, A, B is the same thing as B, A. A, C is the same thing as C, A. A, D is the same thing as D, A. B, C is the same thing as C, B."}, {"video_title": "Handshaking combinations   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So, A, B is the same thing as B, A. A, C is the same thing as C, A. A, D is the same thing as D, A. B, C is the same thing as C, B. B, D is the same thing as D, B. C, D is the same thing as D, C. And so we'd be left with, if we correct for the double counting, we're left with one, two, three, four, five, six combinations, six possible ways of choosing two from a pool of four, especially when you don't care about the order in which you choose them. Thank you."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "It can only take on a finite number of values, and I defined it as the number of workouts I might do in a week. And we calculated the expected value of our random variable X, which you could also denote as the mean of X, and we used the Greek letter mu, which we use for population mean. And all we did is it's the probability weighted sum of the various outcomes. And we got for this random variable with this probability distribution, we got an expected value or a mean of 2.1. What we're going to do now is extend this idea to measuring spread. And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation. The way we are going to do this has parallels with the way that we've calculated variance in the past."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And we got for this random variable with this probability distribution, we got an expected value or a mean of 2.1. What we're going to do now is extend this idea to measuring spread. And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation. The way we are going to do this has parallels with the way that we've calculated variance in the past. So the variance of our random variable X, what we're going to do is take the difference between each outcome and the mean, square that difference, and then we're going to multiply it by the probability of that outcome. So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1. Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "The way we are going to do this has parallels with the way that we've calculated variance in the past. So the variance of our random variable X, what we're going to do is take the difference between each outcome and the mean, square that difference, and then we're going to multiply it by the probability of that outcome. So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1. Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. Then you're going to get plus two minus 2.1 squared times the probability that you get a two, times 0.4. Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1. So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. Then you're going to get plus two minus 2.1 squared times the probability that you get a two, times 0.4. Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1. So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. So this is going to be negative 2.1 squared, which is just 2.1 squared, so I'll just write this as 2.1 squared times 0.1. That's the first term. And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. So this is going to be negative 2.1 squared, which is just 2.1 squared, so I'll just write this as 2.1 squared times 0.1. That's the first term. And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. So that's just going to be the same thing as 1.1 squared, which is 1.21, but I'll just write it out. 1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01. If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. So that's just going to be the same thing as 1.1 squared, which is 1.21, but I'll just write it out. 1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01. If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. This is going to be plus 1.9 squared, 1.9 squared times 0.1, and we get 1.19. So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance. The square root of 1.19, which is equal to, let's just get the calculator back here."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. This is going to be plus 1.9 squared, 1.9 squared times 0.1, and we get 1.19. So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance. The square root of 1.19, which is equal to, let's just get the calculator back here. So we are just going to take the square root of, I'll just type it again, 1.19, and that gives us, so it's approximately 1.09. Approximately 1.09. So let's see if this makes sense."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "The square root of 1.19, which is equal to, let's just get the calculator back here. So we are just going to take the square root of, I'll just type it again, 1.19, and that gives us, so it's approximately 1.09. Approximately 1.09. So let's see if this makes sense. Let me put this all on a number line right over here. So you have the outcome zero, one, two, three, and four. So you have a 10% chance of getting a zero."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So let's see if this makes sense. Let me put this all on a number line right over here. So you have the outcome zero, one, two, three, and four. So you have a 10% chance of getting a zero. So I will draw that like this. Let's just say this is a height of 10%. You have a 15% chance of getting a one, so that'll be 1.5 times higher."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So you have a 10% chance of getting a zero. So I will draw that like this. Let's just say this is a height of 10%. You have a 15% chance of getting a one, so that'll be 1.5 times higher. So it'll look something like this. You have a 40% chance of getting a two, so that's going to be like this. So you have a 40% chance of getting a two."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "You have a 15% chance of getting a one, so that'll be 1.5 times higher. So it'll look something like this. You have a 40% chance of getting a two, so that's going to be like this. So you have a 40% chance of getting a two. You have a 25% chance of getting a three, so it'll look like this. And then you have a 10% chance of getting a four. So it'll look like that."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So you have a 40% chance of getting a two. You have a 25% chance of getting a three, so it'll look like this. And then you have a 10% chance of getting a four. So it'll look like that. So this is a visualization of this discrete probability distribution where I didn't draw a vertical axis here, but this would be.1, this would be.15, this is.25, and that is.4. And then we see that the mean is at 2.1. The mean is at 2.1, which makes sense."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So it'll look like that. So this is a visualization of this discrete probability distribution where I didn't draw a vertical axis here, but this would be.1, this would be.15, this is.25, and that is.4. And then we see that the mean is at 2.1. The mean is at 2.1, which makes sense. Even though this random variable only takes on integer values, you can have a mean that takes on a non-integer value. And then the standard deviation is 1.09. So 1.09 above the mean is going to get us close to 3.2."}, {"video_title": "Variance and standard deviation of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "The mean is at 2.1, which makes sense. Even though this random variable only takes on integer values, you can have a mean that takes on a non-integer value. And then the standard deviation is 1.09. So 1.09 above the mean is going to get us close to 3.2. And 1.09 below the mean is gonna get us close to one. And so this all, at least intuitively, feels reasonable. This mean does seem to be indicative of the central tendency of this distribution."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have four different gum chewers, and they tell us how many bubbles each of them blew. So what I want to do is I want to figure out first the mean of the number of bubbles blown, and then also figure out how dispersed is the data. How much do these vary from the mean? And I'm going to do that by calculating the mean absolute deviation. So pause this video now. Try to calculate the mean of the number of bubbles blown. And then after you do that, see if you can calculate the mean absolute deviation."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I'm going to do that by calculating the mean absolute deviation. So pause this video now. Try to calculate the mean of the number of bubbles blown. And then after you do that, see if you can calculate the mean absolute deviation. All right, so step one, let's figure out the mean. So the mean is just going to be the sum of the number of bubbles blown divided by the number of data points. So Manuela blew four bubbles."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then after you do that, see if you can calculate the mean absolute deviation. All right, so step one, let's figure out the mean. So the mean is just going to be the sum of the number of bubbles blown divided by the number of data points. So Manuela blew four bubbles. So she blew four bubbles. Sophia blew five bubbles. Jada blew six bubbles."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So Manuela blew four bubbles. So she blew four bubbles. Sophia blew five bubbles. Jada blew six bubbles. And Tara blew one bubble. And we have one, two, three, four data points. So let's divide by four."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Jada blew six bubbles. And Tara blew one bubble. And we have one, two, three, four data points. So let's divide by four. And so this is going to be equal to 4 plus 5 is 9, plus 6 is 15, plus 1 is 16. So it's equal to 16 over 4, which is 16 divided by 4 is equal to 4. So the mean number of bubbles blown is 4."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So let's divide by four. And so this is going to be equal to 4 plus 5 is 9, plus 6 is 15, plus 1 is 16. So it's equal to 16 over 4, which is 16 divided by 4 is equal to 4. So the mean number of bubbles blown is 4. And I can do that. Let me actually do this with a bold line right over here. This is the mean number of bubbles blown."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the mean number of bubbles blown is 4. And I can do that. Let me actually do this with a bold line right over here. This is the mean number of bubbles blown. So now what I want to do is I want to figure out the mean absolute deviation. I'll do it right over here. Mad."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is the mean number of bubbles blown. So now what I want to do is I want to figure out the mean absolute deviation. I'll do it right over here. Mad. Mean absolute deviation. And what we want to do is we want to take the mean of how much do each of these data points deviate from the mean. And I know I just used the word mean twice in a sentence, so it might be a little confusing."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Mad. Mean absolute deviation. And what we want to do is we want to take the mean of how much do each of these data points deviate from the mean. And I know I just used the word mean twice in a sentence, so it might be a little confusing. But as we work through it, hopefully it'll make a little bit of sense. So how much does Manuela's, the number of bubbles she blew, how much does that deviate from the mean? Well, Manuela actually blew four bubbles, and four is the mean."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And I know I just used the word mean twice in a sentence, so it might be a little confusing. But as we work through it, hopefully it'll make a little bit of sense. So how much does Manuela's, the number of bubbles she blew, how much does that deviate from the mean? Well, Manuela actually blew four bubbles, and four is the mean. So her deviation, her absolute deviation from the mean is 0. How much did, actually let me just write this over here. So absolute deviation."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, Manuela actually blew four bubbles, and four is the mean. So her deviation, her absolute deviation from the mean is 0. How much did, actually let me just write this over here. So absolute deviation. That's AD. Absolute deviation from the mean. Manuela didn't deviate at all from the mean."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So absolute deviation. That's AD. Absolute deviation from the mean. Manuela didn't deviate at all from the mean. Now let's think about Sophia. Sophia deviates by 1 from the mean. We see that right there."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Manuela didn't deviate at all from the mean. Now let's think about Sophia. Sophia deviates by 1 from the mean. We see that right there. She's 1 above. Now we would say 1 whether it's 1 above or below, because we're saying absolute deviation. So Sophia deviates by 1."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We see that right there. She's 1 above. Now we would say 1 whether it's 1 above or below, because we're saying absolute deviation. So Sophia deviates by 1. Her absolute deviation is 1. And then we have Jada. How much does she deviate from the mean?"}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So Sophia deviates by 1. Her absolute deviation is 1. And then we have Jada. How much does she deviate from the mean? We see it right over here. She deviates by 2. She is 2 more than the mean."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How much does she deviate from the mean? We see it right over here. She deviates by 2. She is 2 more than the mean. And then how much does Tara deviate from the mean? She is at 1, so that is 3 below the mean. So once again, this was 2."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "She is 2 more than the mean. And then how much does Tara deviate from the mean? She is at 1, so that is 3 below the mean. So once again, this was 2. This is 3. So she deviates. Her absolute deviation is 3."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So once again, this was 2. This is 3. So she deviates. Her absolute deviation is 3. And then we want to take the mean of the absolute deviations. That's the M in MAD, in mean absolute deviation. This is Manuela's absolute deviation."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Her absolute deviation is 3. And then we want to take the mean of the absolute deviations. That's the M in MAD, in mean absolute deviation. This is Manuela's absolute deviation. Sophia's absolute deviation. Jada's absolute deviation. Tara's absolute deviation."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "This is Manuela's absolute deviation. Sophia's absolute deviation. Jada's absolute deviation. Tara's absolute deviation. We want the mean of those. So we divide by the number of data points. And we get 0 plus 1 plus 2 plus 3 is 6 over 4, which is the same thing as 1 and 1 half."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Tara's absolute deviation. We want the mean of those. So we divide by the number of data points. And we get 0 plus 1 plus 2 plus 3 is 6 over 4, which is the same thing as 1 and 1 half. Or let me just write it in all the different ways. We could write it as 3 halves or 1 and 1 half or 1.5, which gives us a measure of how much do these data points vary from the mean of 4. Now I know what some of you are thinking."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And we get 0 plus 1 plus 2 plus 3 is 6 over 4, which is the same thing as 1 and 1 half. Or let me just write it in all the different ways. We could write it as 3 halves or 1 and 1 half or 1.5, which gives us a measure of how much do these data points vary from the mean of 4. Now I know what some of you are thinking. Wait, I thought there was a formula associated with the mean absolute deviation. And it seems really complex. It has all of these absolute value signs and whatever else."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now I know what some of you are thinking. Wait, I thought there was a formula associated with the mean absolute deviation. And it seems really complex. It has all of these absolute value signs and whatever else. Well, that's all we did. That was just a, when we write all those absolute value signs, that's just a fancy way of looking at each data point and thinking about, well, how much does it deviate from the mean, whether it's above or below? That's what the absolute value does."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It has all of these absolute value signs and whatever else. Well, that's all we did. That was just a, when we write all those absolute value signs, that's just a fancy way of looking at each data point and thinking about, well, how much does it deviate from the mean, whether it's above or below? That's what the absolute value does. It doesn't matter if it's 3 below. We just say 3. If it's 2 above, we just say 2."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "That's what the absolute value does. It doesn't matter if it's 3 below. We just say 3. If it's 2 above, we just say 2. We don't put a positive or negative on it. But just so you're comfortable seeing how this is the exact same thing you would have done with the formula, let's do it that way as well. So the mean absolute deviation is going to be equal to, well, we'll start with Manuela."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "If it's 2 above, we just say 2. We don't put a positive or negative on it. But just so you're comfortable seeing how this is the exact same thing you would have done with the formula, let's do it that way as well. So the mean absolute deviation is going to be equal to, well, we'll start with Manuela. How many bubbles did she blow? She blew 4. From that, you subtract the mean of 4."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the mean absolute deviation is going to be equal to, well, we'll start with Manuela. How many bubbles did she blow? She blew 4. From that, you subtract the mean of 4. Take the absolute value. That's her absolute deviation. And of course, this does evaluate to this 0 here."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "From that, you subtract the mean of 4. Take the absolute value. That's her absolute deviation. And of course, this does evaluate to this 0 here. Then you take the absolute value. Sophia blew 5 bubbles, and the mean is 4. Then you do that for Jada."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And of course, this does evaluate to this 0 here. Then you take the absolute value. Sophia blew 5 bubbles, and the mean is 4. Then you do that for Jada. Jada blew 6 bubbles. The mean is 4. And then you do it for Tara."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Then you do that for Jada. Jada blew 6 bubbles. The mean is 4. And then you do it for Tara. Tara blew 1 bubble, and the mean is 4. And then you divide it by the number of data points you have. And let me make it very clear."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then you do it for Tara. Tara blew 1 bubble, and the mean is 4. And then you divide it by the number of data points you have. And let me make it very clear. This right over here, this 4 is the mean. This 4 is the mean. So you're taking each of the data points, and you're seeing how far it is away from the mean."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And let me make it very clear. This right over here, this 4 is the mean. This 4 is the mean. So you're taking each of the data points, and you're seeing how far it is away from the mean. You're taking the absolute value, because you just want to figure out the absolute distance. Now you see, or maybe you see, 4 minus 4, this is, let me do this in a different color, 4 minus 4, that is the 0. That is that 0 right over there."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So you're taking each of the data points, and you're seeing how far it is away from the mean. You're taking the absolute value, because you just want to figure out the absolute distance. Now you see, or maybe you see, 4 minus 4, this is, let me do this in a different color, 4 minus 4, that is the 0. That is that 0 right over there. 5 minus 4, absolute value of that, well, that's going to be, let me do this in a new color, this is just going to be 1. This thing is the same thing as that over there. And we were able to see that just by inspecting this graph or this chart."}, {"video_title": "Mean absolute deviation example   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "That is that 0 right over there. 5 minus 4, absolute value of that, well, that's going to be, let me do this in a new color, this is just going to be 1. This thing is the same thing as that over there. And we were able to see that just by inspecting this graph or this chart. And then 6 minus 4, absolute value of that, that's just going to be 2. That 2 is that 2 right over here, which is the same thing as this 2 right over there. And then finally, our 1 minus 4, that's negative 3, but the absolute value of that is just positive 3, which is this positive 3 right over there, which is the distance, this distance right over here."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's start again with a fair coin. And this time, instead of flipping it four times, let's flip it five times. So five flips of this fair coin. And what I want to think about in this video is the probability of getting exactly three heads. And the way I'm going to think about it is, if you have five flips, how many different equally likely possibilities are there? So you're going to have the first flip. Let me draw it over here."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what I want to think about in this video is the probability of getting exactly three heads. And the way I'm going to think about it is, if you have five flips, how many different equally likely possibilities are there? So you're going to have the first flip. Let me draw it over here. First flip, and there's two possibilities there. It could be heads or tails. Second flip, two possibilities there."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me draw it over here. First flip, and there's two possibilities there. It could be heads or tails. Second flip, two possibilities there. Third flip, two possibilities. Fourth flip, two possibilities. Fifth flip, two possibilities."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Second flip, two possibilities there. Third flip, two possibilities. Fourth flip, two possibilities. Fifth flip, two possibilities. So it's 2 times 2 times 2 times 2 times 2. I hope I said that five times. So 2, or you could use 2 to the fifth power, and that is going to be equal to 32 equally likely possibilities."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Fifth flip, two possibilities. So it's 2 times 2 times 2 times 2 times 2. I hope I said that five times. So 2, or you could use 2 to the fifth power, and that is going to be equal to 32 equally likely possibilities. 32, 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, 16 times 2 is 32 possibilities. And to figure out this probability, we really just have to figure out how many of those possibilities involve getting three heads. We could write out all of the 32 possibilities and literally just count the heads, but let's just use that other technique that we just started to explore in that last video."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 2, or you could use 2 to the fifth power, and that is going to be equal to 32 equally likely possibilities. 32, 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, 16 times 2 is 32 possibilities. And to figure out this probability, we really just have to figure out how many of those possibilities involve getting three heads. We could write out all of the 32 possibilities and literally just count the heads, but let's just use that other technique that we just started to explore in that last video. We have five flips here. So let me draw the flips. 1, 2, 3, 4, 5."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could write out all of the 32 possibilities and literally just count the heads, but let's just use that other technique that we just started to explore in that last video. We have five flips here. So let me draw the flips. 1, 2, 3, 4, 5. And we want to have exactly three heads. And I'm going to call those three heads, I'm going to call them, let me do it in pink, heads A, heads B, heads C, just to give them a name. Although what we're going to see later in this video is that we don't want to differentiate between them."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "1, 2, 3, 4, 5. And we want to have exactly three heads. And I'm going to call those three heads, I'm going to call them, let me do it in pink, heads A, heads B, heads C, just to give them a name. Although what we're going to see later in this video is that we don't want to differentiate between them. To us, it makes no difference if we get this ordering, heads A, heads B, heads C, tails, tails, or if we get this ordering, heads A, heads C, or heads B, tails, tails. We can't count these as two different orderings. We can only count this as one."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Although what we're going to see later in this video is that we don't want to differentiate between them. To us, it makes no difference if we get this ordering, heads A, heads B, heads C, tails, tails, or if we get this ordering, heads A, heads C, or heads B, tails, tails. We can't count these as two different orderings. We can only count this as one. So what we're going to do is first come up with all of the different orderings if we cared about the difference between A, B, and C. And then we're going to divide by all of the different ways that you can arrange three different things. So how many ways can we put A, B, and C into these five buckets that we can view as the flips if we cared about A, B, and C? So let's start with A."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "We can only count this as one. So what we're going to do is first come up with all of the different orderings if we cared about the difference between A, B, and C. And then we're going to divide by all of the different ways that you can arrange three different things. So how many ways can we put A, B, and C into these five buckets that we can view as the flips if we cared about A, B, and C? So let's start with A. If we haven't allocated any of these buckets to any of the heads yet, then we could say that A could be in five different buckets. So there's five possibilities where A could be. So let's just say that this is the one that it goes in, although it could be in any one of these five."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's start with A. If we haven't allocated any of these buckets to any of the heads yet, then we could say that A could be in five different buckets. So there's five possibilities where A could be. So let's just say that this is the one that it goes in, although it could be in any one of these five. But if this takes up one of the five, then how many different possibilities can this heads sit in? How many different possibilities are there? Well, then there's only going to be four buckets left."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just say that this is the one that it goes in, although it could be in any one of these five. But if this takes up one of the five, then how many different possibilities can this heads sit in? How many different possibilities are there? Well, then there's only going to be four buckets left. So then there's only four possibilities. And so if this was where heads A goes, then heads B could be in any of the other four. If heads A was in this first one, then heads B could have been in any of the four."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, then there's only going to be four buckets left. So then there's only four possibilities. And so if this was where heads A goes, then heads B could be in any of the other four. If heads A was in this first one, then heads B could have been in any of the four. I'll just do a particular example. Maybe heads B shows up right there. So once we've taken two of the slots, how many spaces do we have for heads C?"}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "If heads A was in this first one, then heads B could have been in any of the four. I'll just do a particular example. Maybe heads B shows up right there. So once we've taken two of the slots, how many spaces do we have for heads C? Well, we only have three spaces left then for heads C. And so it could be in any of these three spaces. And just to show a particular example, it would look like that. And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads?"}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once we've taken two of the slots, how many spaces do we have for heads C? Well, we only have three spaces left then for heads C. And so it could be in any of these three spaces. And just to show a particular example, it would look like that. And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads? You would say it is 5 times 4 times 3. 5 times 4 is 20, times 3 is equal to 60. So you would say there's 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people in five chairs."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads? You would say it is 5 times 4 times 3. 5 times 4 is 20, times 3 is equal to 60. So you would say there's 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people in five chairs. And obviously, there aren't 60 possibilities of getting three heads. In fact, there's only 32 equally likely possibilities. And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you would say there's 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people in five chairs. And obviously, there aren't 60 possibilities of getting three heads. In fact, there's only 32 equally likely possibilities. And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here. And what we need to do is say, well, these aren't different possibilities. We don't have to overcount for all of the different ways you arrange this. And so what we need to do is divide this by all of the different ways that you can arrange three things."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here. And what we need to do is say, well, these aren't different possibilities. We don't have to overcount for all of the different ways you arrange this. And so what we need to do is divide this by all of the different ways that you can arrange three things. So if I have three things that are in three spaces, so here I have heads in the second flip, third flip, and fifth flip. If I have three things in three spaces like this, how many ways can I arrange them? And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces?"}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so what we need to do is divide this by all of the different ways that you can arrange three things. So if I have three things that are in three spaces, so here I have heads in the second flip, third flip, and fifth flip. If I have three things in three spaces like this, how many ways can I arrange them? And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces? Well, A can go into three spaces. It can go into any of the three. A can go into any of the three spaces."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces? Well, A can go into three spaces. It can go into any of the three. A can go into any of the three spaces. Then B would have two spaces left once A takes one of them. And then C would have one space left once A and B take two of them. So there's three times two times one way to arrange three different things."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "A can go into any of the three spaces. Then B would have two spaces left once A takes one of them. And then C would have one space left once A and B take two of them. So there's three times two times one way to arrange three different things. So three times two times one is equal to six. So the number of possibilities of getting three heads is actually going to be this 5 times 4 times 3. Let me write this down in another color."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's three times two times one way to arrange three different things. So three times two times one is equal to six. So the number of possibilities of getting three heads is actually going to be this 5 times 4 times 3. Let me write this down in another color. So the number of possibilities, let's write this for short, is equal to this 5 times 4 times 3 over the number of ways that I can rearrange three things. Because we don't want to over count for all of these. Viewing this arrangement is fundamentally different than this arrangement."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write this down in another color. So the number of possibilities, let's write this for short, is equal to this 5 times 4 times 3 over the number of ways that I can rearrange three things. Because we don't want to over count for all of these. Viewing this arrangement is fundamentally different than this arrangement. So then we want to divide it by 3 times 2. I want to do that same orange color. Dividing it by 3 times 2 times 1."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Viewing this arrangement is fundamentally different than this arrangement. So then we want to divide it by 3 times 2. I want to do that same orange color. Dividing it by 3 times 2 times 1. 3 times 2 times 1. And which gives us, in the numerator, 120 divided by 6. It's 60 divided by 6."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "Dividing it by 3 times 2 times 1. 3 times 2 times 1. And which gives us, in the numerator, 120 divided by 6. It's 60 divided by 6. This is 60. 5 times 4 times 3 is 60. It gives us 60 divided by 6, which gives us 10 possibilities that gives us exactly three heads."}, {"video_title": "Exactly three heads in five flips   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's 60 divided by 6. This is 60. 5 times 4 times 3 is 60. It gives us 60 divided by 6, which gives us 10 possibilities that gives us exactly three heads. And that's of 32 equally likely possibilities. So the probability of getting exactly three heads, well you get exactly three heads in 10 of the 32 equally likely possibilities. So you have a 5 over 16 chance of that happening."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's define the random variable X. So let's say that X is equal to the number of made shots, number of made free throws when taking six free throws. So it's how many of the six do you make? And we're going to assume what we assumed in the first video in this series of these making free throws. So we're gonna assume the 70% free throw probability right over here. So assuming assumptions, assuming 70% free throw free throw percentage. All right, so let's figure out the probabilities of the different values that X could actually take on."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to assume what we assumed in the first video in this series of these making free throws. So we're gonna assume the 70% free throw probability right over here. So assuming assumptions, assuming 70% free throw free throw percentage. All right, so let's figure out the probabilities of the different values that X could actually take on. So let's see, what is the probability, what is the probability that X is equal to zero? That even though you have a 70% free throw percentage, that you make none of the shots. And actually you could calculate this through probably some common sense without using all of these fancy things."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, so let's figure out the probabilities of the different values that X could actually take on. So let's see, what is the probability, what is the probability that X is equal to zero? That even though you have a 70% free throw percentage, that you make none of the shots. And actually you could calculate this through probably some common sense without using all of these fancy things. But just to make things consistent, I'm gonna write it out. So this is going to be, it's going to be equal to six choose zero times 0.7 to the zeroth power times 0.3 to the sixth power. And this right over here is gonna end up being one."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually you could calculate this through probably some common sense without using all of these fancy things. But just to make things consistent, I'm gonna write it out. So this is going to be, it's going to be equal to six choose zero times 0.7 to the zeroth power times 0.3 to the sixth power. And this right over here is gonna end up being one. This over here is going to end up being one. And so you're just gonna be left with 0.3 to the sixth power. And I calculated it ahead of time."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this right over here is gonna end up being one. This over here is going to end up being one. And so you're just gonna be left with 0.3 to the sixth power. And I calculated it ahead of time. So if we just round to the nearest, if we round our percentages to the nearest tenth, this is going to give you approximately, approximately, well if we round the decimal to the nearest thousandth, you're gonna get something like that, which is approximately equal to 0.1% chance of you missing all of them. So roughly, I'm speaking roughly here, one in a thousand, one in a thousand chance of that happening, of missing all six free throws. Now let's keep going."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I calculated it ahead of time. So if we just round to the nearest, if we round our percentages to the nearest tenth, this is going to give you approximately, approximately, well if we round the decimal to the nearest thousandth, you're gonna get something like that, which is approximately equal to 0.1% chance of you missing all of them. So roughly, I'm speaking roughly here, one in a thousand, one in a thousand chance of that happening, of missing all six free throws. Now let's keep going. This is fun. So what is the probability that our random variable is equal to one? Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's keep going. This is fun. So what is the probability that our random variable is equal to one? Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. And I calculated this, and this is approximately 0.01, or we could say 1%. So still a fairly low probability, 10 times more likely than this, roughly, but still a fairly low probability. Let's keep going."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. And I calculated this, and this is approximately 0.01, or we could say 1%. So still a fairly low probability, 10 times more likely than this, roughly, but still a fairly low probability. Let's keep going. So the probability that X is equal to two, well that's what our first video was, essentially. So this is going to be six choose two times 0.7 squared times 0.3 to the fourth power. And we saw that this is approximately going to be 0.06, or we could say 6%."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's keep going. So the probability that X is equal to two, well that's what our first video was, essentially. So this is going to be six choose two times 0.7 squared times 0.3 to the fourth power. And we saw that this is approximately going to be 0.06, or we could say 6%. And obviously you could type these things in a calculator and get a much more precise answer, but just for the sake of just getting a sense of what these probabilities look like, that's why I'm giving these rough estimates. Kind of, I guess you could say to the closest, maybe tenth of a percent. And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we saw that this is approximately going to be 0.06, or we could say 6%. And obviously you could type these things in a calculator and get a much more precise answer, but just for the sake of just getting a sense of what these probabilities look like, that's why I'm giving these rough estimates. Kind of, I guess you could say to the closest, maybe tenth of a percent. And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. But let's keep going. We're obviously going to have to do a few more of these. So let me just make sure I have enough real estate."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. But let's keep going. We're obviously going to have to do a few more of these. So let me just make sure I have enough real estate. All right, so the probability that our random variable is equal to three is going to be six choose three, and I'm sure you could probably fill this out on your own, but I'm going to do it. 0.7 to the third power times 0.3 to the six minus three, which is the third power, which is approximately equal to, well, it's going to be 0.185 or 18.5, 18.5%. So yeah, you know, that's definitely within the realm of possibility."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me just make sure I have enough real estate. All right, so the probability that our random variable is equal to three is going to be six choose three, and I'm sure you could probably fill this out on your own, but I'm going to do it. 0.7 to the third power times 0.3 to the six minus three, which is the third power, which is approximately equal to, well, it's going to be 0.185 or 18.5, 18.5%. So yeah, you know, that's definitely within the realm of possibility. I mean, all of these are in the realm of possibility, but it's starting to be a non-insignificant probability. So now let's do the probability that our random variable is equal to four. Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So yeah, you know, that's definitely within the realm of possibility. I mean, all of these are in the realm of possibility, but it's starting to be a non-insignificant probability. So now let's do the probability that our random variable is equal to four. Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. All right, two more to go. Let's see, I have not used purple as yet. So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. All right, two more to go. Let's see, I have not used purple as yet. So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. That's interesting, one more left. So the probability that I make all of them, of all six, is going to be equal to, is equal to six choose six, and 0.7 to the sixth power times 0.3 to the zeroth power, which is, this right over here is going to be one, this is going to be one, so it's really just 0.7 to the sixth power, and this is approximately 0.118, I calculated that ahead of time, which is 11.8%, and so there's something interesting that's going on here. The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. That's interesting, one more left. So the probability that I make all of them, of all six, is going to be equal to, is equal to six choose six, and 0.7 to the sixth power times 0.3 to the zeroth power, which is, this right over here is going to be one, this is going to be one, so it's really just 0.7 to the sixth power, and this is approximately 0.118, I calculated that ahead of time, which is 11.8%, and so there's something interesting that's going on here. The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. So you have a 70% free throw probability. This is no longer just flipping a fair coin. Where you will see the symmetry is in these coefficients."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. So you have a 70% free throw probability. This is no longer just flipping a fair coin. Where you will see the symmetry is in these coefficients. If you calculate these coefficients, six choose zero is one, six choose six is one. You would see that six choose one is six, and six choose five is six. You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Where you will see the symmetry is in these coefficients. If you calculate these coefficients, six choose zero is one, six choose six is one. You would see that six choose one is six, and six choose five is six. You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. So you definitely see the symmetry in the coefficients, but then these things are weighted by the fact that you're more likely to make something than miss something. If these were both 0.5, then you would also see the symmetry right over here, and you can plot this to essentially visualize what the probability distribution looks like for this example, and I encourage you to do that, to take these different cases, just like we did in that first example with the fair coin, and plot these. But this essentially does give you the probability distribution for the random variable in question."}, {"video_title": "Free throw binomial probability distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. So you definitely see the symmetry in the coefficients, but then these things are weighted by the fact that you're more likely to make something than miss something. If these were both 0.5, then you would also see the symmetry right over here, and you can plot this to essentially visualize what the probability distribution looks like for this example, and I encourage you to do that, to take these different cases, just like we did in that first example with the fair coin, and plot these. But this essentially does give you the probability distribution for the random variable in question. This is, I just wrote it out instead of just visualizing it, but it says, okay, well, so these are the different values that this random variable can take on. It can't take on negative one, or it can't be 15.5, or pi, or one million. These are the only seven values that this random variable can take on, and I've just given you the probabilities, or I guess you could say the rough probabilities, of the random variable taking on each of those seven values."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if someone just says the mean, they're really referring to what we typically, in everyday language, call the average. Sometimes it's called the arithmetic mean because you'll learn that there's other ways of actually calculating a mean. But it's really you just sum up all of the numbers and you divide by the numbers there are. And so it's one way of measuring the central tendency, or the average, I guess we could say. So this is our mean. We want to average 23 plus 29. Or we want to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so it's one way of measuring the central tendency, or the average, I guess we could say. So this is our mean. We want to average 23 plus 29. Or we want to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. And then divide that by the number of numbers. So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. So you want to divide that by 8."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or we want to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. And then divide that by the number of numbers. So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. So you want to divide that by 8. So let's figure out what that actually is. Actually, I'll just get the calculator out for this part. I could do it by hand, but we'll save some time over here."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you want to divide that by 8. So let's figure out what that actually is. Actually, I'll just get the calculator out for this part. I could do it by hand, but we'll save some time over here. So we have 23 plus 29 plus 20 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. So the sum of all the numbers is 206. And then we want to divide 206 by 8."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could do it by hand, but we'll save some time over here. So we have 23 plus 29 plus 20 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. So the sum of all the numbers is 206. And then we want to divide 206 by 8. So if I say 206 divided by 8 gets us 25.75. So the mean is equal to 25.75. So this is one way to kind of measure the center, the central tendency."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we want to divide 206 by 8. So if I say 206 divided by 8 gets us 25.75. So the mean is equal to 25.75. So this is one way to kind of measure the center, the central tendency. Another way is with the median. And this is to pick out the middle number, the median. And to figure out the median, what we want to do is order these numbers from least to greatest."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is one way to kind of measure the center, the central tendency. Another way is with the median. And this is to pick out the middle number, the median. And to figure out the median, what we want to do is order these numbers from least to greatest. So it looks like the smallest number here is 20. Then the next one is 21. Then we go, there's no 22 here."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to figure out the median, what we want to do is order these numbers from least to greatest. So it looks like the smallest number here is 20. Then the next one is 21. Then we go, there's no 22 here. Let's see, there's two 23s, 23 and a 23. So 23 and a 23. And no 24s."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we go, there's no 22 here. Let's see, there's two 23s, 23 and a 23. So 23 and a 23. And no 24s. There's a 25. There's no 26, 27, 28. There is a 29."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And no 24s. There's a 25. There's no 26, 27, 28. There is a 29. And then you have your 32, 32. And then you have your 33, 33. So what's the middle number now that we've ordered it?"}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There is a 29. And then you have your 32, 32. And then you have your 33, 33. So what's the middle number now that we've ordered it? So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. We already knew that. And so there's actually going to be two middles."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's the middle number now that we've ordered it? So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. We already knew that. And so there's actually going to be two middles. If you have an even number, there's actually two numbers that kind of qualify close to the middle. And to actually get the median, we're going to average them. So 23 will be one of them."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so there's actually going to be two middles. If you have an even number, there's actually two numbers that kind of qualify close to the middle. And to actually get the median, we're going to average them. So 23 will be one of them. That by itself can't be the median because there's three less than it and there's four greater than it. Five by itself can't be the median because there's three larger than it and four less than it. So what we do is we take the mean of these two numbers and we pick that as the median."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 23 will be one of them. That by itself can't be the median because there's three less than it and there's four greater than it. Five by itself can't be the median because there's three larger than it and four less than it. So what we do is we take the mean of these two numbers and we pick that as the median. So if you take 23 plus 25 divided by 2, that's 48 over 2, which is equal to 24. So even though 24 isn't one of these numbers, the median is 24. So this is the middle number."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what we do is we take the mean of these two numbers and we pick that as the median. So if you take 23 plus 25 divided by 2, that's 48 over 2, which is equal to 24. So even though 24 isn't one of these numbers, the median is 24. So this is the middle number. So once again, this is one way of thinking about central tendency. If you wanted a number that could somehow represent the middle, and I want to be clear, there's no one way of doing it. This is one way of measuring the middle."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the middle number. So once again, this is one way of thinking about central tendency. If you wanted a number that could somehow represent the middle, and I want to be clear, there's no one way of doing it. This is one way of measuring the middle. Let me put that in quotes. The middle, if you had to represent this data with one number, and this is another way of representing the middle. Then finally, we can think about the mode."}, {"video_title": "Finding mean, median, and mode   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is one way of measuring the middle. Let me put that in quotes. The middle, if you had to represent this data with one number, and this is another way of representing the middle. Then finally, we can think about the mode. And the mode is just the number that shows up the most in this data set. And all of these numbers show up once except we have the 23, it shows up twice. And so because 23 shows up the most, it shows up twice, every other number only shows up once, 23 is our mode."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Sheera drew the line below to show the trend in the data. Assuming the line is correct, what does the line slope of 15 mean? So let's see. The horizontal axis is time studying in hours. Vertical axis is scores on the test. And each of these blue dots represent the time and the score for a given student. So this student right over here spent, I don't know, it looks like they spent about.6 hours studying, and they didn't do too well on the exam."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "The horizontal axis is time studying in hours. Vertical axis is scores on the test. And each of these blue dots represent the time and the score for a given student. So this student right over here spent, I don't know, it looks like they spent about.6 hours studying, and they didn't do too well on the exam. They look like they got below a 45, looks like a 43 or 44 on the exam. This student over here spent almost 4 1 1 1 studying and got, looks like a 94, close to a 95 on the exam. And what Sheera did is try to draw a line that tries to fit this data."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So this student right over here spent, I don't know, it looks like they spent about.6 hours studying, and they didn't do too well on the exam. They look like they got below a 45, looks like a 43 or 44 on the exam. This student over here spent almost 4 1 1 1 studying and got, looks like a 94, close to a 95 on the exam. And what Sheera did is try to draw a line that tries to fit this data. It seems like it does a pretty good job of at least showing the trend in the data. Now, slope of 15 means that if I'm on the line, so let's say I'm here, and if I increase in the horizontal direction by one, so there I increased by the horizontal direction by one, I should be increasing in the vertical direction by 15. And you see that."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And what Sheera did is try to draw a line that tries to fit this data. It seems like it does a pretty good job of at least showing the trend in the data. Now, slope of 15 means that if I'm on the line, so let's say I'm here, and if I increase in the horizontal direction by one, so there I increased by the horizontal direction by one, I should be increasing in the vertical direction by 15. And you see that. We increased by one hour here, we increased by 15% on the test. Now, what that means is that the trend it shows is that in general along this trend, if someone studies an extra hour, then if we're going with that trend, then hey, you know, it seems reasonable that they might expect to see a 15% gain on their test. Now let's see which of these are consistent."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And you see that. We increased by one hour here, we increased by 15% on the test. Now, what that means is that the trend it shows is that in general along this trend, if someone studies an extra hour, then if we're going with that trend, then hey, you know, it seems reasonable that they might expect to see a 15% gain on their test. Now let's see which of these are consistent. In general, students who didn't study at all got scores of about 15 on the test. Well, let's see, this is neither true, we don't see that these are the people who didn't study at all and they didn't get 15 on the test, and that's definitely not what this 15 implies. This doesn't say what the people who didn't study at all get."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Now let's see which of these are consistent. In general, students who didn't study at all got scores of about 15 on the test. Well, let's see, this is neither true, we don't see that these are the people who didn't study at all and they didn't get 15 on the test, and that's definitely not what this 15 implies. This doesn't say what the people who didn't study at all get. So this one is not true. That one is not true. Let's try this one."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "This doesn't say what the people who didn't study at all get. So this one is not true. That one is not true. Let's try this one. If one student studied for one hour more than another student, the student who studied more got exactly 15 more points on the test. Well, this is getting closer to the spirit of what the slope means, but this word exactly is what, at least in my mind, messes this choice up. Because this isn't saying that it's guaranteed that if you study an hour extra that you'll get 15% more on the test."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Let's try this one. If one student studied for one hour more than another student, the student who studied more got exactly 15 more points on the test. Well, this is getting closer to the spirit of what the slope means, but this word exactly is what, at least in my mind, messes this choice up. Because this isn't saying that it's guaranteed that if you study an hour extra that you'll get 15% more on the test. This is just saying that this is the general trend that this line is seeing. So it's not guaranteed. For example, we could find this student here who studied exactly two hours, and if we look at the students who studied for three hours, well, there's no one exactly at three hours, but some of them, so this was, let's see, the student who was at two hours, you go to three hours, there's no one exactly there, but there's gonna be students who got better than what would be expected and students who might get a little bit worse."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Because this isn't saying that it's guaranteed that if you study an hour extra that you'll get 15% more on the test. This is just saying that this is the general trend that this line is seeing. So it's not guaranteed. For example, we could find this student here who studied exactly two hours, and if we look at the students who studied for three hours, well, there's no one exactly at three hours, but some of them, so this was, let's see, the student who was at two hours, you go to three hours, there's no one exactly there, but there's gonna be students who got better than what would be expected and students who might get a little bit worse. Notice, there's points above the trend line and there's points below the trend line. So this exactly, you can't say it's guaranteed an hour more turns into 15%. Let's try this choice."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "For example, we could find this student here who studied exactly two hours, and if we look at the students who studied for three hours, well, there's no one exactly at three hours, but some of them, so this was, let's see, the student who was at two hours, you go to three hours, there's no one exactly there, but there's gonna be students who got better than what would be expected and students who might get a little bit worse. Notice, there's points above the trend line and there's points below the trend line. So this exactly, you can't say it's guaranteed an hour more turns into 15%. Let's try this choice. In general, studying for one extra hour was associated with a 15-point improvement in test score. That feels about right. In general, studying for 15 extra hours was associated with a one-point improvement in test score."}, {"video_title": "Interpreting a trend line   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Let's try this choice. In general, studying for one extra hour was associated with a 15-point improvement in test score. That feels about right. In general, studying for 15 extra hours was associated with a one-point improvement in test score. Now, that would get the slope the other way around, so that's definitely not the case. Let's check our answer. We got it right."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So for example, this one over here in the top left, it's made out of chocolate on the outside, but it doesn't have coconut on the inside. While this one right over here does, is chocolate on the outside, and has coconut on the inside. While this one, whoops, I didn't want to do that. While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut?"}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut?"}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate?"}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two?"}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three."}, {"video_title": "Two-way frequency tables and Venn diagrams   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. What's this three? This is the total amount no chocolate. That's this one plus two."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So I took some screen captures from the Khan Academy exercise on correlation coefficient intuition, and they've given us some correlation coefficients, and we need to match them to the various scatter plots. On that exercise, there's a little interface where we can drag these around in a table to match them to the different scatter plots. And the point isn't to figure out how exactly to calculate these. We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like?"}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that?"}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well."}, {"video_title": "Example Correlation coefficient intuition   Mathematics I   High School Math   Khan Academy.mp3", "Sentence": "And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well. So I would say scatterplot B is a better fit. A linear model works better for scatterplot B than it works for scatterplot D. So I would give the higher r to scatterplot B, and the lower r, r equals 0.65 to scatterplot D. r is equal to 0.65. And once again, that's because with a linear model, it looks like there's a trend, but there's several data points that really, more data points are way off the line in scatterplot D than in the case of scatterplot B."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "It says, Evie read an article that said 6% of teenagers were vegetarians, but she thinks it's higher for students at her school. To test her theory, Evie took a random sample of 25 students at her school, and 20% of them were vegetarians. So just from this first paragraph, some interesting things are being said. It's saying that the true population proportion, if we believe this article, of teenagers that are vegetarian, we could say that is 6%. Now, for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian, so this is at her school, that the true proportion, the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis. And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "It's saying that the true population proportion, if we believe this article, of teenagers that are vegetarian, we could say that is 6%. Now, for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian, so this is at her school, that the true proportion, the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis. And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%. And so to see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%. And so to see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion. And then you figure out what is the probability of getting a sample proportion this high or greater. And if it's lower than a threshold, then you will reject your null hypothesis. And that probability, we call the p-value."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "She took a random sample of 25 students, and you calculate the sample proportion. And then you figure out what is the probability of getting a sample proportion this high or greater. And if it's lower than a threshold, then you will reject your null hypothesis. And that probability, we call the p-value. The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was 6% vegetarians, but you took a sample of 25 students where you got 20%, what is the probability of getting 20% or greater for a sample of 25? Now, there's many ways to approach it, but it looks like she is using a simulation."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "And that probability, we call the p-value. The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was 6% vegetarians, but you took a sample of 25 students where you got 20%, what is the probability of getting 20% or greater for a sample of 25? Now, there's many ways to approach it, but it looks like she is using a simulation. To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation. She simulated 40 samples of n equals 25 students from a large population, where 6% of the students were vegetarian. She recorded the proportion of vegetarians in each sample."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "Now, there's many ways to approach it, but it looks like she is using a simulation. To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation. She simulated 40 samples of n equals 25 students from a large population, where 6% of the students were vegetarian. She recorded the proportion of vegetarians in each sample. Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "She recorded the proportion of vegetarians in each sample. Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. And then they ask us, based on these simulated results, what is the approximate P value of the test? And they say the sample result, the sample proportion here was 20%. We saw that right over here."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. And then they ask us, based on these simulated results, what is the approximate P value of the test? And they say the sample result, the sample proportion here was 20%. We saw that right over here. Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here. And how many of these samples do we get a sample proportion that is greater than or equal to 20%? Well, you could see this is 20% right over here, 20 hundredths."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "We saw that right over here. Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here. And how many of these samples do we get a sample proportion that is greater than or equal to 20%? Well, you could see this is 20% right over here, 20 hundredths. And so you see we have three right over here that meet this constraint. And so that is three out of 40. So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "Well, you could see this is 20% right over here, 20 hundredths. And so you see we have three right over here that meet this constraint. And so that is three out of 40. So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians. And so three 40ths is what? Let's see."}, {"video_title": "Estimating a P-value from a simulation   AP Statistics   Khan Academy.mp3", "Sentence": "So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians. And so three 40ths is what? Let's see. If I multiply both the numerator and the denominator by two and a half, this is approximately equal to, I say two and a half because to go from 40 to 100, and then two and a half times three would be 7.5. So I would say this is approximately 7.5%. And this is actually a multiple choice question."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "She conducted a poll by calling 100 people whose names were randomly sampled from the phone book. Note that mobile phones and unlisted numbers are not in phone books. The senator's office called those numbers until they got a response from all 100 people chosen. The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents?"}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias?"}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here."}, {"video_title": "Example of under coverage introducing bias   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here. She didn't ask 100 people to volunteer. Her team went out and got them from the phone book. So this was definitely a case of undercoverage."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means?"}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance."}, {"video_title": "Statistical significance of experiment   Probability and Statistics   Khan Academy (2).mp3", "Sentence": "It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant. In most studies, in most experiments, the threshold that they think about is the probability of something statistically significant, if the probability of that happening by chance is less than 5%, so this is less than 1%."}, {"video_title": "Interpreting y-intercept in regression model   AP Statistics   Khan Academy.mp3", "Sentence": "She then created the following scatter plot and trend line. So this is salary in millions of dollars and the winning percentage. And so here we have a coach who made over $4 million and it looks like they won over 80% of their games. Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%."}, {"video_title": "Interpreting y-intercept in regression model   AP Statistics   Khan Academy.mp3", "Sentence": "Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that."}, {"video_title": "Interpreting y-intercept in regression model   AP Statistics   Khan Academy.mp3", "Sentence": "There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million."}, {"video_title": "Interpreting y-intercept in regression model   AP Statistics   Khan Academy.mp3", "Sentence": "But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%."}, {"video_title": "Interpreting y-intercept in regression model   AP Statistics   Khan Academy.mp3", "Sentence": "No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that wasn't the case either. The model indicates that teams with coaches who had a salary of zero million dollars will average a winning percentage of approximately 39%. Yeah, this is the closest statement to what we just said, that if you believe that model, and that's a big if, if you believe this model, then this model says someone making zero dollars will get 39%, and this is frankly why you have to be skeptical of models."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the last few videos, we saw that if we had n points, each of them have x and y coordinates. So let me draw n of those points. So let's call this point 1. It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right over here. You would take y1 minus the y value at this point in the line. Well, that point in the line is essentially the y value you get when you substitute x1 into this equation."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right over here. You would take y1 minus the y value at this point in the line. Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, in fact we've proved the formula of how to find these m's and b's. We can find this line, and if we wanted to say, well, you know, how much error is there, we can then calculate it because we now know the m's and the b's. So we can calculate it for a certain set of data."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, in fact we've proved the formula of how to find these m's and b's. We can find this line, and if we wanted to say, well, you know, how much error is there, we can then calculate it because we now know the m's and the b's. So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variation in y is described by the variation in x? And so let's think about this."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variation in y is described by the variation in x? And so let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? So let's think about that. First, let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? So let's think about that. First, let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y? So let's just figure out what the total variation in y is. The total variation, and it's really just a tool for measuring, total variation in y, well, when we think about variation, and this is even true when we talk about variance, which was the mean variation in y, is we think about the square distance from some central tendency, and the best central measure we can have of y is the arithmetic mean. So we could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "How much of the, we could even say total variation, how much of the total variation in y? So let's just figure out what the total variation in y is. The total variation, and it's really just a tool for measuring, total variation in y, well, when we think about variation, and this is even true when we talk about variance, which was the mean variation in y, is we think about the square distance from some central tendency, and the best central measure we can have of y is the arithmetic mean. So we could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values, find their mean, it'll be some value, maybe it's right over here someplace, maybe that is the mean value of all the y's, and so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values, find their mean, it'll be some value, maybe it's right over here someplace, maybe that is the mean value of all the y's, and so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is going to be this distance, just right up to the line. The nth one is going to be the distance from there all the way to the line right over there, and then there are these other points in between. This is the total variation y."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is going to be this distance, just right up to the line. The nth one is going to be the distance from there all the way to the line right over there, and then there are these other points in between. This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distance. So what we want to do is how much of this, how much of the total variation y is described by the variation in x?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distance. So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This right over here tells us what percentage of variation, of the total variation, is not described by the variation in x, by the variation in x, or by the line, or by the regression line. Regression, by the regression line."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This right over here tells us what percentage of variation, of the total variation, is not described by the variation in x, by the variation in x, or by the line, or by the regression line. Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is described, you can either view it as described by the line, or by the variation in x. Is described by the variation, by the variation in x. And this number right here, this is called the coefficient of determination."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is described, you can either view it as described by the line, or by the variation in x. Is described by the variation, by the variation in x. And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that the line, this line over here, it tells us that the line is a really good fit. So if the, let me write it over here."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that the line, this line over here, it tells us that the line is a really good fit. So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here?"}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if this number over here is huge, then that means there's a lot of error between the data points and the line. And so if this number is huge, then this number over here is going to be huge."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if this number over here is huge, then that means there's a lot of error between the data points and the line. And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close to one. And if this whole thing is close to one, the whole coefficient of determination, the whole r squared is going to be close to zero, which makes sense."}, {"video_title": "R-squared or coefficient of determination   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close to one. And if this whole thing is close to one, the whole coefficient of determination, the whole r squared is going to be close to zero, which makes sense. R squared will be close to zero, which makes sense. That tells us that very little of the total variation in y is described by the variation in x, or described by the line. Well, anyway, everything I've been dealing with so far has been a little bit in the abstract."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But he only has enough money to buy at most four packs. Suppose that each pack has probability 0.2 of containing the card Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four."}, {"video_title": "Probability with discrete random variable example   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four. Even if on the fourth, he doesn't get the card he wants, he still has to stop there. So there's a high probability that that's where we end up. There is a little less than 50% chance that he gets the card he's looking for before that point."}, {"video_title": "Matched pairs experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "We have a whole video on it on Khan Academy, but it is an average measure of your blood sugar over roughly a three-month period. So that's the explanatory variable, whether or not you're taking the pill, and the response variable is, well, what does it do to your hemoglobin A1c? We constructed a somewhat classic experiment where we had a control group and a treatment group, and we randomly assigned folks into either the control or the treatment group. And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design."}, {"video_title": "Matched pairs experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated."}, {"video_title": "Matched pairs experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill."}, {"video_title": "Matched pairs experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design."}, {"video_title": "Matched pairs experiment design   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design. Matched, matched pairs, pairs design of an experiment, and it's essentially, instead of going through all of this trouble saying, oh boy, maybe we do block design, all this random sampling, instead, you randomly put people first into either the control or the treatment group, and then we do another round, you measure, and then you do another round where you switch, where the people who are in the treatment go into the control, and the people who are in the control go into the treatment. So we could even extend from what we have here, we could imagine a world where the first three months, we have the 50 people in this treatment group, we have another 50 people in this control group that are taking the placebo, we see what happens to the A1Cs, and then we switch, where this group over here, then, and they don't know, they don't know, first of all, ideally, it's a blind experiment, so they don't even know they were in the treatment groups, and hopefully the pills look identical, so now, that same group, for the next three months, is now going to be the control group, and so they got the medicine for the first three months, and we saw what happens to their A1C, and now they're gonna get the placebo, they're going to get the placebo for the second three months, and then we are going to see what happens to their A1C, and likewise, the other group is going to be switched around. The thing that, the folks that used to be getting the placebo could now get, could now get the treatment."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Let's say we define the random variable capital X as the number of heads we get after three flips of a fair coin. So given that definition of a random variable, what we're going to try to do in this video is think about the probability distribution. So what's the probability of the different possible outcomes or the different possible values for this random variable? And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable?"}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0?"}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head?"}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2?"}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar. Just like that."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "That's right over there. That's 3 eighths. So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete."}, {"video_title": "Constructing a probability distribution for random variable   Khan Academy.mp3", "Sentence": "It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete. The random variable only takes on discrete values. It can't take on any value in between these things. So discrete probability distribution for our random variable X."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you don't want to cut open every watermelon in your watermelon farm or patch or whatever it might be called, because you want to sell most of them. You just want to sample a few watermelons and then take samples of those watermelons to figure out how dense the seeds are, and hope that you can calculate statistics on those samples that are decent estimates of the parameters for the population. So let's start doing that. So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be four minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared, All of that divided by not by 8. Remember, we want the unbiased sample variance."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be four minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared, All of that divided by not by 8. Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. And I could even denote it by this to make it clear that we're dividing by lowercase n minus 1 is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. And I could even denote it by this to make it clear that we're dividing by lowercase n minus 1 is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared. Negative 4 squared is 16."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared. Negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance, so from i equals 1 to n of our unbiased sample variance, so we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance, so from i equals 1 to n of our unbiased sample variance, so we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation?"}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy."}, {"video_title": "Sample standard deviation and bias   Probability and Statistics   Khan Academy.mp3", "Sentence": "Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy. It's actually dependent on how that population is actually distributed. So in statistics, we just define the sample standard deviation. And the one that we typically use is based on the square root of the unbiased sample variance."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean emission of all engines of a new design needs to be below 20 parts per million if the design is to meet new emission requirements. 10 engines are manufactured for testing purposes, and the emission level of each is determined. The emission data is, and they give us 10 data points for the 10 test engines, and I went ahead and calculated the mean of these data points. The sample mean is 17.17, and the standard deviation of these 10 data points right here is 2.98, the sample standard deviation. Does the data supply sufficient evidence to conclude that this type of engine meets the new standard? Assume we are willing to risk a type 1 error with a probability of 0.01, and we'll touch on this in a second. Before we do that, let's just define what our null hypothesis and our alternative hypothesis are going to be."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The sample mean is 17.17, and the standard deviation of these 10 data points right here is 2.98, the sample standard deviation. Does the data supply sufficient evidence to conclude that this type of engine meets the new standard? Assume we are willing to risk a type 1 error with a probability of 0.01, and we'll touch on this in a second. Before we do that, let's just define what our null hypothesis and our alternative hypothesis are going to be. Our null hypothesis can be that we don't meet the standards, that we just barely don't meet the standards, that the mean of our new engines is exactly 20 parts per million, and you essentially want the best possible value where we still don't meet, or the lowest possible value where we still don't meet the standard. Then our alternative hypothesis is no, we do meet the standard, that the true mean for our new engines is below 20 parts per million. To see if the data that we have is sufficient, what we're going to do is assume that this is true."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Before we do that, let's just define what our null hypothesis and our alternative hypothesis are going to be. Our null hypothesis can be that we don't meet the standards, that we just barely don't meet the standards, that the mean of our new engines is exactly 20 parts per million, and you essentially want the best possible value where we still don't meet, or the lowest possible value where we still don't meet the standard. Then our alternative hypothesis is no, we do meet the standard, that the true mean for our new engines is below 20 parts per million. To see if the data that we have is sufficient, what we're going to do is assume that this is true. Given that this is true, if we assume this is true, and the probability of this occurring, and the probability of getting a sample mean of that, is less than 1%, then we will reject the null hypothesis. We are going to reject our null hypothesis if the probability of getting a sample mean of 17.17, given the null hypothesis is true, is less than 1%. Notice, if we do it this way, there will be less than a 1% chance that we are making a type 1 error."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "To see if the data that we have is sufficient, what we're going to do is assume that this is true. Given that this is true, if we assume this is true, and the probability of this occurring, and the probability of getting a sample mean of that, is less than 1%, then we will reject the null hypothesis. We are going to reject our null hypothesis if the probability of getting a sample mean of 17.17, given the null hypothesis is true, is less than 1%. Notice, if we do it this way, there will be less than a 1% chance that we are making a type 1 error. A type 1 error is that we're rejecting it even though it's true. Here, there's only a 1% chance, or less than a 1% chance, that we will reject it if it is true. The next thing we have to think about is what type of distribution we should think about."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Notice, if we do it this way, there will be less than a 1% chance that we are making a type 1 error. A type 1 error is that we're rejecting it even though it's true. Here, there's only a 1% chance, or less than a 1% chance, that we will reject it if it is true. The next thing we have to think about is what type of distribution we should think about. The first thing that rings in my brain is we only have 10 samples here. We have a small sample size right over here. We're going to be dealing with a t distribution and a t statistic."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The next thing we have to think about is what type of distribution we should think about. The first thing that rings in my brain is we only have 10 samples here. We have a small sample size right over here. We're going to be dealing with a t distribution and a t statistic. With that said, let's think of it this way. We can come up with a t statistic that is based on these statistics right over here. The t statistic is going to be 17.17, our sample mean, minus the assumed population mean, minus 20 parts per million, over our sample standard deviation, 2.98."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to be dealing with a t distribution and a t statistic. With that said, let's think of it this way. We can come up with a t statistic that is based on these statistics right over here. The t statistic is going to be 17.17, our sample mean, minus the assumed population mean, minus 20 parts per million, over our sample standard deviation, 2.98. This is really the definition of the t statistic. Hopefully, we see now that this really comes from a z score. The t distribution is kind of an engineered version of the normal distribution using t statistics."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The t statistic is going to be 17.17, our sample mean, minus the assumed population mean, minus 20 parts per million, over our sample standard deviation, 2.98. This is really the definition of the t statistic. Hopefully, we see now that this really comes from a z score. The t distribution is kind of an engineered version of the normal distribution using t statistics. 2.98 divided by the square root of our sample size. We have 10 samples, so it's divided by the square root of 10. This value right here, let me get the calculator out just to get a value in place there."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The t distribution is kind of an engineered version of the normal distribution using t statistics. 2.98 divided by the square root of our sample size. We have 10 samples, so it's divided by the square root of 10. This value right here, let me get the calculator out just to get a value in place there. This is going to be 17.17 minus 20, close parentheses, divided by 2.98 divided by the square root of 10, and then close parentheses. It is almost exactly negative 3. Our t statistic is almost exactly negative 3."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This value right here, let me get the calculator out just to get a value in place there. This is going to be 17.17 minus 20, close parentheses, divided by 2.98 divided by the square root of 10, and then close parentheses. It is almost exactly negative 3. Our t statistic is almost exactly negative 3. Negative 3.00. What we need to figure out, because t statistics have a t distribution, what we need to figure out is the probability of getting this t statistic, or a value of t equal to this or less than this, is that less than 1%. The way we can think about it is we have a t distribution."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our t statistic is almost exactly negative 3. Negative 3.00. What we need to figure out, because t statistics have a t distribution, what we need to figure out is the probability of getting this t statistic, or a value of t equal to this or less than this, is that less than 1%. The way we can think about it is we have a t distribution. Let's say we have a normalized t distribution. The distribution of all the t statistics would be a normalized t distribution. This is the mean of the t distribution."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The way we can think about it is we have a t distribution. Let's say we have a normalized t distribution. The distribution of all the t statistics would be a normalized t distribution. This is the mean of the t distribution. There's going to be some threshold t value right here. This is our threshold t value. This is some threshold t value right over here."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the mean of the t distribution. There's going to be some threshold t value right here. This is our threshold t value. This is some threshold t value right over here. We want a threshold t value such that any t value less than that, or the probability of getting a t value less than that is 1%. That entire area in yellow is 1%. We need to figure out a threshold t value there."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is some threshold t value right over here. We want a threshold t value such that any t value less than that, or the probability of getting a t value less than that is 1%. That entire area in yellow is 1%. We need to figure out a threshold t value there. This is for a t distribution that has n equal to 10, or 10 minus 1 equals 9 degrees of freedom. What is that threshold value over there? Notice that this is a one-sided distribution."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We need to figure out a threshold t value there. This is for a t distribution that has n equal to 10, or 10 minus 1 equals 9 degrees of freedom. What is that threshold value over there? Notice that this is a one-sided distribution. We care about this is 1%, and then all of this stuff over here is going to be 99%. Just the way most t tables are set up, they don't set up a negative t value that is oriented like this. They'll just give you a positive t value that's oriented the other way."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Notice that this is a one-sided distribution. We care about this is 1%, and then all of this stuff over here is going to be 99%. Just the way most t tables are set up, they don't set up a negative t value that is oriented like this. They'll just give you a positive t value that's oriented the other way. The way t tables, and I have one that we're going to use in a second right over here, the way t tables are set up is you have your distribution like this, and they will just give a positive t value. They will give a positive t value over here, some threshold value, where the probability of getting a t value above that is going to be 1%, and the probability of getting a t value below that is going to be 99%. You can see that we know t distributions are symmetric around their mean, so whatever value this is, if this number is 2, then this value is just going to be negative 2."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "They'll just give you a positive t value that's oriented the other way. The way t tables, and I have one that we're going to use in a second right over here, the way t tables are set up is you have your distribution like this, and they will just give a positive t value. They will give a positive t value over here, some threshold value, where the probability of getting a t value above that is going to be 1%, and the probability of getting a t value below that is going to be 99%. You can see that we know t distributions are symmetric around their mean, so whatever value this is, if this number is 2, then this value is just going to be negative 2. We just have to keep that in mind, but the t tables actually help us figure out this value. Let's figure out a t value where the probability of getting a t value below that is 99%. Once again, this is going to be a one-sided situation."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can see that we know t distributions are symmetric around their mean, so whatever value this is, if this number is 2, then this value is just going to be negative 2. We just have to keep that in mind, but the t tables actually help us figure out this value. Let's figure out a t value where the probability of getting a t value below that is 99%. Once again, this is going to be a one-sided situation. Let's look at that over here. One-sided, this is just straight from Wikipedia, we want the cumulative distribution below that t value to be 99%. We have it right over here, 99%."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, this is going to be a one-sided situation. Let's look at that over here. One-sided, this is just straight from Wikipedia, we want the cumulative distribution below that t value to be 99%. We have it right over here, 99%. We have 9 degrees of freedom. We have 10 data points, 10 minus 1 is 9. 9 degrees of freedom."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have it right over here, 99%. We have 9 degrees of freedom. We have 10 data points, 10 minus 1 is 9. 9 degrees of freedom. Our threshold t value here is 2.821, so our threshold t value in the case that we care about, just flip this over, it's completely symmetric, is negative 2.821. What this tells us is the probability of getting a t value less than negative 2.821 is going to be 1%. Now, we got a value that's a good bit less than that."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "9 degrees of freedom. Our threshold t value here is 2.821, so our threshold t value in the case that we care about, just flip this over, it's completely symmetric, is negative 2.821. What this tells us is the probability of getting a t value less than negative 2.821 is going to be 1%. Now, we got a value that's a good bit less than that. We got a t value of negative 3. We got a t value right here, our t statistic of negative 3 right over here. That definitely goes into our, I guess you could call it our area of rejection."}, {"video_title": "Small sample hypothesis test   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, we got a value that's a good bit less than that. We got a t value of negative 3. We got a t value right here, our t statistic of negative 3 right over here. That definitely goes into our, I guess you could call it our area of rejection. This is even less probable than the 1%. We could even figure it out, that the area over here, the probability of getting a t statistic less than negative 3 is even less than, it's a subset of this yellow area right over here. Because the probability of getting the t statistic that we actually got is less than 1%, we can safely reject the null hypothesis and feel pretty good about our alternate hypothesis right over here, that we do meet the emission standards."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker. You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just write J."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying?"}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts. So we counted the jack of hearts twice, so we have to subtract one out of that."}, {"video_title": "Probability with playing cards and Venn diagrams   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts. So we counted the jack of hearts twice, so we have to subtract one out of that. So it's gonna be four plus 13 minus one, or this is going to be 1650 seconds, and both of these things are divisible by four, so this is going to be the same thing as, divide 16 by four, you get four. 52 divided by four is 13. So there's a 413th chance that you get a jack or a hearts."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "Which conditions for constructing this confidence interval did Ali's sample meet? So pause this video, and you can select more than one of these. All right, now let's work through this together. So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval. But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval. But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So n is equal to 30."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample? So that would be the random condition."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample? So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that?"}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal. And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal. And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150. So his sample size was 30 out of 150, which is the same thing as 1 5th of the population, which is the same thing as 20%."}, {"video_title": "Conditions for confidence intervals worked examples   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150. So his sample size was 30 out of 150, which is the same thing as 1 5th of the population, which is the same thing as 20%. And since this is greater than 10%, we are violating the independence condition. We could have met the independence condition if he was sampling with replacement, which it doesn't seem like he is, or if this thing right over here was less than 10%. But we're not meeting that, so we cannot feel good about that constraint."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "And so I'm randomly sampling a bunch of people, measuring their height, measuring their weight, and then for each person, I'm plotting a point that represents their height and weight combination. So for example, let's say I measure someone who is 60 inches tall, that would be five feet tall, and they weigh 100 pounds. And so I'd go to 60 inches and then 100 pounds right over there. So that point right over there is the point 60 comma 100. One way to think about it, height we could say is being measured on our x-axis or plotted along our x-axis, and then weight along our y-axis. And so this point from this person is the point 60 comma 100 representing 60 inches, 100 pounds. And so so far I've done it for one, two, three, four, five, six, seven, eight, nine people."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So that point right over there is the point 60 comma 100. One way to think about it, height we could say is being measured on our x-axis or plotted along our x-axis, and then weight along our y-axis. And so this point from this person is the point 60 comma 100 representing 60 inches, 100 pounds. And so so far I've done it for one, two, three, four, five, six, seven, eight, nine people. And I could keep going, but even with this, I could say, well look, it looks like there's a roughly linear relationship here. It looks like it's positive, that generally speaking, as height increases, so does weight. Maybe I could try to put a line that can approximate this trend."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "And so so far I've done it for one, two, three, four, five, six, seven, eight, nine people. And I could keep going, but even with this, I could say, well look, it looks like there's a roughly linear relationship here. It looks like it's positive, that generally speaking, as height increases, so does weight. Maybe I could try to put a line that can approximate this trend. So let me try to do that. So this is my line tool. I could think about a bunch of lines."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "Maybe I could try to put a line that can approximate this trend. So let me try to do that. So this is my line tool. I could think about a bunch of lines. Something like this seems like it would be, you'd be, most of the data is below the line, so that seems like it's not right. I could do something like, I could do something like this, but that doesn't seem like a good fit. Most of the data seems to be above the line."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "I could think about a bunch of lines. Something like this seems like it would be, you'd be, most of the data is below the line, so that seems like it's not right. I could do something like, I could do something like this, but that doesn't seem like a good fit. Most of the data seems to be above the line. And so, and once again, I'm just eyeballing it here. In the future you will learn better methods of finding a better fit, but this, something like this, and I'm just eyeballing it, looks about right. So that line, you could view this as a regression line."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "Most of the data seems to be above the line. And so, and once again, I'm just eyeballing it here. In the future you will learn better methods of finding a better fit, but this, something like this, and I'm just eyeballing it, looks about right. So that line, you could view this as a regression line. We could view this as y equals mx plus b, where we would have to figure out the slope and the y-intercept, and we could figure it out based on what I just drew, or we could even think of this as weight. Weight is equal to our slope times height, times height plus whatever our y-intercept is, or you could think of it, if you think of the vertical axis as the weight axis, you could think of it as your weight intercept. But either way, this is the model that I'm just, through eyeballing, this is my regression line, something that I'm trying to fit to these points."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So that line, you could view this as a regression line. We could view this as y equals mx plus b, where we would have to figure out the slope and the y-intercept, and we could figure it out based on what I just drew, or we could even think of this as weight. Weight is equal to our slope times height, times height plus whatever our y-intercept is, or you could think of it, if you think of the vertical axis as the weight axis, you could think of it as your weight intercept. But either way, this is the model that I'm just, through eyeballing, this is my regression line, something that I'm trying to fit to these points. But clearly it can't go through, one line won't be able to go through all of these points. There is going to be, for each point, some difference, or not for all of them, but for many of them, some difference between the actual and what would have been predicted by the line. And that idea, the difference between the actual four-point and what would have been predicted given, say, the height, that is called a residual."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "But either way, this is the model that I'm just, through eyeballing, this is my regression line, something that I'm trying to fit to these points. But clearly it can't go through, one line won't be able to go through all of these points. There is going to be, for each point, some difference, or not for all of them, but for many of them, some difference between the actual and what would have been predicted by the line. And that idea, the difference between the actual four-point and what would have been predicted given, say, the height, that is called a residual. Let me write that down. The residual for each of these data points. And so, for example, if I call this right here, if I call that 0.1, the residual for 0.1 is going to be, well, for our variable, for our height variable, 60 inches, the actual here is 100 pounds, and from that we would subtract what would be predicted."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "And that idea, the difference between the actual four-point and what would have been predicted given, say, the height, that is called a residual. Let me write that down. The residual for each of these data points. And so, for example, if I call this right here, if I call that 0.1, the residual for 0.1 is going to be, well, for our variable, for our height variable, 60 inches, the actual here is 100 pounds, and from that we would subtract what would be predicted. And so what would be predicted is right over here. I could just substitute 60 into this equation, so it would be m times 60 plus b. So I could write it as m, maybe let me write it this way, 60m plus b."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "And so, for example, if I call this right here, if I call that 0.1, the residual for 0.1 is going to be, well, for our variable, for our height variable, 60 inches, the actual here is 100 pounds, and from that we would subtract what would be predicted. And so what would be predicted is right over here. I could just substitute 60 into this equation, so it would be m times 60 plus b. So I could write it as m, maybe let me write it this way, 60m plus b. Once again, I would just take the 60 pounds and put it into my model here and say, well, what weight would that have predicted? And I could even, just for the sake of having a number here, I can look, I can, let me get my line tool out and try to get a straight line from that point. So from this point, let me get a straight line."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So I could write it as m, maybe let me write it this way, 60m plus b. Once again, I would just take the 60 pounds and put it into my model here and say, well, what weight would that have predicted? And I could even, just for the sake of having a number here, I can look, I can, let me get my line tool out and try to get a straight line from that point. So from this point, let me get a straight line. So that doesn't look quite straight. Okay, a little bit, okay. So if I, it looks like it's about 150 pounds."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So from this point, let me get a straight line. So that doesn't look quite straight. Okay, a little bit, okay. So if I, it looks like it's about 150 pounds. So my model would have predicted 150 pounds. So the residual here is going to be equal to negative 50. And so a negative residual is when your actual is below your predicted."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So if I, it looks like it's about 150 pounds. So my model would have predicted 150 pounds. So the residual here is going to be equal to negative 50. And so a negative residual is when your actual is below your predicted. So this right over here, this is r one. It is a negative residual. If you had, if you tried to find, let's say this residual right over here for this point, this r two, this would be a positive residual because the actual is larger than what would have actually been predicted."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "And so a negative residual is when your actual is below your predicted. So this right over here, this is r one. It is a negative residual. If you had, if you tried to find, let's say this residual right over here for this point, this r two, this would be a positive residual because the actual is larger than what would have actually been predicted. And so a residual is good for seeing, well, how good does your line, does your regression, does your model fit a given data point or how does a given data point compare to that? But what you probably wanna do is think about some combination of all the residuals and try to minimize it. Now you might say, well, why don't I just add up all the residuals and try to minimize that?"}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "If you had, if you tried to find, let's say this residual right over here for this point, this r two, this would be a positive residual because the actual is larger than what would have actually been predicted. And so a residual is good for seeing, well, how good does your line, does your regression, does your model fit a given data point or how does a given data point compare to that? But what you probably wanna do is think about some combination of all the residuals and try to minimize it. Now you might say, well, why don't I just add up all the residuals and try to minimize that? But that gets tricky because some are positive and some are negative. And so a big negative residual, negative residual could counterbalance a big positive residual and it would look, they would add up to zero and then it would look like there's no residual. So you could just add up the absolute values."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "Now you might say, well, why don't I just add up all the residuals and try to minimize that? But that gets tricky because some are positive and some are negative. And so a big negative residual, negative residual could counterbalance a big positive residual and it would look, they would add up to zero and then it would look like there's no residual. So you could just add up the absolute values. So you could say, well, let me just take the sum of all of the residual, of the absolute value of all of the residuals. And then let me change m and b for my line to minimize this. And that would be a technique of trying to create a regression line."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So you could just add up the absolute values. So you could say, well, let me just take the sum of all of the residual, of the absolute value of all of the residuals. And then let me change m and b for my line to minimize this. And that would be a technique of trying to create a regression line. But another way to do it, and this is actually the most typical way that you will see in statistics, is that people take the sum of the squares of the residuals, the sum of the squares. And when you square something, whether it's negative or positive, it's going to be a positive. So it takes care of that issue of negatives and positives canceling out with each other."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "And that would be a technique of trying to create a regression line. But another way to do it, and this is actually the most typical way that you will see in statistics, is that people take the sum of the squares of the residuals, the sum of the squares. And when you square something, whether it's negative or positive, it's going to be a positive. So it takes care of that issue of negatives and positives canceling out with each other. And when you square a number, things with large residuals are gonna become even larger, relatively speaking. If you square a large, if you think about it this way, let me put regular numbers, one, two, three, four, these are all one apart from each other. But if I were to square them, one, four, nine, 16, they get further and further apart."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "So it takes care of that issue of negatives and positives canceling out with each other. And when you square a number, things with large residuals are gonna become even larger, relatively speaking. If you square a large, if you think about it this way, let me put regular numbers, one, two, three, four, these are all one apart from each other. But if I were to square them, one, four, nine, 16, they get further and further apart. And so something, the larger the residual is, when you square it, when the sum of squares is gonna represent a bigger proportion of the sum. And so what we'll see in future videos is that there's a technique called least squares regression, least squares regression, where you can find an M and a B for a given set of data so it minimizes the sum of the squares of the residual. And that's valuable."}, {"video_title": "Introduction to residuals and least squares regression.mp3", "Sentence": "But if I were to square them, one, four, nine, 16, they get further and further apart. And so something, the larger the residual is, when you square it, when the sum of squares is gonna represent a bigger proportion of the sum. And so what we'll see in future videos is that there's a technique called least squares regression, least squares regression, where you can find an M and a B for a given set of data so it minimizes the sum of the squares of the residual. And that's valuable. And the reason why this is used most is it really tries to take into account things that are significant outliers, things that sit from pretty far away from the model. Something like this is going to really, with a least squares regression, is going to try to be minimized, or it's going to be weighted a little bit heavier because when you square it, it becomes even a bigger factor in this. But this is just a conceptual introduction."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Using the store's total selection, she documented the price of each movie title and how many years it has been since it was featured in movie theaters. She plotted the points below. So let's see what's going on below here. So let's see, it looks like there's two curves that she tries to fit, and I'm assuming we're gonna read about it in a second, but these blue points are the data points. So, for example, this data point right over here shows a movie that the title costs $6 and it has been released for almost two years, a little under two years. This data point right over here, this is a movie that has been released for, looks like, almost four years, looks like maybe three and three quarters years, and they're selling that, looks like, for a dollar or even a little bit less than a dollar. So those are her data points."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's see, it looks like there's two curves that she tries to fit, and I'm assuming we're gonna read about it in a second, but these blue points are the data points. So, for example, this data point right over here shows a movie that the title costs $6 and it has been released for almost two years, a little under two years. This data point right over here, this is a movie that has been released for, looks like, almost four years, looks like maybe three and three quarters years, and they're selling that, looks like, for a dollar or even a little bit less than a dollar. So those are her data points. So once again, she documented the price of each movie title as a function of how long it's been, how many years it's been since it was featured in movie theaters. She is looking for a function that models her data. Since the trend of the data is decreasing and convex, and you see it here, it's decreasing, it's definitely decreasing, and convex, it's opening upwards."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So those are her data points. So once again, she documented the price of each movie title as a function of how long it's been, how many years it's been since it was featured in movie theaters. She is looking for a function that models her data. Since the trend of the data is decreasing and convex, and you see it here, it's decreasing, it's definitely decreasing, and convex, it's opening upwards. If you imagine a curve, it looks like it's opening upwards a little bit like that. So, decreasing and convex. She found a decreasing convex exponential model and a decreasing convex quadratic model."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Since the trend of the data is decreasing and convex, and you see it here, it's decreasing, it's definitely decreasing, and convex, it's opening upwards. If you imagine a curve, it looks like it's opening upwards a little bit like that. So, decreasing and convex. She found a decreasing convex exponential model and a decreasing convex quadratic model. So which of the following functions better fits the data? So, function A, this is an exponential, this is the one in green right over here, and function B, this one right over here is a quadratic, and you can see this one in purple. And so, which one of those better fits the data?"}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "She found a decreasing convex exponential model and a decreasing convex quadratic model. So which of the following functions better fits the data? So, function A, this is an exponential, this is the one in green right over here, and function B, this one right over here is a quadratic, and you can see this one in purple. And so, which one of those better fits the data? And so, if we look at what's going on here, the green function, the exponential one, most of the data points for any given duration, for how long the title's been out, it looks like it's consistently underestimating. That it's always, you know, it's the model's guess or what the model would say the price is, is always, at least for, except for only, essentially except for only one data point right over here, for all of these other data points, it's underestimating what the price would be. The purple model, or the purple function right over here, it is, it has more of a balance between overestimating right over here, and so it's overestimating by a little bit, and underestimating, and its underestimates are closer, and its overestimates are closer than this green model."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so, which one of those better fits the data? And so, if we look at what's going on here, the green function, the exponential one, most of the data points for any given duration, for how long the title's been out, it looks like it's consistently underestimating. That it's always, you know, it's the model's guess or what the model would say the price is, is always, at least for, except for only, essentially except for only one data point right over here, for all of these other data points, it's underestimating what the price would be. The purple model, or the purple function right over here, it is, it has more of a balance between overestimating right over here, and so it's overestimating by a little bit, and underestimating, and its underestimates are closer, and its overestimates are closer than this green model. So I would say that function B is definitely a better model. Use the function of best fits, so we're gonna say function B, to predict the price of a movie that was featured in theaters 5.5 years ago. Round your answer to the nearest cent."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "The purple model, or the purple function right over here, it is, it has more of a balance between overestimating right over here, and so it's overestimating by a little bit, and underestimating, and its underestimates are closer, and its overestimates are closer than this green model. So I would say that function B is definitely a better model. Use the function of best fits, so we're gonna say function B, to predict the price of a movie that was featured in theaters 5.5 years ago. Round your answer to the nearest cent. So 5.5 years ago, that's gonna be right over here, we're gonna go to function B, which is this purple one, so it's gonna be, you know, it's gonna be under a dollar, but we wanna get something to the nearest cent, so let's actually use the actual definition of the function. So this is price as a function of how long the movie has been released, where x is the, how long it's been released, and y is its price. So let's just, if x is 5.5, let's figure out what y is going to be."}, {"video_title": "Comparing models to fit data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Round your answer to the nearest cent. So 5.5 years ago, that's gonna be right over here, we're gonna go to function B, which is this purple one, so it's gonna be, you know, it's gonna be under a dollar, but we wanna get something to the nearest cent, so let's actually use the actual definition of the function. So this is price as a function of how long the movie has been released, where x is the, how long it's been released, and y is its price. So let's just, if x is 5.5, let's figure out what y is going to be. So it's going to be, so y is going to be equal to 0.5 times x squared, so x is 5.5, 5.5 squared, alright? So then we have minus five times x again, so minus five times 5.5, and then we have plus 13, and what does that get us? That gets us 62.5 cents."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "So let's say you put 50 of these magenta marbles. So one, two, three, four, five, six, seven. I'm not going to draw all of them, but you get the general idea. There are going to be 50 magenta marbles. 15 magenta marbles. And there's also going to be 50 blue marbles. So 50 blue marbles."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "There are going to be 50 magenta marbles. 15 magenta marbles. And there's also going to be 50 blue marbles. So 50 blue marbles. And what you do is you have these 100 marbles in there, half of them magenta, half of them blue, and before picking a marble out, and you're going to be blindfolded when you pick a marble out, you shake the bag really good so you can mix them up a little bit. And so if you were to say, well, theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta? I have to feel the need to write the word magenta in magenta."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "So 50 blue marbles. And what you do is you have these 100 marbles in there, half of them magenta, half of them blue, and before picking a marble out, and you're going to be blindfolded when you pick a marble out, you shake the bag really good so you can mix them up a little bit. And so if you were to say, well, theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta? I have to feel the need to write the word magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically, there's 100 equally likely possibilities. There's 100 marbles in the bag."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "I have to feel the need to write the word magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically, there's 100 equally likely possibilities. There's 100 marbles in the bag. And 50 of them involve picking a magenta. So 50 out of 100, and this is the same thing as a 1 half probability. So you could say, well, theoretically, there is a 1 half probability."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "There's 100 marbles in the bag. And 50 of them involve picking a magenta. So 50 out of 100, and this is the same thing as a 1 half probability. So you could say, well, theoretically, there is a 1 half probability. I just did the math. If you say these are 100 equally likely possibilities, 50 of them are picking magenta. Now let's say you actually start doing the experiment."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "So you could say, well, theoretically, there is a 1 half probability. I just did the math. If you say these are 100 equally likely possibilities, 50 of them are picking magenta. Now let's say you actually start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again. And so let's say that after, every time you put your hand in the bag and you take something out of the bag and you observe what it is, we're going to call that an experiment. So after 10 experiments, let's say that you have picked out 7 magenta and 3 blue."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "Now let's say you actually start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again. And so let's say that after, every time you put your hand in the bag and you take something out of the bag and you observe what it is, we're going to call that an experiment. So after 10 experiments, let's say that you have picked out 7 magenta and 3 blue. So does this cause, is this strange that after 10, out of the first 10 experiments, you haven't picked out exactly half of them being magenta. You've picked out 7 magenta, and then the other 3 were blue. Well, no, this is definitely a reasonable thing."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "So after 10 experiments, let's say that you have picked out 7 magenta and 3 blue. So does this cause, is this strange that after 10, out of the first 10 experiments, you haven't picked out exactly half of them being magenta. You've picked out 7 magenta, and then the other 3 were blue. Well, no, this is definitely a reasonable thing. If the true probability of picking out a magenta is 1 half, it's definitely possible that you could still pick out 7 magenta. That just happened to be what your fingers touched. And this isn't a lot of experiments."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "Well, no, this is definitely a reasonable thing. If the true probability of picking out a magenta is 1 half, it's definitely possible that you could still pick out 7 magenta. That just happened to be what your fingers touched. And this isn't a lot of experiments. It's completely reasonable that out of 10, yeah, you could have a, and later on in statistics, we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1 half. Instead of having 5 magenta, it's completely reasonable to have 7 magenta. So this really wouldn't cause me a lot of pause."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "And this isn't a lot of experiments. It's completely reasonable that out of 10, yeah, you could have a, and later on in statistics, we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1 half. Instead of having 5 magenta, it's completely reasonable to have 7 magenta. So this really wouldn't cause me a lot of pause. I still wouldn't say, hey, I still wouldn't question what I did here when I calculated this theoretical probability. But let's say, and let's say you have a lot of time on your hands, and let's say after 10,000 trials here, after 10,000 experiments, and remember, an experiment, you're sticking your hand in the bag without looking, with your fingers kind of feeling around, picks out a marble, and you observe the marble, and you record what you found. And so let's say after 10,000 experiments, you get 7,000 magenta."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "So this really wouldn't cause me a lot of pause. I still wouldn't say, hey, I still wouldn't question what I did here when I calculated this theoretical probability. But let's say, and let's say you have a lot of time on your hands, and let's say after 10,000 trials here, after 10,000 experiments, and remember, an experiment, you're sticking your hand in the bag without looking, with your fingers kind of feeling around, picks out a marble, and you observe the marble, and you record what you found. And so let's say after 10,000 experiments, you get 7,000 magenta. Actually, let me do slightly different numbers. Actually, let me make it even more extreme. Let's say you get 8,000 magenta, and you have 2,000 blue."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "And so let's say after 10,000 experiments, you get 7,000 magenta. Actually, let me do slightly different numbers. Actually, let me make it even more extreme. Let's say you get 8,000 magenta, and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different, and now you have a large number of trials right over here, not just 10. 10 is completely reasonable, that hey, I got 7 magenta and 3 blue instead of 5 and 5, but now you've done 10,000. 10,000 is definitely, you would have expected if this was the true probability, you would have expected that half of these would have been magenta, only 5,000 magenta and 5,000 blue, but you had 8,000 magenta."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "Let's say you get 8,000 magenta, and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different, and now you have a large number of trials right over here, not just 10. 10 is completely reasonable, that hey, I got 7 magenta and 3 blue instead of 5 and 5, but now you've done 10,000. 10,000 is definitely, you would have expected if this was the true probability, you would have expected that half of these would have been magenta, only 5,000 magenta and 5,000 blue, but you had 8,000 magenta. Now this is within the realm of possibility if the true probability of picking a magenta is 1 half, but it's very unlikely that you would have gotten this result with this many experiments, this many trials, if the true probability is 1 half. Here your experimental probability is showing look, out of 10,000 trials, let me write that here, experimental probability, experimental probability here is you had 10,000 trials or 10,000 experiments, I guess you could say, and in 8,000 of them, you got a magenta marble, and so this is going to be 80% or 8 tenths, so 80%, so there seems to be a difference here. The reason why I would take this more seriously is that you had a lot of trials here."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "10,000 is definitely, you would have expected if this was the true probability, you would have expected that half of these would have been magenta, only 5,000 magenta and 5,000 blue, but you had 8,000 magenta. Now this is within the realm of possibility if the true probability of picking a magenta is 1 half, but it's very unlikely that you would have gotten this result with this many experiments, this many trials, if the true probability is 1 half. Here your experimental probability is showing look, out of 10,000 trials, let me write that here, experimental probability, experimental probability here is you had 10,000 trials or 10,000 experiments, I guess you could say, and in 8,000 of them, you got a magenta marble, and so this is going to be 80% or 8 tenths, so 80%, so there seems to be a difference here. The reason why I would take this more seriously is that you had a lot of trials here. You did this 10,000 times. If the true probability was 1 half, it's very low likelihood that you would have gotten this many magenta. So when you think about it, you're like, what's going on here?"}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "The reason why I would take this more seriously is that you had a lot of trials here. You did this 10,000 times. If the true probability was 1 half, it's very low likelihood that you would have gotten this many magenta. So when you think about it, you're like, what's going on here? What are possible explanations for this? This I wouldn't have fretted about. After 10 experiments, not a big deal, but after 10,000, this would have caused me pause."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "So when you think about it, you're like, what's going on here? What are possible explanations for this? This I wouldn't have fretted about. After 10 experiments, not a big deal, but after 10,000, this would have caused me pause. Say, well, why would this happen? I mixed up the bag every time, and there are some different possibilities. Maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settle to the bottom, and you're more likely to pick a magenta marble."}, {"video_title": "Comparing theoretical to experimental probabilites   7th grade   Khan Academy.mp3", "Sentence": "After 10 experiments, not a big deal, but after 10,000, this would have caused me pause. Say, well, why would this happen? I mixed up the bag every time, and there are some different possibilities. Maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settle to the bottom, and you're more likely to pick a magenta marble. Maybe the blue marbles have a slightly different texture to them, in which case maybe they slip out of your hands or they're less likely to be gripped on, and so you're more likely to pick a magenta. I don't know the explanation. I don't know what's going on in that bag, but if I thought theoretically that the probability should be 1 half, because half of the marbles are magenta, but I'm seeing through my experiments that 80% of what I'm picking out, especially if I did 10,000 of it, if I did this 10,000 times, well, this is going to cause me some pause."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "What we're going to do in this video is start to compare distributions. So for example here we have two distributions that show the various temperatures different cities get during the month of January. This is the distribution for Portland, for example they get eight days between one and four degrees Celsius, they get 12 days between four and seven degrees Celsius, so forth and so on, and then this is the distribution for Minneapolis. Now when we make these comparisons, what we're going to focus on is the center of the distributions to compare that, and also the spread. Sometimes people will talk about the variability of the distributions, and so these are the things that we're going to compare. And in making the comparison, we're actually just going to try to eyeball it. We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Now when we make these comparisons, what we're going to focus on is the center of the distributions to compare that, and also the spread. Sometimes people will talk about the variability of the distributions, and so these are the things that we're going to compare. And in making the comparison, we're actually just going to try to eyeball it. We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability. In either of these cases, there are multiple measures in our statistical toolkit."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability. In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of gauge it by looking."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, however you wanna measure it, either mean or median. Now what about the spread or variability? Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, however you wanna measure it, either mean or median. Now what about the spread or variability? Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and similarly, what's contributing to this last column might be a bunch of things at 10.1 degrees, but at most, you have a 12-degree range right over here, while over here, it looks like you have, well, it looks like it's approaching a 27-degree range. So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here. Let's do another example, and we'll use a different representation for the data here."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and similarly, what's contributing to this last column might be a bunch of things at 10.1 degrees, but at most, you have a 12-degree range right over here, while over here, it looks like you have, well, it looks like it's approaching a 27-degree range. So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here. Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit between the lower four and the upper four, so the central tendency right over here is for the final round, is looks like it's around 57.1 seconds, while the, especially if we think about the median, while the central tendency for the semifinal round, let's see, one, two, three, four, five, six, seven, eight, looks like it is right about there, so this is about 57, more than 57.3 seconds, so the semifinal round seems to have a higher central tendency, which is a little bit counterintuitive. You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability? Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit between the lower four and the upper four, so the central tendency right over here is for the final round, is looks like it's around 57.1 seconds, while the, especially if we think about the median, while the central tendency for the semifinal round, let's see, one, two, three, four, five, six, seven, eight, looks like it is right about there, so this is about 57, more than 57.3 seconds, so the semifinal round seems to have a higher central tendency, which is a little bit counterintuitive. You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability? Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeballing it, it looks like it has a higher spread, and there's, of course, times where one distribution could have a higher range, but then it might have a lower standard deviation. For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution. A distribution like this might have a higher range, but lower standard deviation than a distribution like this."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeballing it, it looks like it has a higher spread, and there's, of course, times where one distribution could have a higher range, but then it might have a lower standard deviation. For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution. A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above it. In fact, I can make that even better."}, {"video_title": "Example Comparing distributions   AP Statistics   Khan Academy.mp3", "Sentence": "A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above it. In fact, I can make that even better. A distribution like this would have a lower range, but it would also have a higher standard deviation. So you can't just look at, it's not always the case that just by looking at one of these measures, the range or the standard deviation, you'll know for sure, but in cases like this, it's safe to say when you're looking at it by inspection that look, this green, the final round data does seem to have a higher range, higher variability, and so I'd feel pretty good at this. This is very high-level comparison."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Researchers used these results to test the null hypothesis is that the proportion is 0.5, the alternative hypothesis is that it's greater than 0.5, where P is the true proportion of adults that support the tax increase. They calculated a test statistic of z is approximately equal to 1.84 and a corresponding P value of approximately 0.033. Assuming the conditions for inference were met, which of these is an appropriate conclusion? And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's below our significance level, they haven't told it to us yet, it looks like they're gonna give some in the choices, well, then we would reject the null hypothesis, which would suggest the alternative. If the P value is not lower than this, then we will fail to reject the null hypothesis."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's below our significance level, they haven't told it to us yet, it looks like they're gonna give some in the choices, well, then we would reject the null hypothesis, which would suggest the alternative. If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampling distribution would be our assumed population proportion, how many standard deviations above that mean is this right over here? And that is what this test statistic is. And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean?"}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampling distribution would be our assumed population proportion, how many standard deviations above that mean is this right over here? And that is what this test statistic is. And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we just need to do is compare this P value right over here to the significance level. If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we just need to do is compare this P value right over here to the significance level. If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis. So let's look at these choices. And if you didn't answer it the first time, I encourage you to pause the video again."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis. So let's look at these choices. And if you didn't answer it the first time, I encourage you to pause the video again. So at the alpha is equal to 0.01 significance level, they should conclude that more than 50% of adults support the tax increase. So if the alpha is 1 hundredth, the P value right over here is over 3 hundredths. It's roughly 3.3%."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And if you didn't answer it the first time, I encourage you to pause the video again. So at the alpha is equal to 0.01 significance level, they should conclude that more than 50% of adults support the tax increase. So if the alpha is 1 hundredth, the P value right over here is over 3 hundredths. It's roughly 3.3%. So this is a situation where our P value, our P value is greater than or equal to alpha. In fact, it's definitely greater than alpha here. And so here we would fail to reject, we would fail to reject our null hypothesis."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "It's roughly 3.3%. So this is a situation where our P value, our P value is greater than or equal to alpha. In fact, it's definitely greater than alpha here. And so here we would fail to reject, we would fail to reject our null hypothesis. And so we wouldn't conclude that more than 50% of adults support the tax increase. Because remember, our null hypothesis is that 50% do, and we're failing to reject this. So that's not gonna be true."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And so here we would fail to reject, we would fail to reject our null hypothesis. And so we wouldn't conclude that more than 50% of adults support the tax increase. Because remember, our null hypothesis is that 50% do, and we're failing to reject this. So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis, that the true proportion is 50%."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis, that the true proportion is 50%. At the alpha is equal to, so at the alpha equals to 5 hundredth significance level, they should conclude that more than 50% of adults support the tax increase. Well yeah, in this situation, we have our P value, which is 0.033. It is indeed less than our significance level, in which case we reject, reject the null hypothesis."}, {"video_title": "Making conclusions in a test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "We just failed to reject this null hypothesis, that the true proportion is 50%. At the alpha is equal to, so at the alpha equals to 5 hundredth significance level, they should conclude that more than 50% of adults support the tax increase. Well yeah, in this situation, we have our P value, which is 0.033. It is indeed less than our significance level, in which case we reject, reject the null hypothesis. And if we reject the null hypothesis, that would suggest the alternative, that the true proportion is greater than 50%. And so I would pick this choice right over here. And then choice D, at that same significance level, they should conclude that less than 50% of adults support the tax increase."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Thomas's favorite colors are blue and green. He has one blue shirt, one green shirt, one blue hat, one green scarf, one blue pair of pants, and one green pair of pants. Thomas selects one of these garments at random. Let A be the event that he selects a blue garment, and let B be the event that he chooses a shirt. Which of the following statements are true? And they all, let's see, before I even read them, they all deal with probability of event A, probability of event B, probability of B given A, probability of A given B, probability of A and B. So actually, let's just calculate these things ahead of time before we even look at these right over here."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let A be the event that he selects a blue garment, and let B be the event that he chooses a shirt. Which of the following statements are true? And they all, let's see, before I even read them, they all deal with probability of event A, probability of event B, probability of B given A, probability of A given B, probability of A and B. So actually, let's just calculate these things ahead of time before we even look at these right over here. So let's just think about probability of A. The probability of A, that's the probability that he picks a blue, that he selects a blue garment. So how many equally likely outcomes are there?"}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So actually, let's just calculate these things ahead of time before we even look at these right over here. So let's just think about probability of A. The probability of A, that's the probability that he picks a blue, that he selects a blue garment. So how many equally likely outcomes are there? Well, there's one, two, three, four, five, six equally likely outcomes. And how many involve selecting a blue garment? Well, there's one, two, three of the equally likely outcomes involve selecting a blue garment."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many equally likely outcomes are there? Well, there's one, two, three, four, five, six equally likely outcomes. And how many involve selecting a blue garment? Well, there's one, two, three of the equally likely outcomes involve selecting a blue garment. So he has a three-sixths or one-half probability of selecting a blue garment. Now let's put the probability of B. What's probability of B?"}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's one, two, three of the equally likely outcomes involve selecting a blue garment. So he has a three-sixths or one-half probability of selecting a blue garment. Now let's put the probability of B. What's probability of B? And I'll do this in a neutral color since we're just saying that B is just the event that he chooses a shirt. So once again, there's six possible items, equally likely outcomes here. And which involve a shirt?"}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's probability of B? And I'll do this in a neutral color since we're just saying that B is just the event that he chooses a shirt. So once again, there's six possible items, equally likely outcomes here. And which involve a shirt? Well, there's one, there is two. So it looks like two of the six involve picking a shirt. Or we could say the probability of B is equal to one-third."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "And which involve a shirt? Well, there's one, there is two. So it looks like two of the six involve picking a shirt. Or we could say the probability of B is equal to one-third. Now what's the probability of A given B? Let's write that down. What's the probability of, let me just do it in a new color."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or we could say the probability of B is equal to one-third. Now what's the probability of A given B? Let's write that down. What's the probability of, let me just do it in a new color. What's the probability, probability of A given B? I'll do those in the colors. A, given that B has happened."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's the probability of, let me just do it in a new color. What's the probability, probability of A given B? I'll do those in the colors. A, given that B has happened. So this is saying, what's the probability, what's the probability, probability of A given B is the probability that he picks a blue garment given that he has picked a shirt. So this, the given B, that restricts our outcomes to these two. And so the probability that he's picked a blue item, well that's one out of the two equally likely ones."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "A, given that B has happened. So this is saying, what's the probability, what's the probability, probability of A given B is the probability that he picks a blue garment given that he has picked a shirt. So this, the given B, that restricts our outcomes to these two. And so the probability that he's picked a blue item, well that's one out of the two equally likely ones. So there is a one-half probability that he picks a blue garment given that he's picked a shirt. And that's because there's one blue shirt and one green shirt. Now let's look at the probability of B given A. Probability of B given A."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so the probability that he's picked a blue item, well that's one out of the two equally likely ones. So there is a one-half probability that he picks a blue garment given that he's picked a shirt. And that's because there's one blue shirt and one green shirt. Now let's look at the probability of B given A. Probability of B given A. So assuming that we've picked a blue garment, so assuming we've picked a blue garment, so it's either that one, that one, or that one, what's the probability that we have also chosen a shirt? Well, there's one, two, three possibilities, equally likely possibilities where we have a blue garment, and only one of those involves a shirt. So probability of B given A is one-third."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's look at the probability of B given A. Probability of B given A. So assuming that we've picked a blue garment, so assuming we've picked a blue garment, so it's either that one, that one, or that one, what's the probability that we have also chosen a shirt? Well, there's one, two, three possibilities, equally likely possibilities where we have a blue garment, and only one of those involves a shirt. So probability of B given A is one-third. And then finally, we could think about probability of A and B. So the probability of A and B. So this is the probability of picking a blue shirt."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So probability of B given A is one-third. And then finally, we could think about probability of A and B. So the probability of A and B. So this is the probability of picking a blue shirt. So only one out of the six equally likely outcomes is a blue shirt. So this one right over here is going to be one over six. So now that we've figured out all of that, let's see if we can answer these questions."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the probability of picking a blue shirt. So only one out of the six equally likely outcomes is a blue shirt. So this one right over here is going to be one over six. So now that we've figured out all of that, let's see if we can answer these questions. The probability of A given B equals the probability of A, and that does work out. Probability of A given B is one-half, and that's the same thing as the probability of A. The probability that Thomas selects a blue garment given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So now that we've figured out all of that, let's see if we can answer these questions. The probability of A given B equals the probability of A, and that does work out. Probability of A given B is one-half, and that's the same thing as the probability of A. The probability that Thomas selects a blue garment given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment. Yep, that's exactly. So I guess the words are just rephrasing what they wrote here in, I guess, more mathy notation. So this is absolutely true."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability that Thomas selects a blue garment given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment. Yep, that's exactly. So I guess the words are just rephrasing what they wrote here in, I guess, more mathy notation. So this is absolutely true. The probability of B given A is equal to the probability of B. Yep, probability of B given A is one-third, and the probability of B is one-third. The probability that Thomas selects a shirt given that he has chosen a blue garment is equal to the probability that Thomas selects a shirt. Yep, that's right."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is absolutely true. The probability of B given A is equal to the probability of B. Yep, probability of B given A is one-third, and the probability of B is one-third. The probability that Thomas selects a shirt given that he has chosen a blue garment is equal to the probability that Thomas selects a shirt. Yep, that's right. Events A and B are independent events. So two events are independent if the, well, let me write it more in math notation. These are independent if the probability of A given B is equal to the probability of A."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yep, that's right. Events A and B are independent events. So two events are independent if the, well, let me write it more in math notation. These are independent if the probability of A given B is equal to the probability of A. Then we could say A and B are independent because the probability of A, if this is true, then this means the probability of A given B actually isn't dependent on whether B happened or not. It's the same thing as the probability of A. This would lead to these events being independent."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are independent if the probability of A given B is equal to the probability of A. Then we could say A and B are independent because the probability of A, if this is true, then this means the probability of A given B actually isn't dependent on whether B happened or not. It's the same thing as the probability of A. This would lead to these events being independent. Also, if you had probability of B given A is equal to the probability of B, same argument. That would mean they're independent. Or if we said that the probability of A and B is equal to the probability of A times the probability of B, then these would also mean they're independent."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would lead to these events being independent. Also, if you had probability of B given A is equal to the probability of B, same argument. That would mean they're independent. Or if we said that the probability of A and B is equal to the probability of A times the probability of B, then these would also mean they're independent. We know that this one's true. The probability of A and B is one-sixth, and the probability of A times the probability of B is one-half times one-third, which is one-sixth. So all of these are clearly true, so we can say that A and B are independent."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if we said that the probability of A and B is equal to the probability of A times the probability of B, then these would also mean they're independent. We know that this one's true. The probability of A and B is one-sixth, and the probability of A times the probability of B is one-half times one-third, which is one-sixth. So all of these are clearly true, so we can say that A and B are independent. The probability of A is independent of whether B has happened or not, and the probability of B happening is independent of whether A has happened or not. The outcome of events A and B are dependent on each other. No, that's the opposite of saying that they're independent, so we're not going to, we can cross that out."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "So all of these are clearly true, so we can say that A and B are independent. The probability of A is independent of whether B has happened or not, and the probability of B happening is independent of whether A has happened or not. The outcome of events A and B are dependent on each other. No, that's the opposite of saying that they're independent, so we're not going to, we can cross that out. Probability of A and B is equal to probability of A times probability of B. We already said that to be true. One-sixth is one-half times one-third."}, {"video_title": "Analyzing event probability for independence   Probability and Statistics   Khan Academy.mp3", "Sentence": "No, that's the opposite of saying that they're independent, so we're not going to, we can cross that out. Probability of A and B is equal to probability of A times probability of B. We already said that to be true. One-sixth is one-half times one-third. The probability that Tom selects a blue garment that is a shirt is equal to the probability that Tom selects a blue garment multiplied by the probability that he selects a shirt. Yep, that's absolutely right. So actually, a lot of these statements are true."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "The scores of the first four rounds and the lowest round are shown in the following dot plot. And we see it right over here, the lowest round she scores an 80. She also scores a 90 once, a 92 once, a 94 once, and a 96 once. It was discovered that Anna broke some rules when she scored 80. So that score, so I guess cheating didn't help her. So that score will be removed from the data set. So they removed that 80 right over there, or just left with the scores from the other four rounds."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "It was discovered that Anna broke some rules when she scored 80. So that score, so I guess cheating didn't help her. So that score will be removed from the data set. So they removed that 80 right over there, or just left with the scores from the other four rounds. How will the removal of the lowest round affect the mean and the median? So let's actually think about the median first. So the median is the middle number."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So they removed that 80 right over there, or just left with the scores from the other four rounds. How will the removal of the lowest round affect the mean and the median? So let's actually think about the median first. So the median is the middle number. So over here, when you had five data points, the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So the median is the middle number. So over here, when you had five data points, the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers, you look at the middle two numbers."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers, you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "When you're trying to find the median of an even number of numbers, you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is gonna be halfway between 92 and 94, which is 93. So the median, the median is 93."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is gonna be halfway between 92 and 94, which is 93. So the median, the median is 93. Median is 93. So removing the lowest data point, in this case, increased the median. So the median, let me write it down here."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So the median, the median is 93. Median is 93. So removing the lowest data point, in this case, increased the median. So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean?"}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean? What's going to happen to the mean? Well, one way to think about it, without even doing any calculations, is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that, the mean is going to go up. The mean is going to go up."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "Now what's going to happen to the mean? What's going to happen to the mean? Well, one way to think about it, without even doing any calculations, is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that, the mean is going to go up. The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean, your mean is going to go down, because you don't have that large number anymore. If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean, your mean is going to go down, because you don't have that large number anymore. If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're going to add 80 plus 90 plus 92 plus 94 plus 96."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're going to add 80 plus 90 plus 92 plus 94 plus 96. Those are our data points. And that gets us, two plus four is six, plus six is 12. And then we have one plus eight is nine."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So we're going to add 80 plus 90 plus 92 plus 94 plus 96. Those are our data points. And that gets us, two plus four is six, plus six is 12. And then we have one plus eight is nine. And we essentially, this is, so these are nine, and you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 450, 452."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "And then we have one plus eight is nine. And we essentially, this is, so these are nine, and you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 450, 452. So that's the sum of the scores of these five rounds. And then you divide it by the number of rounds you have. So it'd be 452 divided by five."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So this is going to be 450, 452. So that's the sum of the scores of these five rounds. And then you divide it by the number of rounds you have. So it'd be 452 divided by five. So 452 divided by five is going to give us, five goes into, doesn't go into four, it goes into 45 nine times. Nine times five is 45. You subtract, you get zero, bring down the two."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So it'd be 452 divided by five. So 452 divided by five is going to give us, five goes into, doesn't go into four, it goes into 45 nine times. Nine times five is 45. You subtract, you get zero, bring down the two. Five goes into two zero times. Zero times five is, zero times five is zero. Subtract, you have two left over."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "You subtract, you get zero, bring down the two. Five goes into two zero times. Zero times five is, zero times five is zero. Subtract, you have two left over. So you can say that the mean here, the mean here is 90 and 2 5ths. Maybe, not nine and 2 5ths. 90 and 2 5ths."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "Subtract, you have two left over. So you can say that the mean here, the mean here is 90 and 2 5ths. Maybe, not nine and 2 5ths. 90 and 2 5ths. So the mean is right around here. So that's the mean of these data points right over there. And if you remove it, what is the mean going to be?"}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "90 and 2 5ths. So the mean is right around here. So that's the mean of these data points right over there. And if you remove it, what is the mean going to be? So here we're just going to take our 90 plus our 92 plus our 94 plus our 96. Add them together. So let's see, two plus four plus six is 12."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "And if you remove it, what is the mean going to be? So here we're just going to take our 90 plus our 92 plus our 94 plus our 96. Add them together. So let's see, two plus four plus six is 12. And you add these together, you're going to get 37. 372 divided by four. Because I have four data points now, not five."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "So let's see, two plus four plus six is 12. And you add these together, you're going to get 37. 372 divided by four. Because I have four data points now, not five. Four goes into three, let me do this in a place where you can see it. So four goes into 372. Goes into 37 nine times."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "Because I have four data points now, not five. Four goes into three, let me do this in a place where you can see it. So four goes into 372. Goes into 37 nine times. Nine times four is 36. Subtract, you get a one. Bring down the two."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "Goes into 37 nine times. Nine times four is 36. Subtract, you get a one. Bring down the two. It goes exactly three times. Three times four is 12. You have no remainder."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "Bring down the two. It goes exactly three times. Three times four is 12. You have no remainder. So the median and the mean here are both. So this is also the mean. The mean here is also 93."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "You have no remainder. So the median and the mean here are both. So this is also the mean. The mean here is also 93. So you see that the median, the median went from 92 to 93. It increased. The mean went from 90 and 2 5ths to 93."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "The mean here is also 93. So you see that the median, the median went from 92 to 93. It increased. The mean went from 90 and 2 5ths to 93. So the mean increased by more than the median. They both increased, but the mean increased by more. And it makes sense, because this number was way, way below all of these over here."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "The mean went from 90 and 2 5ths to 93. So the mean increased by more than the median. They both increased, but the mean increased by more. And it makes sense, because this number was way, way below all of these over here. So you can imagine, if you take this out, the mean should increase by a good amount. But let's see which of these choices are what we just described. But the mean and the median will decrease."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "And it makes sense, because this number was way, way below all of these over here. So you can imagine, if you take this out, the mean should increase by a good amount. But let's see which of these choices are what we just described. But the mean and the median will decrease. Nope. But the mean and the median will decrease. Nope."}, {"video_title": "Impact on median and mean when removing lowest value example   6th grade   Khan Academy.mp3", "Sentence": "But the mean and the median will decrease. Nope. But the mean and the median will decrease. Nope. But the mean and the median will increase, but the mean will increase by more than the median. That's exactly, that's exactly what happened. The mean went from 90 and 2 5ths, or 90.4, went from 90.4, or 90 and 2 5ths, to 93."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "As we go further in our statistical careers, it's going to be valuable to assume that certain distributions are normal distributions or sometimes to assume that they are binomial distributions because if we can do that, we can make all sorts of interesting inferences about them when we make that assumption. But one of the key things about normal distributions or binomial distributions is we assume that they're the sum or they can be viewed as the sum of a bunch of independent trials. So we have to assume that trials are independent. Now that is reasonable in a lot of situations, but sometimes, let's say you're conducting a survey of people exiting a mall, and in that case, and let's say you're saying whether they have done their taxes already, if they're exiting the mall, it's hard to do these samples with replacement. They're leaving the mall. You can't say, hey, hey, wait, I just asked you a question. Now you've answered it."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now that is reasonable in a lot of situations, but sometimes, let's say you're conducting a survey of people exiting a mall, and in that case, and let's say you're saying whether they have done their taxes already, if they're exiting the mall, it's hard to do these samples with replacement. They're leaving the mall. You can't say, hey, hey, wait, I just asked you a question. Now you've answered it. Now go back into the mall because I want each trial to be truly independent. But we all know it feels intuitive that, hey, if there are 10,000 people in the mall and I'm going to sample 10 of them, does it really matter that it's truly independent? Doesn't it matter that we're just kind of close to being independent?"}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now you've answered it. Now go back into the mall because I want each trial to be truly independent. But we all know it feels intuitive that, hey, if there are 10,000 people in the mall and I'm going to sample 10 of them, does it really matter that it's truly independent? Doesn't it matter that we're just kind of close to being independent? And because of that idea and because we do wanna make inferences based on things being close to a binomial distribution or a normal distribution, we have something called the 10% rule. And the 10% rule says that if our sample, if our sample is less than or equal to 10% of the population, then it is okay to assume approximate independence, approximate independence. And there are some fairly sophisticated ways of coming up with this 10% threshold."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Doesn't it matter that we're just kind of close to being independent? And because of that idea and because we do wanna make inferences based on things being close to a binomial distribution or a normal distribution, we have something called the 10% rule. And the 10% rule says that if our sample, if our sample is less than or equal to 10% of the population, then it is okay to assume approximate independence, approximate independence. And there are some fairly sophisticated ways of coming up with this 10% threshold. People could have picked 9%. They could have picked 10.1%. But 10% is a nice round number."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And there are some fairly sophisticated ways of coming up with this 10% threshold. People could have picked 9%. They could have picked 10.1%. But 10% is a nice round number. And if we look at some tangible examples, it seems to do a pretty good job. So for example, right over here, let's let x be the number of boys from three trials of selecting from a classroom of n students where 50% of the class is a boy and 50% of the class is a girl. And so what we have over here is we have a bunch of different n's."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But 10% is a nice round number. And if we look at some tangible examples, it seems to do a pretty good job. So for example, right over here, let's let x be the number of boys from three trials of selecting from a classroom of n students where 50% of the class is a boy and 50% of the class is a girl. And so what we have over here is we have a bunch of different n's. What if we have 20 students in the class? What if we have 30? What if we have 100?"}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so what we have over here is we have a bunch of different n's. What if we have 20 students in the class? What if we have 30? What if we have 100? What if we have 10,000? And so we could find the probability that we select three boys with replacement in each of these scenarios. And we could also find the probability that we select three boys without replacement."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "What if we have 100? What if we have 10,000? And so we could find the probability that we select three boys with replacement in each of these scenarios. And we could also find the probability that we select three boys without replacement. And then we could think about what proportion is our sample size of the entire population. And then we could say, does the 10% rule actually make sense? So this first column where we are picking three boys with replacement, in this case, because we are replacing, each of these trials are independent, are truly independent."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And we could also find the probability that we select three boys without replacement. And then we could think about what proportion is our sample size of the entire population. And then we could say, does the 10% rule actually make sense? So this first column where we are picking three boys with replacement, in this case, because we are replacing, each of these trials are independent, are truly independent. And if our trials are independent, then x would be truly a binomial variable. Here, we aren't independent because we are not replacing, so not independent. And so officially, in this column right over here when we're not replacing, x would not be considered a binomial random variable."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this first column where we are picking three boys with replacement, in this case, because we are replacing, each of these trials are independent, are truly independent. And if our trials are independent, then x would be truly a binomial variable. Here, we aren't independent because we are not replacing, so not independent. And so officially, in this column right over here when we're not replacing, x would not be considered a binomial random variable. But let's see if there's a threshold where if our sample size is a small enough percentage of our entire population where we would feel not so bad about assuming x is close to being binomial. So in all of the cases where you have independent trials and 50% of the population is boys, 50% is girls, well, you're going to amount to 1 1\u20442 times 1 1\u20442 times 1 1\u20442. So in all of those situations, you have a 12.5% chance that x is going to be equal to three."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so officially, in this column right over here when we're not replacing, x would not be considered a binomial random variable. But let's see if there's a threshold where if our sample size is a small enough percentage of our entire population where we would feel not so bad about assuming x is close to being binomial. So in all of the cases where you have independent trials and 50% of the population is boys, 50% is girls, well, you're going to amount to 1 1\u20442 times 1 1\u20442 times 1 1\u20442. So in all of those situations, you have a 12.5% chance that x is going to be equal to three. And in this case, x would be a binomial variable. But look over here. When three is a fairly large percentage of our population, in this case, it is 15%, the percent chance of getting three boys without replacement is 10.5%, which is reasonably different from 12 1\u20442%."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So in all of those situations, you have a 12.5% chance that x is going to be equal to three. And in this case, x would be a binomial variable. But look over here. When three is a fairly large percentage of our population, in this case, it is 15%, the percent chance of getting three boys without replacement is 10.5%, which is reasonably different from 12 1\u20442%. It is 2% different, but 2% relative to 12 1\u20442%. So that's someplace in between 10 and 20% difference in terms of the probability. So this is a reasonably big difference."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "When three is a fairly large percentage of our population, in this case, it is 15%, the percent chance of getting three boys without replacement is 10.5%, which is reasonably different from 12 1\u20442%. It is 2% different, but 2% relative to 12 1\u20442%. So that's someplace in between 10 and 20% difference in terms of the probability. So this is a reasonably big difference. But as we increase the population size without increasing the sample size, we see that these numbers get closer and closer to each other, all the way so that if you have 10,000 people in your population and you're only doing three trials, that the numbers get very, very close. This is actually 12.49 something percent. But if you round to the nearest 10th of a percent, you see that they are close."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this is a reasonably big difference. But as we increase the population size without increasing the sample size, we see that these numbers get closer and closer to each other, all the way so that if you have 10,000 people in your population and you're only doing three trials, that the numbers get very, very close. This is actually 12.49 something percent. But if you round to the nearest 10th of a percent, you see that they are close. So I think most people would say, all right, if your sample is 3 10,000th of the population, that you'd feel pretty good treating this column without replacement as being pretty close to being a binomial variable. And most people would say, all right, this first scenario where your sample size is 15% of your population, you wouldn't feel so good treating this without replacement column as a binomial random variable. But where do you draw the line?"}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But if you round to the nearest 10th of a percent, you see that they are close. So I think most people would say, all right, if your sample is 3 10,000th of the population, that you'd feel pretty good treating this column without replacement as being pretty close to being a binomial variable. And most people would say, all right, this first scenario where your sample size is 15% of your population, you wouldn't feel so good treating this without replacement column as a binomial random variable. But where do you draw the line? And as we alluded to earlier in the video, the line is typically drawn at 10%. That if your sample size is less than or equal to 10% of your population, it's not unreasonable to treat your random variable, even though it's not officially binomial, to say, okay, maybe it is, maybe I can functionally treat it as binomial, and then from there I can make all of the powerful inferences that we tend to do in statistics. With that said, the lower the percentage the sample is of the population, the better."}, {"video_title": "10% Rule of assuming  independence  between trials   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But where do you draw the line? And as we alluded to earlier in the video, the line is typically drawn at 10%. That if your sample size is less than or equal to 10% of your population, it's not unreasonable to treat your random variable, even though it's not officially binomial, to say, okay, maybe it is, maybe I can functionally treat it as binomial, and then from there I can make all of the powerful inferences that we tend to do in statistics. With that said, the lower the percentage the sample is of the population, the better. Now to be clear, that's not saying that small sample sizes are better than large sample sizes. In statistics, large sample sizes tend to be a lot better than small sample sizes. But if you wanna make this independence assumption, so to speak, even when it's not exactly true, you want your sample to be a small percentage of the population."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "What we're going to do in this video is calculate a typical measure of how well the actual data points agree with a model, in this case, a linear model. And there's several names for it. We could consider this to be the standard deviation of the residuals, and that's essentially what we're going to calculate. You could also call it the root mean square error, and you'll see why it's called this, because this really describes how we calculate it. So what we're going to do is look at the residuals for each of these points, and then we're going to find the standard deviation of them. So just as a bit of review, the ith residual is going to be equal to the ith y value for a given x minus the predicted y value for a given x. Now when I say y hat right over here, this just says what would the linear regression predict for a given x?"}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "You could also call it the root mean square error, and you'll see why it's called this, because this really describes how we calculate it. So what we're going to do is look at the residuals for each of these points, and then we're going to find the standard deviation of them. So just as a bit of review, the ith residual is going to be equal to the ith y value for a given x minus the predicted y value for a given x. Now when I say y hat right over here, this just says what would the linear regression predict for a given x? And this is the actual y for a given x. So for example, and we've done this in other videos, this is all review, the residual here, when x is equal to one, we have y is equal to one, but what was predicted by the model is 2.5 times one minus two, which is.5. So one minus.5, so this residual here, this residual is equal to one minus 0.5, which is equal to 0.5, and it's a positive 0.5."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "Now when I say y hat right over here, this just says what would the linear regression predict for a given x? And this is the actual y for a given x. So for example, and we've done this in other videos, this is all review, the residual here, when x is equal to one, we have y is equal to one, but what was predicted by the model is 2.5 times one minus two, which is.5. So one minus.5, so this residual here, this residual is equal to one minus 0.5, which is equal to 0.5, and it's a positive 0.5. And if the actual point is above the model, you are going to have a positive residual. Now the residual over here, you also have the actual point being higher than the model. So this is also going to be a positive residual."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "So one minus.5, so this residual here, this residual is equal to one minus 0.5, which is equal to 0.5, and it's a positive 0.5. And if the actual point is above the model, you are going to have a positive residual. Now the residual over here, you also have the actual point being higher than the model. So this is also going to be a positive residual. And once again, when x is equal to three, the actual y is six. The predicted y is 2.5 times three, which is 7.5 minus two, which is 5.5. So you have six minus 5.5."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "So this is also going to be a positive residual. And once again, when x is equal to three, the actual y is six. The predicted y is 2.5 times three, which is 7.5 minus two, which is 5.5. So you have six minus 5.5. So here I'll write residual is equal to six minus 5.5, which is equal to 0.5. So once again, you have a positive residual. Now for this point that sits right on the model, the actual is the predicted."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "So you have six minus 5.5. So here I'll write residual is equal to six minus 5.5, which is equal to 0.5. So once again, you have a positive residual. Now for this point that sits right on the model, the actual is the predicted. When x is two, the actual is three, and what was predicted by the model is three. So the residual here is equal to the actual is three and the predicted is three, so it's equal to zero. And then last but not least, you have this data point where the residual is going to be the actual when x is equal to two is two minus the predicted."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "Now for this point that sits right on the model, the actual is the predicted. When x is two, the actual is three, and what was predicted by the model is three. So the residual here is equal to the actual is three and the predicted is three, so it's equal to zero. And then last but not least, you have this data point where the residual is going to be the actual when x is equal to two is two minus the predicted. Well, when x is equal to two, you have 2.5 times two, which is equal to five minus two is equal to three. So two minus three is equal to negative one. And so when your actual is below your regression line, you're going to have a negative residual."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And then last but not least, you have this data point where the residual is going to be the actual when x is equal to two is two minus the predicted. Well, when x is equal to two, you have 2.5 times two, which is equal to five minus two is equal to three. So two minus three is equal to negative one. And so when your actual is below your regression line, you're going to have a negative residual. So this is going to be negative one right over there. Now we can calculate the standard deviation of the residuals. We're going to take this first residual, which is 0.5, and we're going to square it."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And so when your actual is below your regression line, you're going to have a negative residual. So this is going to be negative one right over there. Now we can calculate the standard deviation of the residuals. We're going to take this first residual, which is 0.5, and we're going to square it. We're going to add it to the second residual right over here. I'll use this blue, this teal color. That's zero."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "We're going to take this first residual, which is 0.5, and we're going to square it. We're going to add it to the second residual right over here. I'll use this blue, this teal color. That's zero. Gonna square that. Then we have this third residual, which is negative one. So plus negative one squared."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "That's zero. Gonna square that. Then we have this third residual, which is negative one. So plus negative one squared. And then finally we have that fourth residual, which is 0.5 squared. 0.5 squared. So once again, we took each of the residuals, which you could view as the distance between the points and what the model would predict."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "So plus negative one squared. And then finally we have that fourth residual, which is 0.5 squared. 0.5 squared. So once again, we took each of the residuals, which you could view as the distance between the points and what the model would predict. We are squaring them. When you take a typical standard deviation, you're taking the distance between a point and the mean. Here we're taking the distance between a point and what the model would have predicted."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "So once again, we took each of the residuals, which you could view as the distance between the points and what the model would predict. We are squaring them. When you take a typical standard deviation, you're taking the distance between a point and the mean. Here we're taking the distance between a point and what the model would have predicted. But we're squaring each of those residuals and adding them all up together. And just like we do with the sample standard deviation, we are now going to divide by one less than the number of residuals we just squared and added. So we have four residuals."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "Here we're taking the distance between a point and what the model would have predicted. But we're squaring each of those residuals and adding them all up together. And just like we do with the sample standard deviation, we are now going to divide by one less than the number of residuals we just squared and added. So we have four residuals. We're gonna divide by four minus one, which is equal to, of course, three. You could view this part as a mean of the squared errors. And now we're gonna take the square root of it."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "So we have four residuals. We're gonna divide by four minus one, which is equal to, of course, three. You could view this part as a mean of the squared errors. And now we're gonna take the square root of it. So let's see. This is going to be equal to square root of, this is 0.25. 0.25."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And now we're gonna take the square root of it. So let's see. This is going to be equal to square root of, this is 0.25. 0.25. This is just zero. This is going to be positive one. And then this 0.5 squared is going to be 0.25."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "0.25. This is just zero. This is going to be positive one. And then this 0.5 squared is going to be 0.25. 0.25. All of that over three. Now this numerator is going to be 1.5 over three."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And then this 0.5 squared is going to be 0.25. 0.25. All of that over three. Now this numerator is going to be 1.5 over three. So this is going to be equal to, 1.5 is exactly half of three. So we could say this is equal to the square root of 1.5. This is one over the square root of two."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "Now this numerator is going to be 1.5 over three. So this is going to be equal to, 1.5 is exactly half of three. So we could say this is equal to the square root of 1.5. This is one over the square root of two. One divided by the square root of two, which gets us to, so if we round to the nearest thousandths, it's roughly 0.707. So approximately 0.707. And if you wanted to visualize that, one standard deviation of the residuals below the line would look like this."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "This is one over the square root of two. One divided by the square root of two, which gets us to, so if we round to the nearest thousandths, it's roughly 0.707. So approximately 0.707. And if you wanted to visualize that, one standard deviation of the residuals below the line would look like this. And one standard deviation above the line for any given x value would go one standard deviation of the residuals above it. It would look something like that. And this is obviously just a hand-drawn approximation."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And if you wanted to visualize that, one standard deviation of the residuals below the line would look like this. And one standard deviation above the line for any given x value would go one standard deviation of the residuals above it. It would look something like that. And this is obviously just a hand-drawn approximation. But you do see that this does seem to be roughly indicative of the typical residual. Now it's worth noting, sometimes people will say it's the average residual. And it depends how you think about the word average."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And this is obviously just a hand-drawn approximation. But you do see that this does seem to be roughly indicative of the typical residual. Now it's worth noting, sometimes people will say it's the average residual. And it depends how you think about the word average. Because we are squaring the residuals. So outliers, things that are really far from the line, when you square it, are going to have disproportionate impact here. If you didn't want to have that behavior, we could have done something like find the mean of the absolute residuals."}, {"video_title": "Standard deviation of residuals or Root-mean-square error (RMSD).mp3", "Sentence": "And it depends how you think about the word average. Because we are squaring the residuals. So outliers, things that are really far from the line, when you square it, are going to have disproportionate impact here. If you didn't want to have that behavior, we could have done something like find the mean of the absolute residuals. That actually in some ways would have been a simpler one. But this is a standard way of people trying to figure out how much a model disagrees with the actual data. And so you can imagine, the lower this number is, the better the fit of the model."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what it tells us is we can start off with any distribution that has a well-defined mean and variance, and if it has a well-defined variance, it has a well-defined standard deviation. And it could be a continuous distribution or a discrete one. I'll draw a discrete one just because it's easier to imagine, at least for the purposes of this video. So let's say I have a discrete probability distribution function, and I want to be very careful not to make it look anything close to a normal distribution, because I want to show you the power of the central limit theorem. So let's say I have a distribution, let's say I can take on values 1 through 6. 1, 2, 3, 4, 5, 6, it's some kind of crazy dice that's very likely to get a 1. Let's say it's impossible, let me make that a straight line."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say I have a discrete probability distribution function, and I want to be very careful not to make it look anything close to a normal distribution, because I want to show you the power of the central limit theorem. So let's say I have a distribution, let's say I can take on values 1 through 6. 1, 2, 3, 4, 5, 6, it's some kind of crazy dice that's very likely to get a 1. Let's say it's impossible, let me make that a straight line. You're very high likelihood of getting a 1. Let's say it's impossible to get a 2. Let's say it's an okay likelihood of getting a 3 or a 4."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say it's impossible, let me make that a straight line. You're very high likelihood of getting a 1. Let's say it's impossible to get a 2. Let's say it's an okay likelihood of getting a 3 or a 4. Let's say it's impossible to get a 5. Let's say it's very likely to get a 6 like that. So that's my probability distribution function."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say it's an okay likelihood of getting a 3 or a 4. Let's say it's impossible to get a 5. Let's say it's very likely to get a 6 like that. So that's my probability distribution function. If I were to draw a mean, this is symmetric, so maybe the mean would be something like that. It would be halfway, so that would be my mean right there, the standard deviation. Maybe it would be that far and that far above and below the mean."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's my probability distribution function. If I were to draw a mean, this is symmetric, so maybe the mean would be something like that. It would be halfway, so that would be my mean right there, the standard deviation. Maybe it would be that far and that far above and below the mean. But that's my discrete probability distribution function. Now what I'm going to do here, instead of just taking samples of this random variable that's described by this probability distribution function, I'm going to take samples of it, but I'm going to average the samples, and then look at those samples and see the frequency of the averages that I get. And when I say average, I mean the mean."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe it would be that far and that far above and below the mean. But that's my discrete probability distribution function. Now what I'm going to do here, instead of just taking samples of this random variable that's described by this probability distribution function, I'm going to take samples of it, but I'm going to average the samples, and then look at those samples and see the frequency of the averages that I get. And when I say average, I mean the mean. Let me define something. Let's say my sample size, and I could put any number here, but let's say first off we try a sample size of n is equal to 4. And what that means is I'm going to take 4 samples from this."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And when I say average, I mean the mean. Let me define something. Let's say my sample size, and I could put any number here, but let's say first off we try a sample size of n is equal to 4. And what that means is I'm going to take 4 samples from this. So let's say the first time I take 4 samples, so my sample size is 4, let's say I get a 1, let's say I get another 1, and let's say I get a 3, and I get a 6. So that right there is my first sample of sample size 4. I know the terminology can get confusing because this is a sample that's made up of 4 samples."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what that means is I'm going to take 4 samples from this. So let's say the first time I take 4 samples, so my sample size is 4, let's say I get a 1, let's say I get another 1, and let's say I get a 3, and I get a 6. So that right there is my first sample of sample size 4. I know the terminology can get confusing because this is a sample that's made up of 4 samples. But when we talk about the sample mean and the sampling distribution of the sample mean, which we're going to talk more and more about over the next few videos, normally the sample refers to the set of samples from your distribution, and the sample size tells you how many you actually took from your distribution. But the terminology can be very confusing because you could easily view one of these as a sample. But we're taking 4 samples from here."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I know the terminology can get confusing because this is a sample that's made up of 4 samples. But when we talk about the sample mean and the sampling distribution of the sample mean, which we're going to talk more and more about over the next few videos, normally the sample refers to the set of samples from your distribution, and the sample size tells you how many you actually took from your distribution. But the terminology can be very confusing because you could easily view one of these as a sample. But we're taking 4 samples from here. We have a sample size of 4. And what I'm going to do is I'm going to average them. So let's say the mean, I want to be very careful when I say average, the mean of this first sample of size 4 is what?"}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we're taking 4 samples from here. We have a sample size of 4. And what I'm going to do is I'm going to average them. So let's say the mean, I want to be very careful when I say average, the mean of this first sample of size 4 is what? 1 plus 1 is 2, 2 plus 3 is 5, 5 plus 6 is 11, 11 divided by 4 is 2.75. That is my first sample mean for my first sample of size 4. Let me do another one."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say the mean, I want to be very careful when I say average, the mean of this first sample of size 4 is what? 1 plus 1 is 2, 2 plus 3 is 5, 5 plus 6 is 11, 11 divided by 4 is 2.75. That is my first sample mean for my first sample of size 4. Let me do another one. My second sample of size 4, let's say that I get a 3, a 4, let's say I get another 3, and let's say I get a 1. I just didn't happen to get a 6 that time. And notice, I can't get a 2 or a 5."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do another one. My second sample of size 4, let's say that I get a 3, a 4, let's say I get another 3, and let's say I get a 1. I just didn't happen to get a 6 that time. And notice, I can't get a 2 or a 5. It's impossible for this distribution. The chance of getting a 2 or a 5 is 0, so I can't have any 2s or 5s over here. So for the second sample of sample size 4, my sample mean, so my second sample mean is going to be 3 plus 4 is 7, 7 plus 3 is 10, plus 1 is 11, 11 divided by 4, once again is 2.75."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And notice, I can't get a 2 or a 5. It's impossible for this distribution. The chance of getting a 2 or a 5 is 0, so I can't have any 2s or 5s over here. So for the second sample of sample size 4, my sample mean, so my second sample mean is going to be 3 plus 4 is 7, 7 plus 3 is 10, plus 1 is 11, 11 divided by 4, once again is 2.75. Let me do one more, because I really want to make it clear what we're doing here. So I do one more. Actually, we're going to do a gazillion more, but let me just do one more in detail."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for the second sample of sample size 4, my sample mean, so my second sample mean is going to be 3 plus 4 is 7, 7 plus 3 is 10, plus 1 is 11, 11 divided by 4, once again is 2.75. Let me do one more, because I really want to make it clear what we're doing here. So I do one more. Actually, we're going to do a gazillion more, but let me just do one more in detail. So let's say my third sample of sample size 4, I get, so I'm going to literally take 4 samples. So my sample is made up of 4 samples from this original crazy distribution. Let's say I get a 1, a 1, and a 6, and a 6."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, we're going to do a gazillion more, but let me just do one more in detail. So let's say my third sample of sample size 4, I get, so I'm going to literally take 4 samples. So my sample is made up of 4 samples from this original crazy distribution. Let's say I get a 1, a 1, and a 6, and a 6. And so my third sample mean is going to be 1 plus 1 is 2, 2 plus 6 is 8, 8 plus 6 is 14, 14 divided by 4, 14 divided by 4 is what? 3.5. And as I find each of these sample means, so for each of my samples of sample size 4, I figure out a mean."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I get a 1, a 1, and a 6, and a 6. And so my third sample mean is going to be 1 plus 1 is 2, 2 plus 6 is 8, 8 plus 6 is 14, 14 divided by 4, 14 divided by 4 is what? 3.5. And as I find each of these sample means, so for each of my samples of sample size 4, I figure out a mean. And as I do each of them, I'm going to plot it on a frequency distribution. And this is all going to amaze you in a few seconds. So I plot this all on a frequency distribution."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And as I find each of these sample means, so for each of my samples of sample size 4, I figure out a mean. And as I do each of them, I'm going to plot it on a frequency distribution. And this is all going to amaze you in a few seconds. So I plot this all on a frequency distribution. So I say, okay, on my first sample, my first sample mean was 2.75. So I'm plotting the actual frequency of the sample means I get for each sample. So 2.75, I got it one time, so I'll put a little plot there."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I plot this all on a frequency distribution. So I say, okay, on my first sample, my first sample mean was 2.75. So I'm plotting the actual frequency of the sample means I get for each sample. So 2.75, I got it one time, so I'll put a little plot there. So that's from that one right there. And I got, the next time, I also got a 2.75. That's a 2.75 there, so I got twice."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 2.75, I got it one time, so I'll put a little plot there. So that's from that one right there. And I got, the next time, I also got a 2.75. That's a 2.75 there, so I got twice. So I'll plot the frequency right there. Then I got a 3.5, so all the possible values, I could have a 3, I could have a 3.25, I could have a 3.5. So then I have the 3.5, so I'll plot it right there."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's a 2.75 there, so I got twice. So I'll plot the frequency right there. Then I got a 3.5, so all the possible values, I could have a 3, I could have a 3.25, I could have a 3.5. So then I have the 3.5, so I'll plot it right there. And what I'm going to do is I'm going to keep taking these samples. Maybe I'll take 10,000 of them. So I'm going to keep taking these samples."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So then I have the 3.5, so I'll plot it right there. And what I'm going to do is I'm going to keep taking these samples. Maybe I'll take 10,000 of them. So I'm going to keep taking these samples. So I go all the way to S, you know, 10,000. I just do a bunch of these. And what it's going to look like over time is each of these, I'm going to make it a dot because I'm going to have to zoom out."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm going to keep taking these samples. So I go all the way to S, you know, 10,000. I just do a bunch of these. And what it's going to look like over time is each of these, I'm going to make it a dot because I'm going to have to zoom out. So if I look at it like this, over time, still it's all the values that it might be able to take on. You know, 2.75 might be here. So this first dot is going to be right there."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what it's going to look like over time is each of these, I'm going to make it a dot because I'm going to have to zoom out. So if I look at it like this, over time, still it's all the values that it might be able to take on. You know, 2.75 might be here. So this first dot is going to be right there. And that second one is going to be right there. And that one at 3.5 is going to look right there. But I'm going to do it 10,000 times because I'm going to have 10,000 dots."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this first dot is going to be right there. And that second one is going to be right there. And that one at 3.5 is going to look right there. But I'm going to do it 10,000 times because I'm going to have 10,000 dots. And let's say as I do it, I'm going to just keep plotting them. I'm just going to keep plotting the frequencies. I'm just going to keep plotting them over and over and over again."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I'm going to do it 10,000 times because I'm going to have 10,000 dots. And let's say as I do it, I'm going to just keep plotting them. I'm just going to keep plotting the frequencies. I'm just going to keep plotting them over and over and over again. And what you're going to see is as I take many, many samples of size 4, I'm going to have something that's going to start kind of approximating a normal distribution. So each of these dots represent an incidence of a sample mean. So as I keep adding on this column right here, that means I kept getting the sample mean 2.75."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just going to keep plotting them over and over and over again. And what you're going to see is as I take many, many samples of size 4, I'm going to have something that's going to start kind of approximating a normal distribution. So each of these dots represent an incidence of a sample mean. So as I keep adding on this column right here, that means I kept getting the sample mean 2.75. So over time, I'm going to have something that's starting to approximate a normal distribution. And that is the neat thing about the central limit theorem. So the central limit to N, this was the case for, so in orange, that's the case for N is equal to 4."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So as I keep adding on this column right here, that means I kept getting the sample mean 2.75. So over time, I'm going to have something that's starting to approximate a normal distribution. And that is the neat thing about the central limit theorem. So the central limit to N, this was the case for, so in orange, that's the case for N is equal to 4. This was for sample size of 4. Now, if I did the same thing with the sample size of maybe 20. So in this case, instead of just taking 4 samples from my original crazy distribution, every sample, I take 20 instances of my random variable and I average those 20, and then I plot the sample mean on here."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the central limit to N, this was the case for, so in orange, that's the case for N is equal to 4. This was for sample size of 4. Now, if I did the same thing with the sample size of maybe 20. So in this case, instead of just taking 4 samples from my original crazy distribution, every sample, I take 20 instances of my random variable and I average those 20, and then I plot the sample mean on here. So in that case, I'm going to have a distribution that looks like this. And we'll discuss this in more videos. But it turns out if I were to plot the sample, 10,000 of the sample means here, I'm going to have something that, two things, it's going to even more closely approximate a normal distribution, and we're going to see in future videos it's actually going to have a smaller, well, let me be clear, it's going to have the same mean."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in this case, instead of just taking 4 samples from my original crazy distribution, every sample, I take 20 instances of my random variable and I average those 20, and then I plot the sample mean on here. So in that case, I'm going to have a distribution that looks like this. And we'll discuss this in more videos. But it turns out if I were to plot the sample, 10,000 of the sample means here, I'm going to have something that, two things, it's going to even more closely approximate a normal distribution, and we're going to see in future videos it's actually going to have a smaller, well, let me be clear, it's going to have the same mean. So that's the mean, this is going to have the same mean. But it's going to have a smaller standard deviation. So I want to, well, I should plot these from the bottom because you kind of stack it."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it turns out if I were to plot the sample, 10,000 of the sample means here, I'm going to have something that, two things, it's going to even more closely approximate a normal distribution, and we're going to see in future videos it's actually going to have a smaller, well, let me be clear, it's going to have the same mean. So that's the mean, this is going to have the same mean. But it's going to have a smaller standard deviation. So I want to, well, I should plot these from the bottom because you kind of stack it. One, you get one, then another instance, then another instance. But this is going to more and more approach a normal distribution. So the reality is, and this is what's super cool about the central limit theorem, as your sample size becomes larger, or you can even say as it approaches infinity, but you really don't have to get that close to infinity to really get close to a normal distribution."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I want to, well, I should plot these from the bottom because you kind of stack it. One, you get one, then another instance, then another instance. But this is going to more and more approach a normal distribution. So the reality is, and this is what's super cool about the central limit theorem, as your sample size becomes larger, or you can even say as it approaches infinity, but you really don't have to get that close to infinity to really get close to a normal distribution. So if you have a sample size of 10 or 20, you're already getting very close to a normal distribution. In fact, about as good an approximation as we see in our everyday life. What's cool is we can start with some crazy distribution."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the reality is, and this is what's super cool about the central limit theorem, as your sample size becomes larger, or you can even say as it approaches infinity, but you really don't have to get that close to infinity to really get close to a normal distribution. So if you have a sample size of 10 or 20, you're already getting very close to a normal distribution. In fact, about as good an approximation as we see in our everyday life. What's cool is we can start with some crazy distribution. This has nothing to do with a normal distribution. But if we have a sample size, this was n equals 4, but if we have a sample size of n equals 10 or n equals 100, and we were to take 100 of these instead of 4 here and average them and then plot that average, the frequency of it, then we would take 100 again, average them, take the mean, plot that again. And if we were to do that a bunch of times, in fact, if we were to do that an infinite time, we would find that, especially if we had an infinite sample size, we would find a perfect normal distribution."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's cool is we can start with some crazy distribution. This has nothing to do with a normal distribution. But if we have a sample size, this was n equals 4, but if we have a sample size of n equals 10 or n equals 100, and we were to take 100 of these instead of 4 here and average them and then plot that average, the frequency of it, then we would take 100 again, average them, take the mean, plot that again. And if we were to do that a bunch of times, in fact, if we were to do that an infinite time, we would find that, especially if we had an infinite sample size, we would find a perfect normal distribution. That's the crazy thing. And it doesn't apply just to taking the sample mean. Here we took the sample mean every time, but you could have also taken the sample sum."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we were to do that a bunch of times, in fact, if we were to do that an infinite time, we would find that, especially if we had an infinite sample size, we would find a perfect normal distribution. That's the crazy thing. And it doesn't apply just to taking the sample mean. Here we took the sample mean every time, but you could have also taken the sample sum. The central limit theorem would have still applied. But that's what's so super useful about it. Because in life, there's all sorts of processes out there, proteins bumping into each other, people doing crazy things, humans interacting in weird ways, and you don't know the probability distribution functions for any of those things."}, {"video_title": "Central limit theorem   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here we took the sample mean every time, but you could have also taken the sample sum. The central limit theorem would have still applied. But that's what's so super useful about it. Because in life, there's all sorts of processes out there, proteins bumping into each other, people doing crazy things, humans interacting in weird ways, and you don't know the probability distribution functions for any of those things. But what the central limit theorem tells us is that if we add a bunch of those actions together, assuming that they all have the same distribution, or if we were to take the mean of all of those actions together, and if we were to plot the frequency of those means, we do get a normal distribution. And that's frankly why the normal distribution shows up so much in statistics, and why, frankly, it's a very good approximation for the sum or the means of a lot of processes. Normal distribution."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you can ignore the question right here. You can ignore all of this. I'm just using that same data to come up with a 95% confidence interval for the actual mean emission for this new engine design. So we want to find a 95% confidence interval. And as you can imagine, because we only have 10 samples right here, we're going to want to use a t distribution. And right down here, I have a t table. And we want a 95% confidence interval."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we want to find a 95% confidence interval. And as you can imagine, because we only have 10 samples right here, we're going to want to use a t distribution. And right down here, I have a t table. And we want a 95% confidence interval. So we want to think about the range of t values that 95% of t values will fall under. So let's think about it this way. So let me draw a t distribution."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we want a 95% confidence interval. So we want to think about the range of t values that 95% of t values will fall under. So let's think about it this way. So let me draw a t distribution. Let me draw a t distribution right over here. So a t distribution looks very similar to a normal distribution, but it has fatter tails. This end and this end will be fatter than in a normal distribution."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me draw a t distribution. Let me draw a t distribution right over here. So a t distribution looks very similar to a normal distribution, but it has fatter tails. This end and this end will be fatter than in a normal distribution. And then we want to find an interval. So this is a normalized t distribution. The mean is going to be 0."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This end and this end will be fatter than in a normal distribution. And then we want to find an interval. So this is a normalized t distribution. The mean is going to be 0. And we want to find an interval of t values between some negative value here and some positive value here that contains 95% of the probability. So this right here has to be 95%. And to figure what these critical t values are at this end and this end, we can just use a t table."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean is going to be 0. And we want to find an interval of t values between some negative value here and some positive value here that contains 95% of the probability. So this right here has to be 95%. And to figure what these critical t values are at this end and this end, we can just use a t table. And we're going to use the two-sided version of this because we're symmetric around the center. So you look at the two-sided. We want a 95% confidence interval."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to figure what these critical t values are at this end and this end, we can just use a t table. And we're going to use the two-sided version of this because we're symmetric around the center. So you look at the two-sided. We want a 95% confidence interval. So we're going to look right over here, 95% confidence interval. We have 10 data points, which means we have 9 degrees of freedom. So 9 degrees of freedom for our 10 data points."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want a 95% confidence interval. So we're going to look right over here, 95% confidence interval. We have 10 data points, which means we have 9 degrees of freedom. So 9 degrees of freedom for our 10 data points. We just took 10 minus 1. So if we look over here, for that type, so for a t distribution with 9 degrees of freedom, you're going to have 95% of the probability is going to be contained within a t value of, so the t value is going to be between negative. So this value right here is 2.262."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 9 degrees of freedom for our 10 data points. We just took 10 minus 1. So if we look over here, for that type, so for a t distribution with 9 degrees of freedom, you're going to have 95% of the probability is going to be contained within a t value of, so the t value is going to be between negative. So this value right here is 2.262. And this value right here is negative 2.262. That's what this right here tells us. That if you contain all the values that are less than 2.262 away from the center of your t distribution, you will contain 95% of the probability."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this value right here is 2.262. And this value right here is negative 2.262. That's what this right here tells us. That if you contain all the values that are less than 2.262 away from the center of your t distribution, you will contain 95% of the probability. So that is our t distribution right there. Let me make it very clear. This is our t distribution."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "That if you contain all the values that are less than 2.262 away from the center of your t distribution, you will contain 95% of the probability. So that is our t distribution right there. Let me make it very clear. This is our t distribution. So if you randomly pick a t value from this t distribution, it has a 95% chance of being within this far from the mean. Or maybe we should write it this way. If I pick a random t value, if I take a random t statistic, there's a 95% chance that a random t statistic is going to be less than 2.262 and greater than negative 2.262."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our t distribution. So if you randomly pick a t value from this t distribution, it has a 95% chance of being within this far from the mean. Or maybe we should write it this way. If I pick a random t value, if I take a random t statistic, there's a 95% chance that a random t statistic is going to be less than 2.262 and greater than negative 2.262. 95% chance. Now, when we took this sample, we can also derive a random t statistic from this. We have our sample mean and our sample standard deviation."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I pick a random t value, if I take a random t statistic, there's a 95% chance that a random t statistic is going to be less than 2.262 and greater than negative 2.262. 95% chance. Now, when we took this sample, we can also derive a random t statistic from this. We have our sample mean and our sample standard deviation. Our sample mean here is 17.17. Figured that out in the last video. Just add these up, divide by 10."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have our sample mean and our sample standard deviation. Our sample mean here is 17.17. Figured that out in the last video. Just add these up, divide by 10. And our sample standard deviation here is 2.98. So the t statistic that we can derive from this information right over here, so let me write it over here. The t statistic that we can derive from this."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just add these up, divide by 10. And our sample standard deviation here is 2.98. So the t statistic that we can derive from this information right over here, so let me write it over here. The t statistic that we can derive from this. And you can view this t statistic as being a random sample from a t distribution. A t distribution with 9 degrees of freedom. So the t statistic that we can derive from that is going to be our mean, 17.17, minus the true mean of our population."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The t statistic that we can derive from this. And you can view this t statistic as being a random sample from a t distribution. A t distribution with 9 degrees of freedom. So the t statistic that we can derive from that is going to be our mean, 17.17, minus the true mean of our population. Or actually, we say the true mean of our sampling distribution, which is also going to be the same as the true mean of our population. That's our population mean over there. Divided by s, which is 2.98, over the square root of our number of samples."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the t statistic that we can derive from that is going to be our mean, 17.17, minus the true mean of our population. Or actually, we say the true mean of our sampling distribution, which is also going to be the same as the true mean of our population. That's our population mean over there. Divided by s, which is 2.98, over the square root of our number of samples. We've seen this multiple times. This right here is the t statistic. So by taking this sample, you can say that we've randomly sampled a t statistic from this 9 degree of freedom t distribution."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Divided by s, which is 2.98, over the square root of our number of samples. We've seen this multiple times. This right here is the t statistic. So by taking this sample, you can say that we've randomly sampled a t statistic from this 9 degree of freedom t distribution. So there's a 95% chance that this thing right over here is going to be less than 2.262 and greater than negative 2.262. So the 95% probability still applies to this right here. Now, we just have to do some math, calculate these things."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So by taking this sample, you can say that we've randomly sampled a t statistic from this 9 degree of freedom t distribution. So there's a 95% chance that this thing right over here is going to be less than 2.262 and greater than negative 2.262. So the 95% probability still applies to this right here. Now, we just have to do some math, calculate these things. Let me get my calculator out. Let me just calculate this denominator right over here. So we have 2.98 divided by the square root of 10."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, we just have to do some math, calculate these things. Let me get my calculator out. Let me just calculate this denominator right over here. So we have 2.98 divided by the square root of 10. So that's 0.9423. So what I'm going to do is I'm going to multiply both sides of this equation by this expression right over here. So if I do that, let me just do that right over."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have 2.98 divided by the square root of 10. So that's 0.9423. So what I'm going to do is I'm going to multiply both sides of this equation by this expression right over here. So if I do that, let me just do that right over. So if I multiply this entire, this is really two equations, or two inequalities, I should say. That this quantity is greater than this quantity and that this quantity is greater than that quantity. But we can operate on all of them at the same time, this entire inequality."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I do that, let me just do that right over. So if I multiply this entire, this is really two equations, or two inequalities, I should say. That this quantity is greater than this quantity and that this quantity is greater than that quantity. But we can operate on all of them at the same time, this entire inequality. So what we want to do is multiply this entire inequality by this value right over here. And we just calculated that that value, let me write it over here, that 2.98, I'll write it over here, 2.98 over the square root of 10 is equal to 0.942. So if I multiply this entire inequality by 0.942, I get, on this left-hand side over here, I have negative 2.262 times 0.942."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we can operate on all of them at the same time, this entire inequality. So what we want to do is multiply this entire inequality by this value right over here. And we just calculated that that value, let me write it over here, that 2.98, I'll write it over here, 2.98 over the square root of 10 is equal to 0.942. So if I multiply this entire inequality by 0.942, I get, on this left-hand side over here, I have negative 2.262 times 0.942. And it's a positive number that we're multiplying the whole inequality by, so the inequality signs are still going to be in the same direction, is less than, well, we're multiplying this whole expression by the same expression in the denominator, so it'll cancel out. So we're just going to be less than 17.17 minus our population mean, which is going to be less than 2.262 times, once again, 0.942. Let me scroll over to the right a little bit."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I multiply this entire inequality by 0.942, I get, on this left-hand side over here, I have negative 2.262 times 0.942. And it's a positive number that we're multiplying the whole inequality by, so the inequality signs are still going to be in the same direction, is less than, well, we're multiplying this whole expression by the same expression in the denominator, so it'll cancel out. So we're just going to be less than 17.17 minus our population mean, which is going to be less than 2.262 times, once again, 0.942. Let me scroll over to the right a little bit. 0.942. Just to be clear, I'm just multiplying both or all three sides of this inequality by this number right over here. In the middle, this cancels out."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me scroll over to the right a little bit. 0.942. Just to be clear, I'm just multiplying both or all three sides of this inequality by this number right over here. In the middle, this cancels out. So if I multiply, I'll just write it right here, 0.942, 0.942, 0.942, this and this is the same number, so that's why those cancel out. And now let's hit the calculator to figure out what these numbers are. So if we have the 0.942 times 2.262, so we're going to say times 2.262 is 2.13."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the middle, this cancels out. So if I multiply, I'll just write it right here, 0.942, 0.942, 0.942, this and this is the same number, so that's why those cancel out. And now let's hit the calculator to figure out what these numbers are. So if we have the 0.942 times 2.262, so we're going to say times 2.262 is 2.13. So this is going to be, so this number right over here on the right-hand side, this number on the right-hand side is 2.13. This number on the left is just the negative of that, so it's negative 2.13. And then we still have our inequalities is going to be less than 17.17 minus the mean, which is less than 2.13."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if we have the 0.942 times 2.262, so we're going to say times 2.262 is 2.13. So this is going to be, so this number right over here on the right-hand side, this number on the right-hand side is 2.13. This number on the left is just the negative of that, so it's negative 2.13. And then we still have our inequalities is going to be less than 17.17 minus the mean, which is less than 2.13. Now what I want to do is I actually want to solve for this mean. And I don't like that negative sign in the mean. I'd rather have this swapped around."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we still have our inequalities is going to be less than 17.17 minus the mean, which is less than 2.13. Now what I want to do is I actually want to solve for this mean. And I don't like that negative sign in the mean. I'd rather have this swapped around. I'd rather have the mean minus 17.17. So what I'm going to do is multiply this entire inequality by negative 1. If you do that, if you multiply the entire thing times negative 1, this quantity right here, this negative 2.13 will become a positive 2.13."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'd rather have this swapped around. I'd rather have the mean minus 17.17. So what I'm going to do is multiply this entire inequality by negative 1. If you do that, if you multiply the entire thing times negative 1, this quantity right here, this negative 2.13 will become a positive 2.13. But since we are multiplying an inequality by a negative number, you have to swap the inequality sign. So this less than will become a greater than. This negative mu will become a positive mu."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you do that, if you multiply the entire thing times negative 1, this quantity right here, this negative 2.13 will become a positive 2.13. But since we are multiplying an inequality by a negative number, you have to swap the inequality sign. So this less than will become a greater than. This negative mu will become a positive mu. This positive 17.17 will become a negative 17.17. We have to swap this inequality sign as well. And this positive 2.13 will become a negative 2.13."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This negative mu will become a positive mu. This positive 17.17 will become a negative 17.17. We have to swap this inequality sign as well. And this positive 2.13 will become a negative 2.13. And we're almost there. We just want to solve for mu, have this inequality expressed in terms of mu. So what we can do is now just add 17.17 to all three sides of this inequality."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this positive 2.13 will become a negative 2.13. And we're almost there. We just want to solve for mu, have this inequality expressed in terms of mu. So what we can do is now just add 17.17 to all three sides of this inequality. And we are left with 2.13 plus 17.17 is greater than mu minus 17.17 plus 17.17 is just going to be mu, which is greater than. So this is greater than mu, which is greater than negative 2.13 plus 17.17. Or a more natural way to write it, since we actually have a bunch of greater than signs, that this is actually the largest number."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what we can do is now just add 17.17 to all three sides of this inequality. And we are left with 2.13 plus 17.17 is greater than mu minus 17.17 plus 17.17 is just going to be mu, which is greater than. So this is greater than mu, which is greater than negative 2.13 plus 17.17. Or a more natural way to write it, since we actually have a bunch of greater than signs, that this is actually the largest number. And this is actually the smallest number. And this over here is actually the largest number. It's actually flip."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or a more natural way to write it, since we actually have a bunch of greater than signs, that this is actually the largest number. And this is actually the smallest number. And this over here is actually the largest number. It's actually flip. You can just rewrite this inequality the other way. So now we can write. Well, actually, let's just figure out what these values are."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually flip. You can just rewrite this inequality the other way. So now we can write. Well, actually, let's just figure out what these values are. So we have 2.13 plus 17.17. So that is the high end of our range. So that is 19.3."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, actually, let's just figure out what these values are. So we have 2.13 plus 17.17. So that is the high end of our range. So that is 19.3. So this value right over here. So this is 19. Let me do it in that same color."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that is 19.3. So this value right over here. So this is 19. Let me do it in that same color. This value right here is 19.3 is going to be greater than mu, which is going to be greater than negative 2.13 plus 17.17. Or we could have 17.17 minus 2.13, which gives us 15.04. And remember, the whole thing, all of this, we started with, there was a 95% chance that a random t statistic will fall in this interval."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do it in that same color. This value right here is 19.3 is going to be greater than mu, which is going to be greater than negative 2.13 plus 17.17. Or we could have 17.17 minus 2.13, which gives us 15.04. And remember, the whole thing, all of this, we started with, there was a 95% chance that a random t statistic will fall in this interval. We had a random t statistic. And all we did is a bunch of math. So there's a 95% chance that any of these steps are true."}, {"video_title": "T-statistic confidence interval   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And remember, the whole thing, all of this, we started with, there was a 95% chance that a random t statistic will fall in this interval. We had a random t statistic. And all we did is a bunch of math. So there's a 95% chance that any of these steps are true. And so there's a 95% chance that this is true. There's a 95% chance that the true population mean, which is the same thing as the mean of the sampling distribution of the sample mean, there's a 95% chance, or we're confident that there's a 95% chance, that it will fall in this interval. And we're done."}, {"video_title": "Mean (expected value) of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And as you can see, X can take on only a finite number of values, zero, one, two, three, or four, and so because there's a finite number of values here, we would call this a discrete random variable. And you can see that this is a valid probability distribution because the combined probability is one. .1 plus 0.15 plus 0.4 plus 0.25 plus 0.1 is one, and none of these are negative probabilities, which wouldn't have made sense. But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X."}, {"video_title": "Mean (expected value) of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1."}, {"video_title": "Mean (expected value) of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4."}, {"video_title": "Mean (expected value) of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1."}, {"video_title": "Mean (expected value) of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers."}, {"video_title": "Mean (expected value) of a discrete random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. How can you have 2.1 workouts in a week? What is 0.1 of a workout? Well, this isn't saying that in a given week, you would expect me to work out exactly 2.1 times, but this is valuable because you could say, well, in 10 weeks, you would expect me to do roughly 21 workouts."}, {"video_title": "Reading bar charts basic example   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "At Hogwarts, there are four houses, Gryffindor, Hufflepuff, Ravenclaw, and Slytherin. The bar chart below shows the number of house points that each house received today. How many house points did Hufflepuff receive? So if we look at this bar chart, we can assume that this column right over here is Hufflepuff, because it's the only one that starts with an H. This will be Gryffindor. This is Ravenclaw. And this must be Slytherin. And so if we look at Huff and Puff's bar chart, it looks like they have three points scored today."}, {"video_title": "Reading bar charts basic example   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So if we look at this bar chart, we can assume that this column right over here is Hufflepuff, because it's the only one that starts with an H. This will be Gryffindor. This is Ravenclaw. And this must be Slytherin. And so if we look at Huff and Puff's bar chart, it looks like they have three points scored today. So let's put three points. Let's try one more. So here, they're saying how many house points did Slytherin receive?"}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "A club of nine people wants to choose a board of three officers. A president, a vice president, and a secretary. How many ways are there to choose the board from the nine people? Now, we're going to assume that one person can't hold more than one office. That if I'm picked for president, that I'm no longer a valid person for vice president or secretary. So let's just think about the three different positions. So you have the president, you have the vice president, VP, and then you have the secretary."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now, we're going to assume that one person can't hold more than one office. That if I'm picked for president, that I'm no longer a valid person for vice president or secretary. So let's just think about the three different positions. So you have the president, you have the vice president, VP, and then you have the secretary. Now, let's say that we go for the president first. It actually doesn't matter. Let's say we're picking the president slot first and we haven't appointed any other slots yet."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So you have the president, you have the vice president, VP, and then you have the secretary. Now, let's say that we go for the president first. It actually doesn't matter. Let's say we're picking the president slot first and we haven't appointed any other slots yet. How many possibilities are there for president? Well, the club has nine people, so there's nine possibilities for president. Now, we're going to pick one of those nine."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let's say we're picking the president slot first and we haven't appointed any other slots yet. How many possibilities are there for president? Well, the club has nine people, so there's nine possibilities for president. Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices, right? Because someone's going to be president. So one of the nine is going to be president."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices, right? Because someone's going to be president. So one of the nine is going to be president. There's nine possibilities, but one of the nine is going to be president. So you take that person aside. He or she is now the president."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So one of the nine is going to be president. There's nine possibilities, but one of the nine is going to be president. So you take that person aside. He or she is now the president. How many people are left to be vice president? Well, now there's only eight possible candidates for vice president. Eight possibilities."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "He or she is now the president. How many people are left to be vice president? Well, now there's only eight possible candidates for vice president. Eight possibilities. Now, he or she also goes aside. Now, how many people are left for secretary? Well, now there's only seven possibilities for secretary."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Eight possibilities. Now, he or she also goes aside. Now, how many people are left for secretary? Well, now there's only seven possibilities for secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the nine for president times the eight for vice president times the seven for secretary. You didn't have to do it this way. You could have picked secretary first and there would have been nine choices."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, now there's only seven possibilities for secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the nine for president times the eight for vice president times the seven for secretary. You didn't have to do it this way. You could have picked secretary first and there would have been nine choices. And then you could have picked vice president. There would have still been eight choices. Then you could have picked president last and there would have only been seven choices."}, {"video_title": "Example Ways to pick officers   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "You could have picked secretary first and there would have been nine choices. And then you could have picked vice president. There would have still been eight choices. Then you could have picked president last and there would have only been seven choices. But either way, you would have gotten nine times eight times seven. And that is, let's see, nine times eight is 72. 72 times seven is 14."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "A nutritionist wants to estimate the average caloric content of the burritos at a popular restaurant. They obtain a random sample of 14 burritos and measure their caloric content. Their sample data are roughly symmetric with a mean of 700 calories and a standard deviation of 50 calories. Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos? So pause this video and see if you can figure it out. All right, what's going on here? So there's a population of burritos here."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos? So pause this video and see if you can figure it out. All right, what's going on here? So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but doesn't know the true population parameter here, the population mean. And so they take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but doesn't know the true population parameter here, the population mean. And so they take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. And they wanna use this data to construct a 95% confidence interval. And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. And they wanna use this data to construct a 95% confidence interval. And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. And if we use that, if we used sigma, which is a population parameter, then we could use a z-statistic right over here. We would use a z-distribution."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. And if we use that, if we used sigma, which is a population parameter, then we could use a z-statistic right over here. We would use a z-distribution. But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be?"}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "We would use a z-distribution. But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus, plus or minus. So what would be our critical value for a 95% confidence interval?"}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus, plus or minus. So what would be our critical value for a 95% confidence interval? Well, we will just get out our t-table. And with a t-table, remember, you have to care about degrees of freedom. And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So what would be our critical value for a 95% confidence interval? Well, we will just get out our t-table. And with a t-table, remember, you have to care about degrees of freedom. And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. So that's gonna be 14 minus one is equal to 13. So we have 13 degrees of freedom that we have to keep in mind when we look at our t-table. So let's look at our t-table."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. So that's gonna be 14 minus one is equal to 13. So we have 13 degrees of freedom that we have to keep in mind when we look at our t-table. So let's look at our t-table. So 95% confidence interval and 13 degrees of freedom. So degrees of freedom right over here. So we have 13 degrees of freedom."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So let's look at our t-table. So 95% confidence interval and 13 degrees of freedom. So degrees of freedom right over here. So we have 13 degrees of freedom. So that is this row right over here. And if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see if I'll draw it really small, little small distribution right over here. So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1\u20442% at the tails, 2 1\u20442%."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So we have 13 degrees of freedom. So that is this row right over here. And if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see if I'll draw it really small, little small distribution right over here. So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1\u20442% at the tails, 2 1\u20442%. So what you wanna look for is a tail probability of 2 1\u20442%. So that is this right over here,.025, that's 2 1\u20442%. And so there you go, that is our critical value, 2.160."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1\u20442% at the tails, 2 1\u20442%. So what you wanna look for is a tail probability of 2 1\u20442%. So that is this right over here,.025, that's 2 1\u20442%. And so there you go, that is our critical value, 2.160. So this, so this part right over here, so this is going to be two, let me do that in a darker color. This is going to be 2.160 times, what's our sample standard deviation? It's 50 over the square root of n, square root of 14."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so there you go, that is our critical value, 2.160. So this, so this part right over here, so this is going to be two, let me do that in a darker color. This is going to be 2.160 times, what's our sample standard deviation? It's 50 over the square root of n, square root of 14. So all of our choices have the 700 there. So we just need to figure out what our margin of error, this part of it, and we could use a calculator for that. Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "It's 50 over the square root of n, square root of 14. So all of our choices have the 700 there. So we just need to figure out what our margin of error, this part of it, and we could use a calculator for that. Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here, I think. 28.86, so this part right over here is approximately 28.86. That's our margin of error."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here, I think. 28.86, so this part right over here is approximately 28.86. That's our margin of error. And we see out of all of these choices here, if we round to the nearest tenth, that'd be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "That's our margin of error. And we see out of all of these choices here, if we round to the nearest tenth, that'd be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. So there we have it. We have established our 95% confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval?"}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. So there we have it. We have established our 95% confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval? And here we have to think, well, did we take a random sample and they tell us that they obtained a random sample of 14 burritos, so we check that one. Is the sampling distribution roughly normal?"}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval? And here we have to think, well, did we take a random sample and they tell us that they obtained a random sample of 14 burritos, so we check that one. Is the sampling distribution roughly normal? Well, either you take, if you take over 30 samples, then it would be, but here we only took 14. But they do tell us that the sample data is roughly symmetric. And so if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal."}, {"video_title": "Example constructing a t interval for a mean   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Is the sampling distribution roughly normal? Well, either you take, if you take over 30 samples, then it would be, but here we only took 14. But they do tell us that the sample data is roughly symmetric. And so if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal. And then the last condition is the independence condition. And here, if we aren't sampling with replacement, and it doesn't look like we are, if we're not sampling with replacement, this has to be less than 10% of, this has to be less than 10% of the population of burritos. And we're assuming that there's going to be more than 140 burritos that the universe, that the population, that this popular restaurant makes."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "In previous videos, we took this bivariate data and we calculated the correlation coefficient. And just as a bit of a review, we have the formula here. And it looks a bit intimidating, but in that video we saw all it is is an average of the product of the z-scores for each of those pairs. And as we said, if r is equal to one, you have a perfect positive correlation. If r is equal to negative one, you have a perfect negative correlation. And if r is equal to zero, you don't have a correlation. But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And as we said, if r is equal to one, you have a perfect positive correlation. If r is equal to negative one, you have a perfect negative correlation. And if r is equal to zero, you don't have a correlation. But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points. So before I do that, let's just visualize some of the statistics that we have here for these data points. We clearly have the four data points plotted, but let's plot the statistics for x."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points. So before I do that, let's just visualize some of the statistics that we have here for these data points. We clearly have the four data points plotted, but let's plot the statistics for x. So the sample mean and the sample standard deviation for x are here in red. And actually, let me box these off in red so that you know that's what is going on here. So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "We clearly have the four data points plotted, but let's plot the statistics for x. So the sample mean and the sample standard deviation for x are here in red. And actually, let me box these off in red so that you know that's what is going on here. So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. So we have x equals two right over here. And then this is one sample standard deviation above the mean. This is one sample standard deviation below the mean."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. So we have x equals two right over here. And then this is one sample standard deviation above the mean. This is one sample standard deviation below the mean. And then we could do the same thing for the y variables. So the mean is three. And this is one sample standard deviation for y above the mean."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "This is one sample standard deviation below the mean. And then we could do the same thing for the y variables. So the mean is three. And this is one sample standard deviation for y above the mean. And this is one sample standard deviation for y below the mean. And visualizing these means, especially their intersection, and also their standard deviations, will help us build an intuition for the equation of the least squares line. So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And this is one sample standard deviation for y above the mean. And this is one sample standard deviation for y below the mean. And visualizing these means, especially their intersection, and also their standard deviations, will help us build an intuition for the equation of the least squares line. So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. For the regression line, we'll put a little hat over it. So this, you would literally say y hat. This tells you that this is a regression line that we're trying to fit to these points."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. For the regression line, we'll put a little hat over it. So this, you would literally say y hat. This tells you that this is a regression line that we're trying to fit to these points. First, what is going to be the slope? Well, the slope is going to be r times the ratio between the sample standard deviation in the y direction over the sample standard deviation in the x direction. This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "This tells you that this is a regression line that we're trying to fit to these points. First, what is going to be the slope? Well, the slope is going to be r times the ratio between the sample standard deviation in the y direction over the sample standard deviation in the x direction. This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept? Well, like you first learned in Algebra I, you can calculate the y-intercept if you already know the slope by saying, well, what point is definitely going to be on my line? And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept? Well, like you first learned in Algebra I, you can calculate the y-intercept if you already know the slope by saying, well, what point is definitely going to be on my line? And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. So you're definitely going to go through that point. So before I even calculate for this particular example where in previous videos we calculated the r to be 0.946, or roughly equal to that, let's just think about what's going on. So our least squares line is definitely going to go through that point."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. So you're definitely going to go through that point. So before I even calculate for this particular example where in previous videos we calculated the r to be 0.946, or roughly equal to that, let's just think about what's going on. So our least squares line is definitely going to go through that point. Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x. So if you were to start at this point, and if you were to run your standard deviation of x and rise your standard deviation of y, well, with a perfect positive correlation, your line would look like this. And that makes a lot of sense because you're looking at your spread of y over your spread of x."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So our least squares line is definitely going to go through that point. Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x. So if you were to start at this point, and if you were to run your standard deviation of x and rise your standard deviation of y, well, with a perfect positive correlation, your line would look like this. And that makes a lot of sense because you're looking at your spread of y over your spread of x. If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x. That has parallels to when you first learn about slope. Change in y over change in x."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And that makes a lot of sense because you're looking at your spread of y over your spread of x. If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x. That has parallels to when you first learn about slope. Change in y over change in x. Here you're seeing the, you could say the average spread in y over the average spread in x. And this would be the case when r is one. So let me write that down."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Change in y over change in x. Here you're seeing the, you could say the average spread in y over the average spread in x. And this would be the case when r is one. So let me write that down. This would be the case if r is equal to one. What if r were equal to negative one? It would look like this."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So let me write that down. This would be the case if r is equal to one. What if r were equal to negative one? It would look like this. That would be our line if we had a perfect negative correlation. Now what if r were zero? Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "It would look like this. That would be our line if we had a perfect negative correlation. Now what if r were zero? Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. So you would just go through that right over there. But now let's think about this scenario. In this scenario, our r is 0.946."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. So you would just go through that right over there. But now let's think about this scenario. In this scenario, our r is 0.946. So we have a fairly strong correlation. This is pretty close to one. And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y?"}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "In this scenario, our r is 0.946. So we have a fairly strong correlation. This is pretty close to one. And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? Well, you would move up r times the standard deviation of y. And as we said, if r was one, you would get all the way up to this perfect correlation line but here it's 0.946. So you would get up about 95% of the way to that."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? Well, you would move up r times the standard deviation of y. And as we said, if r was one, you would get all the way up to this perfect correlation line but here it's 0.946. So you would get up about 95% of the way to that. And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points. I'm not proving it here in this video. But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So you would get up about 95% of the way to that. And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points. I'm not proving it here in this video. But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. It actually makes intuitive sense. Let's calculate it for this particular set of data. M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. It actually makes intuitive sense. Let's calculate it for this particular set of data. M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. We can get our calculator out to calculate that. So we have 0.946 times 2.160 divided by 0.816. It gets us to 2.50."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. We can get our calculator out to calculate that. So we have 0.946 times 2.160 divided by 0.816. It gets us to 2.50. Let's just round to the nearest hundredth for simplicity here. So this is approximately equal to 2.50. And so how do we figure out the y-intercept?"}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "It gets us to 2.50. Let's just round to the nearest hundredth for simplicity here. So this is approximately equal to 2.50. And so how do we figure out the y-intercept? Well, remember, we go through this point. So we're going to have 2.50 times our x-mean. So our x-mean is two times two."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And so how do we figure out the y-intercept? Well, remember, we go through this point. So we're going to have 2.50 times our x-mean. So our x-mean is two times two. Remember, this right over here is our x-mean. Plus b is going to be equal to our y-mean. Our y-mean, we see right over here, is three."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So our x-mean is two times two. Remember, this right over here is our x-mean. Plus b is going to be equal to our y-mean. Our y-mean, we see right over here, is three. And so what do we get? We get three is equal to five plus b. Five plus b."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Our y-mean, we see right over here, is three. And so what do we get? We get three is equal to five plus b. Five plus b. And so what is b? Well, if you subtract five from both sides, you get b is equal to negative two. And so there you have it."}, {"video_title": "Calculating the equation of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Five plus b. And so what is b? Well, if you subtract five from both sides, you get b is equal to negative two. And so there you have it. The equation for our regression line. We deserve a little bit of a drum roll here. We would say y-hat, the hat tells us that this is the equation for a regression line, is equal to 2.50 times x minus two."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's ask ourselves some interesting questions about alphabets in the English language. And in case you don't remember, or are in the mood to count, there are 26 alphabets. So if you go A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, and Z, you get 26. 26 alphabets. Now let's ask some interesting questions. So given that there are 26 alphabets in the English language, how many possible three-letter words are there? And we're not gonna be thinking about phonetics or how hard it is to pronounce it."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "26 alphabets. Now let's ask some interesting questions. So given that there are 26 alphabets in the English language, how many possible three-letter words are there? And we're not gonna be thinking about phonetics or how hard it is to pronounce it. So for example, the word, the word zigget would be a legitimate word in this example. Or the word, the word, the word skudge would be a legitimate word in this example. So how many possible three-letter words are there in the English language?"}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're not gonna be thinking about phonetics or how hard it is to pronounce it. So for example, the word, the word zigget would be a legitimate word in this example. Or the word, the word, the word skudge would be a legitimate word in this example. So how many possible three-letter words are there in the English language? I encourage you to pause the video and try to think about it. All right, I assume you've had a go at it. So let's just think about it."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many possible three-letter words are there in the English language? I encourage you to pause the video and try to think about it. All right, I assume you've had a go at it. So let's just think about it. For three-letter words, there's three spaces. So how many possibilities are there for the first one? Well, there's 26 possible letters for the first one."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just think about it. For three-letter words, there's three spaces. So how many possibilities are there for the first one? Well, there's 26 possible letters for the first one. Anything from A to Z would be completely fine. Now how many possibilities for the second one? And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's 26 possible letters for the first one. Anything from A to Z would be completely fine. Now how many possibilities for the second one? And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. But that's not the case right over here, because we can repeat letters. I didn't say that all the letters had to be different. So for example, the word, the word, ha, would also be a legitimate word in our example right over here."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. But that's not the case right over here, because we can repeat letters. I didn't say that all the letters had to be different. So for example, the word, the word, ha, would also be a legitimate word in our example right over here. So we have 26 possibilities for the second letter, and we have 26 possibilities for the third letter. So we're going to have, and I don't know what this is, 26 to the third power possibilities, or 26 times 26 times 26, and you can figure out what that is. That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, the word, the word, ha, would also be a legitimate word in our example right over here. So we have 26 possibilities for the second letter, and we have 26 possibilities for the third letter. So we're going to have, and I don't know what this is, 26 to the third power possibilities, or 26 times 26 times 26, and you can figure out what that is. That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. Now let's ask a different question. What if we said, how many possible three-letter words are there if we want all different letters? So we want all different letters."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. Now let's ask a different question. What if we said, how many possible three-letter words are there if we want all different letters? So we want all different letters. So these all have to be different letters. Different, different letters. And once again, pause the video and see if you can think it through."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we want all different letters. So these all have to be different letters. Different, different letters. And once again, pause the video and see if you can think it through. All right. So this is where permutations start to be useful. Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "And once again, pause the video and see if you can think it through. All right. So this is where permutations start to be useful. Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. So in this situation, well, if we went in order, we could have 26 different letters for the first one, 26 different possibilities for the first one. You know, I'm always starting with that one, but there's nothing special about the one on the left. We could say that the one on the right, there's 26 possibilities."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. So in this situation, well, if we went in order, we could have 26 different letters for the first one, 26 different possibilities for the first one. You know, I'm always starting with that one, but there's nothing special about the one on the left. We could say that the one on the right, there's 26 possibilities. Well, for each of those possibilities, for each of those 26 possibilities, there might be 25 possibilities for what we put in the middle one if we say we're gonna figure out the middle one next. And then for each of these 25 times 26 possibilities for where we figured out two of the letters, there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled. And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could say that the one on the right, there's 26 possibilities. Well, for each of those possibilities, for each of those 26 possibilities, there might be 25 possibilities for what we put in the middle one if we say we're gonna figure out the middle one next. And then for each of these 25 times 26 possibilities for where we figured out two of the letters, there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled. And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. It's just about, well, let's just think in terms of, let's fill out one of the buckets first. Hey, we have the most possibilities for that. Once we use something up, then for each of those possibilities, we'll have one less for the next bucket."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. It's just about, well, let's just think in terms of, let's fill out one of the buckets first. Hey, we have the most possibilities for that. Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. And so I could do 24 times 25 times 26, but just so I don't fully confuse you, I'll go back to what I have been doing. 26 possibilities for the leftmost one. For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. And so I could do 24 times 25 times 26, but just so I don't fully confuse you, I'll go back to what I have been doing. 26 possibilities for the leftmost one. For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. And then for the last bucket, you're going to have 24 possibilities. So this is going to be 26 times 25 times 24, whatever that happens to be. And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. And then for the last bucket, you're going to have 24 possibilities. So this is going to be 26 times 25 times 24, whatever that happens to be. And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. And this is 26 if we just blindly apply the formula, which I never suggest doing. It'd be 26 factorial over 26 minus three factorial, which would be 26 factorial over 23 factorial, which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial. And so the whole point of this video, there's two points, is one, as soon as someone says, oh, how many different letters could you form or something like that, you just don't blindly do permutations or combinations."}, {"video_title": "Possible three letter words   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. And this is 26 if we just blindly apply the formula, which I never suggest doing. It'd be 26 factorial over 26 minus three factorial, which would be 26 factorial over 23 factorial, which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial. And so the whole point of this video, there's two points, is one, as soon as someone says, oh, how many different letters could you form or something like that, you just don't blindly do permutations or combinations. You think about, well, what is being asked in the question here? I really just have to take 26 times 26 times 26. The other thing I want to point out, and I know I keep pointing it out and it's probably getting tiring to you, is even when permutations are applicable, in my brain at least, it's always more valuable to just try to reason through the problem as opposed to just saying, oh, there's this formula that I remember from weeks or years ago in my life that had n factorial and k factorial and I have to memorize it, I have to look it up, always much more useful to just reason it through."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "A random sample of 200 computers show that 12 computers have the defect. What critical value, z star, should Elena use to construct this confidence interval? So before I even ask you to pause this video, let me just give you a little reminder of what a critical value is. Remember, the whole point behind confidence intervals are we have some true population parameter. In this case, it is the proportion of computers that have a defect. So there's some true population proportion. We don't know what that is, but we try to estimate it."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "Remember, the whole point behind confidence intervals are we have some true population parameter. In this case, it is the proportion of computers that have a defect. So there's some true population proportion. We don't know what that is, but we try to estimate it. We take a sample. In this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "We don't know what that is, but we try to estimate it. We take a sample. In this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion. But then we also wanna construct a confidence interval. And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one. Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "We take a random sample, and then we estimate this by calculating the sample proportion. But then we also wanna construct a confidence interval. And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one. Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic. And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic. And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value. And then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "So the number of standard deviations we wanna go, that is our critical value. And then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star. And we're gonna think about which z star, because that's essentially the question, the critical value. So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "So it's that one sample proportion that she was able to calculate, plus or minus z star. And we're gonna think about which z star, because that's essentially the question, the critical value. So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area. So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area. So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. Now, all we have to really do is look it up on a z table, but even there we have to be careful. And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. Now, all we have to really do is look it up on a z table, but even there we have to be careful. And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "Because a lot of z tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%. But remember, this is going to be symmetric on the left and the right."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%. But remember, this is going to be symmetric on the left and the right. So you're gonna want 3% not shaded in over here, and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%. But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "But remember, this is going to be symmetric on the left and the right. So you're gonna want 3% not shaded in over here, and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%. But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So with that out of the way, let's look that up. What z gives us, fills us, fills in 97% of the area? So I got a z table."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So with that out of the way, let's look that up. What z gives us, fills us, fills in 97% of the area? So I got a z table. This is actually the one that you would see if you were, say, taking AP Statistics. And we would just look up, where do we get to 97%? And so it is 97%, looks like it is right about here."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "So I got a z table. This is actually the one that you would see if you were, say, taking AP Statistics. And we would just look up, where do we get to 97%? And so it is 97%, looks like it is right about here. That looks like the closest number. This is 6 10,000ths above it. This is only 1 10,000th below it."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "And so it is 97%, looks like it is right about here. That looks like the closest number. This is 6 10,000ths above it. This is only 1 10,000th below it. And so this is, let's see, you would look at the row first. If we look at the row, it is 1.8, 1.88 is our z. So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "This is only 1 10,000th below it. And so this is, let's see, you would look at the row first. If we look at the row, it is 1.8, 1.88 is our z. So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side. And so this would be 94%. So this would be 94%."}, {"video_title": "Critical value (z ) for a given confidence level   AP Statistics   Khan Academy.mp3", "Sentence": "So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side. And so this would be 94%. So this would be 94%. But to answer their question, what critical value z star? Well, this is going to be 1.88. And we're done."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll assume it's a quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of..."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got heads on the second flip. So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got heads on the second flip. So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. We know the probability of getting heads on the first flip is one-half, and the probability of getting heads on the second flip is one-half. And so we have one-half times one-half, which is equal to 1 4th, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. We know the probability of getting heads on the first flip is one-half, and the probability of getting heads on the second flip is one-half. And so we have one-half times one-half, which is equal to 1 4th, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails, so this exact series of events. So I'm not saying in any order, two tails and a head."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails, so this exact series of events. So I'm not saying in any order, two tails and a head. I'm saying in this exact order, the first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm not saying in any order, two tails and a head. I'm saying in this exact order, the first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events. So this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we know these are all independent events. So this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all over the different scenarios again."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all over the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's try it all over the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, heads."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. We have 8 equally likely possibilities."}, {"video_title": "Compound probability of independent events   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could get tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. We have 8 equally likely possibilities. And the tail, heads, tails is exactly one of them. It is this possibility right over here. So it is one of 8 equally likely possibilities."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "It's all for free. So problem number four, and it's, at least in my mind, pretty good practice. For a standard normal distribution, place the following in order from smallest to largest. So let's see, percentage of data below 1, negative 1. OK, let's draw our standard normal distribution. So a standard normal distribution is one where the mean is, sorry, that's due to the standard deviation, is one where the mean, mu for mean, is where the mean is equal to 0 and the standard deviation is equal to 1. So let me draw that standard normal distribution."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So let's see, percentage of data below 1, negative 1. OK, let's draw our standard normal distribution. So a standard normal distribution is one where the mean is, sorry, that's due to the standard deviation, is one where the mean, mu for mean, is where the mean is equal to 0 and the standard deviation is equal to 1. So let me draw that standard normal distribution. So let me draw the axis right like that. Let me see if I can draw a nice looking bell curve. There's the bell curve right there."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So let me draw that standard normal distribution. So let me draw the axis right like that. Let me see if I can draw a nice looking bell curve. There's the bell curve right there. You get the idea. And this is a standard normal distribution, so the mean, or you can kind of view the center point right here. It's not skewed."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "There's the bell curve right there. You get the idea. And this is a standard normal distribution, so the mean, or you can kind of view the center point right here. It's not skewed. This is the mean is going to be 0 right there. And the standard deviation is 1. So if we go one standard deviation to the right, that is going to be 1."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "It's not skewed. This is the mean is going to be 0 right there. And the standard deviation is 1. So if we go one standard deviation to the right, that is going to be 1. If you go two standard deviations, it's going to be 2, three standard deviations, 3, just like that. One standard deviation to the left is going to be minus 1. Two standard deviations to the left will be minus 2, and so on and so forth."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So if we go one standard deviation to the right, that is going to be 1. If you go two standard deviations, it's going to be 2, three standard deviations, 3, just like that. One standard deviation to the left is going to be minus 1. Two standard deviations to the left will be minus 2, and so on and so forth. Minus 3 will be three standard deviations to the left, because the standard deviation is 1. So let's see if we can answer this question. So what's the percentage of data below 1?"}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Two standard deviations to the left will be minus 2, and so on and so forth. Minus 3 will be three standard deviations to the left, because the standard deviation is 1. So let's see if we can answer this question. So what's the percentage of data below 1? So the percentage, the part A, that's this stuff right here. So everything below 1, so it's all of, well not just that little center portion, it keeps going. Everything below 1, right?"}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So what's the percentage of data below 1? So the percentage, the part A, that's this stuff right here. So everything below 1, so it's all of, well not just that little center portion, it keeps going. Everything below 1, right? Percentage of data below 1. So this is another situation where we should use the empirical rule. Never hurts to get more practice."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Everything below 1, right? Percentage of data below 1. So this is another situation where we should use the empirical rule. Never hurts to get more practice. Empirical rule, or maybe the better way to remember the empirical rule is just the 68, 95, 99.7 rule. And I call that a better way because it essentially gives you the rule. These are just the numbers that you have to essentially memorize."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Never hurts to get more practice. Empirical rule, or maybe the better way to remember the empirical rule is just the 68, 95, 99.7 rule. And I call that a better way because it essentially gives you the rule. These are just the numbers that you have to essentially memorize. And if you have a calculator or normal distribution table, you don't have to do this. But sometimes in class, or people want you to estimate percentages, and so it's good to do, you can impress people if you can do this in your head. So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "These are just the numbers that you have to essentially memorize. And if you have a calculator or normal distribution table, you don't have to do this. But sometimes in class, or people want you to estimate percentages, and so it's good to do, you can impress people if you can do this in your head. So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1. So the empirical rule tells us that this middle area between one standard deviation to the left and one standard deviation to the right, that right there is 68%. We saw that in the previous video as well. That's what the empirical rule tells us."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1. So the empirical rule tells us that this middle area between one standard deviation to the left and one standard deviation to the right, that right there is 68%. We saw that in the previous video as well. That's what the empirical rule tells us. Now, if that 68% we saw in the last video, that everything else combined, it all has to add up to 1, or to 100%, that this left-hand tail, let me draw it a little bit, this part right here, plus this part right here, has to add up, when you add it to 68, has to add up to 1, or to 100%. So those two combined are 32%. 32 plus 68 is 100."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "That's what the empirical rule tells us. Now, if that 68% we saw in the last video, that everything else combined, it all has to add up to 1, or to 100%, that this left-hand tail, let me draw it a little bit, this part right here, plus this part right here, has to add up, when you add it to 68, has to add up to 1, or to 100%. So those two combined are 32%. 32 plus 68 is 100. Now, this is symmetrical. These two things are the exact same. So if they add up to 32, this right here is 16%, and this right here is 16%."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "32 plus 68 is 100. Now, this is symmetrical. These two things are the exact same. So if they add up to 32, this right here is 16%, and this right here is 16%. Now, the question, they want us to know the area of everything, let me do it in a new color, everything less than 1. The percentage of data below 1. So everything to the left of this point."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So if they add up to 32, this right here is 16%, and this right here is 16%. Now, the question, they want us to know the area of everything, let me do it in a new color, everything less than 1. The percentage of data below 1. So everything to the left of this point. So it's the 68%, it's right there, so it's 68%, which is this middle area within one standard deviation, plus this left branch right there. So 68 plus 16%, which is what? That's equal to 84%."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So everything to the left of this point. So it's the 68%, it's right there, so it's 68%, which is this middle area within one standard deviation, plus this left branch right there. So 68 plus 16%, which is what? That's equal to 84%. So this part A is 84%. They're going to want us to put it in order eventually, but it's good to just solve it. That's really the hard part."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "That's equal to 84%. So this part A is 84%. They're going to want us to put it in order eventually, but it's good to just solve it. That's really the hard part. Once we know the numbers, ordering is pretty easy. Part B. The percentage of data below minus 1."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "That's really the hard part. Once we know the numbers, ordering is pretty easy. Part B. The percentage of data below minus 1. So minus 1 is right there. So they really just want us to figure out this area right here, the percentage of data below minus 1. Well, that's going to be 16%."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "The percentage of data below minus 1. So minus 1 is right there. So they really just want us to figure out this area right here, the percentage of data below minus 1. Well, that's going to be 16%. We just figured that out. And you could have already known, just without even knowing the empirical, just looking at a normal distribution, that this entire area, that part B is a subset of part A, so it's going to be a smaller number. So if you just had to order things, you could have made that intuition, but it's good to do practice with the empirical rule."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Well, that's going to be 16%. We just figured that out. And you could have already known, just without even knowing the empirical, just looking at a normal distribution, that this entire area, that part B is a subset of part A, so it's going to be a smaller number. So if you just had to order things, you could have made that intuition, but it's good to do practice with the empirical rule. Now, part C. They want to know what's the mean. Well, that's the easiest thing. The mean of a standard normal distribution, by definition, is 0."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So if you just had to order things, you could have made that intuition, but it's good to do practice with the empirical rule. Now, part C. They want to know what's the mean. Well, that's the easiest thing. The mean of a standard normal distribution, by definition, is 0. So number C is 0. D. The standard deviation. Well, by definition, the standard deviation for the standard normal distribution is 1."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "The mean of a standard normal distribution, by definition, is 0. So number C is 0. D. The standard deviation. Well, by definition, the standard deviation for the standard normal distribution is 1. So this is 1 right here. This is easier than I thought it would be. Part E. The percentage of data above 2."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Well, by definition, the standard deviation for the standard normal distribution is 1. So this is 1 right here. This is easier than I thought it would be. Part E. The percentage of data above 2. So they want the percentage of data above 2. So we know from the 68, 95, 99.7 rule that if we want to know how much data is within 2 standard deviations. So let me do it in a new color."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Part E. The percentage of data above 2. So they want the percentage of data above 2. So we know from the 68, 95, 99.7 rule that if we want to know how much data is within 2 standard deviations. So let me do it in a new color. So if we're looking for, from this, let me do it in a more vibrant color, green. If we're looking from this point to this point, so it's within 2 standard deviations, right? The standard deviation here is 1."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So let me do it in a new color. So if we're looking for, from this, let me do it in a more vibrant color, green. If we're looking from this point to this point, so it's within 2 standard deviations, right? The standard deviation here is 1. If we're looking within 2 standard deviations, that whole area right there, by the empirical rule, is 95% within 2 standard deviations. This is 95%. Which tells us that everything else combined."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "The standard deviation here is 1. If we're looking within 2 standard deviations, that whole area right there, by the empirical rule, is 95% within 2 standard deviations. This is 95%. Which tells us that everything else combined. So if you take this yellow portion right here, and this yellow portion right here. So everything beyond 2 standard deviations in either direction, that has to be the remainder. So everything in the middle was 95 within 2 standard deviations."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Which tells us that everything else combined. So if you take this yellow portion right here, and this yellow portion right here. So everything beyond 2 standard deviations in either direction, that has to be the remainder. So everything in the middle was 95 within 2 standard deviations. So that has to be 5%. If you add that tail and that tail together, everything to the left and right of 2 standard deviations. Well, I've made the argument before."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So everything in the middle was 95 within 2 standard deviations. So that has to be 5%. If you add that tail and that tail together, everything to the left and right of 2 standard deviations. Well, I've made the argument before. Everything is symmetrical. This and this are equal. So this right here is 2.5%."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Well, I've made the argument before. Everything is symmetrical. This and this are equal. So this right here is 2.5%. And this right here is also 2.5%. So they're asking us the percentage of data above 2. That's this tail."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So this right here is 2.5%. And this right here is also 2.5%. So they're asking us the percentage of data above 2. That's this tail. Just this tail right here. The percentage of data more than 2 standard deviations away from the mean. So that's 2.5%."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "That's this tail. Just this tail right here. The percentage of data more than 2 standard deviations away from the mean. So that's 2.5%. I'll do it in a darker color. 2.5%. Now they're asking us, let's see, place the following in order from smallest to largest."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So that's 2.5%. I'll do it in a darker color. 2.5%. Now they're asking us, let's see, place the following in order from smallest to largest. So there's a little bit of ambiguity here. Because if they're saying the percentage of data below 1, do they want us to say, well, it's 84%. So should we consider the answer to part A, 84?"}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Now they're asking us, let's see, place the following in order from smallest to largest. So there's a little bit of ambiguity here. Because if they're saying the percentage of data below 1, do they want us to say, well, it's 84%. So should we consider the answer to part A, 84? Or should we consider, if they said the fraction of data below 1, I would say 0.84. So it depends on how they want to interpret it. Same way here."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So should we consider the answer to part A, 84? Or should we consider, if they said the fraction of data below 1, I would say 0.84. So it depends on how they want to interpret it. Same way here. The percentage of data below minus 1. I could say the answer is 16. 16 is the percentage below minus 1."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Same way here. The percentage of data below minus 1. I could say the answer is 16. 16 is the percentage below minus 1. But the actual number, if I said the fraction of data below minus 1, I would say 0.16. So this actually would be very different in how we order it. Similarly, if someone asked me the percentage, I'd say, oh, that's 2.5."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "16 is the percentage below minus 1. But the actual number, if I said the fraction of data below minus 1, I would say 0.16. So this actually would be very different in how we order it. Similarly, if someone asked me the percentage, I'd say, oh, that's 2.5. But the actual number is 0.025. That's the actual fraction or the actual decimal. So I mean, this is just ordering numbers."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Similarly, if someone asked me the percentage, I'd say, oh, that's 2.5. But the actual number is 0.025. That's the actual fraction or the actual decimal. So I mean, this is just ordering numbers. So I shouldn't fixate on this too much. But let's just say that they're going with the decimals. So if we wanted to do it that way, they want to do it from smallest to largest."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So I mean, this is just ordering numbers. So I shouldn't fixate on this too much. But let's just say that they're going with the decimals. So if we wanted to do it that way, they want to do it from smallest to largest. The smallest number we have here is c. That's 0. And then the next smallest is e, which is 0.025. Then the next smallest is b, which is 0.16."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "So if we wanted to do it that way, they want to do it from smallest to largest. The smallest number we have here is c. That's 0. And then the next smallest is e, which is 0.025. Then the next smallest is b, which is 0.16. And then the next one after that is a, which is 0.84. And then the largest would be the standard deviation, d. So the answer is c bad. And obviously, the order would be different if the answer to this, instead of saying it was 0.84, if you said it was 84, because you're asking for the percentage."}, {"video_title": "k12.org exercise Standard normal distribution and the empirical   Khan Academy.mp3", "Sentence": "Then the next smallest is b, which is 0.16. And then the next one after that is a, which is 0.84. And then the largest would be the standard deviation, d. So the answer is c bad. And obviously, the order would be different if the answer to this, instead of saying it was 0.84, if you said it was 84, because you're asking for the percentage. So a little bit of ambiguity. If you had a question like this on the exam, I would clarify that with the teacher. But hopefully you found this useful."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So for example, a 1 and a 1, that's doubles. 2 and a 2, that is doubles. A 3 and a 3, a 4 and a 4, a 5 and a 5, a 6 and a 6. All of those are instances of doubles. So the event in question is rolling doubles on two six-sided dice numbered from 1 to 6. So let's think about all of the possible outcomes. Or another way to think about it, let's think about the sample space here."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "All of those are instances of doubles. So the event in question is rolling doubles on two six-sided dice numbered from 1 to 6. So let's think about all of the possible outcomes. Or another way to think about it, let's think about the sample space here. So what can we roll on the first die? So let me write this as die number 1. What are the possible rolls?"}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Or another way to think about it, let's think about the sample space here. So what can we roll on the first die? So let me write this as die number 1. What are the possible rolls? Well, they're numbered from 1 to 6. It's a six-sided die. So I can get a 1, a 2, a 3, a 4, a 5, or a 6."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "What are the possible rolls? Well, they're numbered from 1 to 6. It's a six-sided die. So I can get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the second die. So die number 2. Well, exact same thing."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So I can get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the second die. So die number 2. Well, exact same thing. I could get a 1, a 2, a 3, a 4, a 5, or a 6. Now, given these possible outcomes for each of the die, we can now think of the outcomes for both die. So for example, in this, let me draw a grid here, just to make it a little bit neater."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Well, exact same thing. I could get a 1, a 2, a 3, a 4, a 5, or a 6. Now, given these possible outcomes for each of the die, we can now think of the outcomes for both die. So for example, in this, let me draw a grid here, just to make it a little bit neater. So let me draw a line there, and then a line right over there. Let me draw, actually, several of these, just so that we can really do this a little bit clearer. So let me draw a full grid."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So for example, in this, let me draw a grid here, just to make it a little bit neater. So let me draw a line there, and then a line right over there. Let me draw, actually, several of these, just so that we can really do this a little bit clearer. So let me draw a full grid. All right. And then let me draw the vertical lines. Only a few more left."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let me draw a full grid. All right. And then let me draw the vertical lines. Only a few more left. There we go. Now, all of this top row, these are the outcomes where I roll a 1 on the first die. So I roll a 1 on the first die."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Only a few more left. There we go. Now, all of this top row, these are the outcomes where I roll a 1 on the first die. So I roll a 1 on the first die. These are all of those outcomes. And this would be I run a 1 on the second die, but I'll fill that in later. These are all the outcomes where I roll a 2 on the first die."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So I roll a 1 on the first die. These are all of those outcomes. And this would be I run a 1 on the second die, but I'll fill that in later. These are all the outcomes where I roll a 2 on the first die. This is where I roll a 3 on the first die. 4, I think you get the idea, on the first die. And then a 5 on the first die."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "These are all the outcomes where I roll a 2 on the first die. This is where I roll a 3 on the first die. 4, I think you get the idea, on the first die. And then a 5 on the first die. 5. And then finally, this last row is all the outcomes where I roll a 6 on the first die. Now, we can go through the columns."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then a 5 on the first die. 5. And then finally, this last row is all the outcomes where I roll a 6 on the first die. Now, we can go through the columns. And this first column is where we roll a 1 on the second die. This is where we roll a 2 on the second die. Let's draw that out, write it out, fill in the chart."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Now, we can go through the columns. And this first column is where we roll a 1 on the second die. This is where we roll a 2 on the second die. Let's draw that out, write it out, fill in the chart. Here's where we roll a 3 on the second die. This is a comma that I'm doing between the two numbers. Here's where we have a 4."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Let's draw that out, write it out, fill in the chart. Here's where we roll a 3 on the second die. This is a comma that I'm doing between the two numbers. Here's where we have a 4. And then here's where we roll a 5 on the second die. Just filling this in. And then filling this in."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Here's where we have a 4. And then here's where we roll a 5 on the second die. Just filling this in. And then filling this in. This last column is where we roll a 6 on the second die. Now, every one of these represents a possible outcome. This outcome is where we roll a 1 on the first die and a 1 on the second die."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then filling this in. This last column is where we roll a 6 on the second die. Now, every one of these represents a possible outcome. This outcome is where we roll a 1 on the first die and a 1 on the second die. This outcome is where we roll a 3 on the first die, a 2 on the second die. This outcome is where we roll a 4 on the first die and a 5 on the second die. And you can see here, there are 36 possible outcomes."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "This outcome is where we roll a 1 on the first die and a 1 on the second die. This outcome is where we roll a 3 on the first die, a 2 on the second die. This outcome is where we roll a 4 on the first die and a 5 on the second die. And you can see here, there are 36 possible outcomes. 6 times 6 possible outcomes. Now, with this out of the way, how many of these outcomes satisfy our criteria, satisfy the criteria of rolling doubles on two six-sided dice? How many of these outcomes are essentially described by our event?"}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And you can see here, there are 36 possible outcomes. 6 times 6 possible outcomes. Now, with this out of the way, how many of these outcomes satisfy our criteria, satisfy the criteria of rolling doubles on two six-sided dice? How many of these outcomes are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling a 1 and a 1. It's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "How many of these outcomes are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling a 1 and a 1. It's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6 events satisfy this event or are the outcomes that are consistent with this event. Now, given that, let's answer our question. What is the probability of rolling doubles on two six-sided die numbered from 1 to 6?"}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "It's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6 events satisfy this event or are the outcomes that are consistent with this event. Now, given that, let's answer our question. What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? Well, the probability is going to be equal to the number of outcomes that satisfy our criteria, or the number of outcomes for this event, which are 6. We just figured that out. Over the total number of outcomes, over the size of our sample space."}, {"video_title": "Die rolling probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? Well, the probability is going to be equal to the number of outcomes that satisfy our criteria, or the number of outcomes for this event, which are 6. We just figured that out. Over the total number of outcomes, over the size of our sample space. So this right over here, we have 36 total outcomes. So we have 36 outcomes. And if you simplify this, 6 over 36 is the same thing as 1, 6."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "Ludwig got a score of 47.5 points on the exam. What proportion of exam scores are higher than Ludwig's score? Give your answer correct to four decimal places. So let's just visualize what's going on here. So the scores are normally distributed. So it would look like this. So the distribution would look something like that."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just visualize what's going on here. So the scores are normally distributed. So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So the distribution would look something like that. Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So that would be 40 points right over there. Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough. And so this would be 43. This would be 37 right over here."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "That would be one standard deviation below the mean. And once again, this is just very rough. And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "Because that's what z-tables give us. They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of all, let's figure out the z-score for 47.5."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of all, let's figure out the z-score for 47.5. So let's see. We would take 47.5 and we would subtract the mean. So this is his score."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So first of all, let's figure out the z-score for 47.5. So let's see. We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So this is his score. We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5. So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "That's 7.5. So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "Well, each standard deviation is three. So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So here we go."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So here we go. And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So here we go. And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hundredths here."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So I'm gonna go down to 2.5. Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062. So that's the proportion."}, {"video_title": "Standard normal table for proportion above   AP Statistics   Khan Academy.mp3", "Sentence": "One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062. So that's the proportion. If you thought of it in percent, it would be.62% scores higher than Ludwig. And that makes sense, because Ludwig scored over two standard deviations, two and a half standard deviations above the mean. So our answer here is 0.0062."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "In a previous video, we talked about trying to estimate a population mean with a sample mean, and then constructing a confidence interval about that sample mean. And we talked about different scenarios. We could use a z-table plus the true population standard deviation, and that actually would construct pretty valid confidence intervals. But the problem is you don't know the population standard deviation. And so you might try to use a z-table to find your critical values, plus the sample standard deviation. But what we talked about is that this doesn't actually do a good job of calculating our confidence intervals. And we're going to see that experimentally in a few seconds."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "But the problem is you don't know the population standard deviation. And so you might try to use a z-table to find your critical values, plus the sample standard deviation. But what we talked about is that this doesn't actually do a good job of calculating our confidence intervals. And we're going to see that experimentally in a few seconds. And so instead, we have something called a t-statistic, where if we want our critical value, we use a t-table instead of a z-table. And then we use that in conjunction with our sample standard deviation. And then all of a sudden, we are actually going to have pretty good confidence intervals."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And we're going to see that experimentally in a few seconds. And so instead, we have something called a t-statistic, where if we want our critical value, we use a t-table instead of a z-table. And then we use that in conjunction with our sample standard deviation. And then all of a sudden, we are actually going to have pretty good confidence intervals. To make this a little bit more real, let's look at a simulation. So this is a scratch pad on Khan Academy made by Khan Academy user Charlotte Allen. And the whole point there is to see what our confidence intervals look like with these different scenarios."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And then all of a sudden, we are actually going to have pretty good confidence intervals. To make this a little bit more real, let's look at a simulation. So this is a scratch pad on Khan Academy made by Khan Academy user Charlotte Allen. And the whole point there is to see what our confidence intervals look like with these different scenarios. So let's say we have a true population mean of 2.0 for some, let's say it's the average number, the mean number of apples people eat a day. The true population mean is two. That seems high, but maybe it's in a certain country that has a lot of apples."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And the whole point there is to see what our confidence intervals look like with these different scenarios. So let's say we have a true population mean of 2.0 for some, let's say it's the average number, the mean number of apples people eat a day. The true population mean is two. That seems high, but maybe it's in a certain country that has a lot of apples. And let's say we know that the population standard deviation is 0.5. And we're gonna create confidence intervals with the goal of having a 95% confidence level. And we're gonna take sample sizes of 12."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "That seems high, but maybe it's in a certain country that has a lot of apples. And let's say we know that the population standard deviation is 0.5. And we're gonna create confidence intervals with the goal of having a 95% confidence level. And we're gonna take sample sizes of 12. So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it. And so let's just draw a bunch of samples here. And so we do see that it looks like it is roughly 95% when we keep making these samples and constructing these confidence intervals, that 95% of the time, these confidence intervals contain our true population mean."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And we're gonna take sample sizes of 12. So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it. And so let's just draw a bunch of samples here. And so we do see that it looks like it is roughly 95% when we keep making these samples and constructing these confidence intervals, that 95% of the time, these confidence intervals contain our true population mean. So these look like good confidence intervals. But what we've talked about is, normally when you're doing this type of thing, this type of inferential statistics, you don't know the population standard deviation. You don't know sigma."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so we do see that it looks like it is roughly 95% when we keep making these samples and constructing these confidence intervals, that 95% of the time, these confidence intervals contain our true population mean. So these look like good confidence intervals. But what we've talked about is, normally when you're doing this type of thing, this type of inferential statistics, you don't know the population standard deviation. You don't know sigma. So instead, you might be tempted to use z with our sample standard deviations. But if you look at that for these exact same samples we just calculated, notice now when we did it over and over again, we've done this 625 times, in this scenario where we keep calculating the confidence intervals with z and s, the true population mean is contained in the intervals only 92.2% of the time. And we could keep going."}, {"video_title": "Simulation showing value of t statistic   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "You don't know sigma. So instead, you might be tempted to use z with our sample standard deviations. But if you look at that for these exact same samples we just calculated, notice now when we did it over and over again, we've done this 625 times, in this scenario where we keep calculating the confidence intervals with z and s, the true population mean is contained in the intervals only 92.2% of the time. And we could keep going. So we have a much lower hit rate than we would hope to have if we were actually using z and sigma. Now what's neat is if we use t, use a t-table, notice this is getting much closer. And this is neat because with a t-table and something that we can actually get from the sample, the sample standard deviation, we're actually able to have a pretty close hit rate to what we would have if we actually knew the population standard deviation."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I want to build on what we did on the last video a little bit. So let's say we have two random variables. So I have random variable X, and let me draw its probability distribution. And actually, it doesn't have to be normal, but I'll just draw it as a normal distribution. So this is the distribution of random variable X. This is the population mean of random variable X. And then it has some type of standard deviation."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, it doesn't have to be normal, but I'll just draw it as a normal distribution. So this is the distribution of random variable X. This is the population mean of random variable X. And then it has some type of standard deviation. Or actually, let me just focus on the variance. So it has some variance right here for random variable X. Now let's say, let me just write this is X, the distribution for X."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then it has some type of standard deviation. Or actually, let me just focus on the variance. So it has some variance right here for random variable X. Now let's say, let me just write this is X, the distribution for X. And let's say we have another random variable, random variable Y. Let's do the same thing for it. Let's draw its distribution."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's say, let me just write this is X, the distribution for X. And let's say we have another random variable, random variable Y. Let's do the same thing for it. Let's draw its distribution. And let me draw the parameters for that distribution. So it has some true mean, some population mean for the random variable Y. And it has some variance right over here."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's draw its distribution. And let me draw the parameters for that distribution. So it has some true mean, some population mean for the random variable Y. And it has some variance right over here. So it has some variance to this distribution. And I've drawn it roughly normal. Once again, we don't have to assume that it's normal, because we're going to assume when we go to the next level that when we take the samples, we're taking enough samples that the central limit theorem will actually apply."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it has some variance right over here. So it has some variance to this distribution. And I've drawn it roughly normal. Once again, we don't have to assume that it's normal, because we're going to assume when we go to the next level that when we take the samples, we're taking enough samples that the central limit theorem will actually apply. But with that said, let's think about the sampling distributions of each of these random variables. So let's think about the sampling distribution of the sample mean of X. When the sample size, and let's say the sample size over here is going to be equal to n. And actually over here, I'm going, well I'll stay in green right now."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, we don't have to assume that it's normal, because we're going to assume when we go to the next level that when we take the samples, we're taking enough samples that the central limit theorem will actually apply. But with that said, let's think about the sampling distributions of each of these random variables. So let's think about the sampling distribution of the sample mean of X. When the sample size, and let's say the sample size over here is going to be equal to n. And actually over here, I'm going, well I'll stay in green right now. So what is that going to look like? Well it's going to be some distribution. And now we're assuming that n is a fairly large number."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "When the sample size, and let's say the sample size over here is going to be equal to n. And actually over here, I'm going, well I'll stay in green right now. So what is that going to look like? Well it's going to be some distribution. And now we're assuming that n is a fairly large number. So this is going to be a normal distribution, or can be approximated with a normal distribution. Notice I drew it having a, let me shift it over a little bit. I'm going to draw it a little bit narrow, because we learned from the central limit theorem that the standard deviation of this thing, that the standard deviation, so let me draw the mean."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now we're assuming that n is a fairly large number. So this is going to be a normal distribution, or can be approximated with a normal distribution. Notice I drew it having a, let me shift it over a little bit. I'm going to draw it a little bit narrow, because we learned from the central limit theorem that the standard deviation of this thing, that the standard deviation, so let me draw the mean. So the population mean of the sampling distribution is going to be, we're going to denote it with this X bar. That tells us the distribution of the means when the sample size is n. And we know that this is going to be the same thing as the population mean for that random variable. And we know from the central limit theorem that the variance of the sampling distribution, or often called the standard error of the mean, is going to be equal to the population variance divided by this n right over here."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to draw it a little bit narrow, because we learned from the central limit theorem that the standard deviation of this thing, that the standard deviation, so let me draw the mean. So the population mean of the sampling distribution is going to be, we're going to denote it with this X bar. That tells us the distribution of the means when the sample size is n. And we know that this is going to be the same thing as the population mean for that random variable. And we know from the central limit theorem that the variance of the sampling distribution, or often called the standard error of the mean, is going to be equal to the population variance divided by this n right over here. And if you wanted the standard deviation of this, you just take the square root of both sides. Now let's do the same thing for random variable y. So let's take the sampling distribution of the sample mean, but here we're talking about y, random variable y."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we know from the central limit theorem that the variance of the sampling distribution, or often called the standard error of the mean, is going to be equal to the population variance divided by this n right over here. And if you wanted the standard deviation of this, you just take the square root of both sides. Now let's do the same thing for random variable y. So let's take the sampling distribution of the sample mean, but here we're talking about y, random variable y. And let's just say it has a different sample size. It doesn't have to be a different one, but it just shows you that it doesn't have to be the same. So it has a sample size."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's take the sampling distribution of the sample mean, but here we're talking about y, random variable y. And let's just say it has a different sample size. It doesn't have to be a different one, but it just shows you that it doesn't have to be the same. So it has a sample size. Let's say it has a sample size of m. So let me draw its distribution right over here. Once again, it'll be a narrower distribution than the population distribution. And it will be approximately normal, assuming that we have a large enough sample size."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it has a sample size. Let's say it has a sample size of m. So let me draw its distribution right over here. Once again, it'll be a narrower distribution than the population distribution. And it will be approximately normal, assuming that we have a large enough sample size. And its mean, the sampling distribution of the sample mean, is going to be the same thing as the population mean. We've seen that multiple times. Same thing as the population mean."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it will be approximately normal, assuming that we have a large enough sample size. And its mean, the sampling distribution of the sample mean, is going to be the same thing as the population mean. We've seen that multiple times. Same thing as the population mean. And its variance, so the variance over here, so the variance for the sample means, or the standard error of the mean, actually this isn't the standard error. This is the, I guess you could, well, standard error would be the square root of this. So if I call this the standard error mean, that's wrong."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Same thing as the population mean. And its variance, so the variance over here, so the variance for the sample means, or the standard error of the mean, actually this isn't the standard error. This is the, I guess you could, well, standard error would be the square root of this. So if I call this the standard error mean, that's wrong. The standard error of the mean is the square root of this. It's the standard deviation. This is the variance of the mean."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I call this the standard error mean, that's wrong. The standard error of the mean is the square root of this. It's the standard deviation. This is the variance of the mean. The variance of the mean, don't want to confuse you. So the variance of the mean here is going to be the exact same thing. It's going to be the variance of the population divided by our sample size."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the variance of the mean. The variance of the mean, don't want to confuse you. So the variance of the mean here is going to be the exact same thing. It's going to be the variance of the population divided by our sample size. And everything we've done so far is complete review. It's a little different, because I'm actually doing it with two different random variables. And I'm doing it with two different random variables for a reason."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be the variance of the population divided by our sample size. And everything we've done so far is complete review. It's a little different, because I'm actually doing it with two different random variables. And I'm doing it with two different random variables for a reason. Because now I'm going to define a new random variable. I'm now going to define a new random variable. That is, well, we could just call it z."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm doing it with two different random variables for a reason. Because now I'm going to define a new random variable. I'm now going to define a new random variable. That is, well, we could just call it z. We'll just call it z. But z is equal to the difference of our sample means. And let me stay with the colors."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is, well, we could just call it z. We'll just call it z. But z is equal to the difference of our sample means. And let me stay with the colors. It's equal to the x sample mean minus the y sample mean. So what does that really mean? Well, to get a sample mean, or at least for this distribution, you're taking n samples from this population over here."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let me stay with the colors. It's equal to the x sample mean minus the y sample mean. So what does that really mean? Well, to get a sample mean, or at least for this distribution, you're taking n samples from this population over here. Maybe n is 10. You're taking 10 samples and finding its mean. That sample mean is a random variable."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, to get a sample mean, or at least for this distribution, you're taking n samples from this population over here. Maybe n is 10. You're taking 10 samples and finding its mean. That sample mean is a random variable. You could view that sample mean. Let's say you take 10 samples from here and you get 9.2 when you find their mean. That 9.2 can be viewed as a sample from this distribution right over here."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "That sample mean is a random variable. You could view that sample mean. Let's say you take 10 samples from here and you get 9.2 when you find their mean. That 9.2 can be viewed as a sample from this distribution right over here. Same thing if this right here is m. Or if m right here is 12. You're taking 12 samples, taking its mean. And that sample mean, maybe it's 15.2, could be viewed as a sample from this distribution, as a sample from the sampling distribution."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "That 9.2 can be viewed as a sample from this distribution right over here. Same thing if this right here is m. Or if m right here is 12. You're taking 12 samples, taking its mean. And that sample mean, maybe it's 15.2, could be viewed as a sample from this distribution, as a sample from the sampling distribution. So what z is, z is a random variable where you're taking n samples from this distribution up here, this population distribution, taking its mean. Then you're taking m samples from this population distribution up here, taking its mean, and then finding the difference between that mean and that mean. So it's another random variable."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that sample mean, maybe it's 15.2, could be viewed as a sample from this distribution, as a sample from the sampling distribution. So what z is, z is a random variable where you're taking n samples from this distribution up here, this population distribution, taking its mean. Then you're taking m samples from this population distribution up here, taking its mean, and then finding the difference between that mean and that mean. So it's another random variable. But what is the distribution of z? What is going to be the distribution of z? So let's draw it."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's another random variable. But what is the distribution of z? What is going to be the distribution of z? So let's draw it. Let's draw it like this. Well, there's a couple of things we immediately know about z. And we kind of came up with this in the last video."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's draw it. Let's draw it like this. Well, there's a couple of things we immediately know about z. And we kind of came up with this in the last video. So the mean of z, instead of writing z, I'm just going to write the mean of x. Let me do that same shade of green. The mean of x bar, which is the mean of x minus, or a sample from the sampling distribution of x, or the sample mean of x, minus the sample mean of y."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we kind of came up with this in the last video. So the mean of z, instead of writing z, I'm just going to write the mean of x. Let me do that same shade of green. The mean of x bar, which is the mean of x minus, or a sample from the sampling distribution of x, or the sample mean of x, minus the sample mean of y. So the mean of this is going to be equal to, and we saw this in the last video. In fact, I think I still have the work up here. Yeah, I still have the work right up here."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean of x bar, which is the mean of x minus, or a sample from the sampling distribution of x, or the sample mean of x, minus the sample mean of y. So the mean of this is going to be equal to, and we saw this in the last video. In fact, I think I still have the work up here. Yeah, I still have the work right up here. The mean of the difference is going to be the difference of the means. The mean of the difference is the same thing as the difference of the means. So the mean of this new distribution right over here is going to be the same thing as it's going to be the mean of our sample mean minus the mean of our sample mean of y."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yeah, I still have the work right up here. The mean of the difference is going to be the difference of the means. The mean of the difference is the same thing as the difference of the means. So the mean of this new distribution right over here is going to be the same thing as it's going to be the mean of our sample mean minus the mean of our sample mean of y. And this might seem a little abstract in this video. In the next video, we're actually going to do this with concrete numbers. And hopefully, it'll make a little bit more sense."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the mean of this new distribution right over here is going to be the same thing as it's going to be the mean of our sample mean minus the mean of our sample mean of y. And this might seem a little abstract in this video. In the next video, we're actually going to do this with concrete numbers. And hopefully, it'll make a little bit more sense. Just so you know where we're going with this, the point of this is so that we can eventually do some inferential statistics about differences of means. How likely is a difference of means of two samples, random chance or not random chance? Or what is a confidence interval of the difference of means?"}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And hopefully, it'll make a little bit more sense. Just so you know where we're going with this, the point of this is so that we can eventually do some inferential statistics about differences of means. How likely is a difference of means of two samples, random chance or not random chance? Or what is a confidence interval of the difference of means? That's what this is all building up to. So anyway, we know the mean of this distribution right over here. And what's the variance of this distribution?"}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or what is a confidence interval of the difference of means? That's what this is all building up to. So anyway, we know the mean of this distribution right over here. And what's the variance of this distribution? And we came up with that result in the last video. If we're taking essentially the difference of two random variables, the variance is going to be the sum of those two random variables. And the whole point of that video is to show you that, hey, it's not the difference of the variances."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what's the variance of this distribution? And we came up with that result in the last video. If we're taking essentially the difference of two random variables, the variance is going to be the sum of those two random variables. And the whole point of that video is to show you that, hey, it's not the difference of the variances. It's the sum of the variances. So the variance of this new distribution, and I haven't drawn the distribution yet, the variance of this new distribution, I'll just write x bar minus y bar, is going to be equal to the sum of the variances of each of these distributions. The variance of x bar plus the variance of y bar."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the whole point of that video is to show you that, hey, it's not the difference of the variances. It's the sum of the variances. So the variance of this new distribution, and I haven't drawn the distribution yet, the variance of this new distribution, I'll just write x bar minus y bar, is going to be equal to the sum of the variances of each of these distributions. The variance of x bar plus the variance of y bar. Now, actually, let me just draw this here, just so we can visualize another distribution. Although, all I'm going to draw is another normal distribution. So this is its mean."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The variance of x bar plus the variance of y bar. Now, actually, let me just draw this here, just so we can visualize another distribution. Although, all I'm going to draw is another normal distribution. So this is its mean. So the mean over here, let me scroll down a little bit. So the mean over here, mean of x bar minus y bar, is going to be equal to the difference of these means over here. So I have to rewrite it."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is its mean. So the mean over here, let me scroll down a little bit. So the mean over here, mean of x bar minus y bar, is going to be equal to the difference of these means over here. So I have to rewrite it. And then let me draw the curve. And notice, I'm drawing a fatter curve. I'm drawing a fatter curve than either one."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I have to rewrite it. And then let me draw the curve. And notice, I'm drawing a fatter curve. I'm drawing a fatter curve than either one. And why am I doing that? Because the variance here is the sum of the variances here. So we're going to have a fatter curve."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm drawing a fatter curve than either one. And why am I doing that? Because the variance here is the sum of the variances here. So we're going to have a fatter curve. It's going to have a bigger variance or a bigger standard deviation than either of these. So then we have some variance here. Variance of x bar minus y bar."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're going to have a fatter curve. It's going to have a bigger variance or a bigger standard deviation than either of these. So then we have some variance here. Variance of x bar minus y bar. Now, what are these in terms of the original population distribution? Well, we came up with those results right over here. We know what the standard deviation."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Variance of x bar minus y bar. Now, what are these in terms of the original population distribution? Well, we came up with those results right over here. We know what the standard deviation. We know that this thing is the same thing as the variance of the population distribution divided by n. We've done this multiple, multiple times. So this is going to be equal to, what's this going to be equal to? This part right here is the same thing as the variance of our population distribution."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know what the standard deviation. We know that this thing is the same thing as the variance of the population distribution divided by n. We've done this multiple, multiple times. So this is going to be equal to, what's this going to be equal to? This part right here is the same thing as the variance of our population distribution. And the x just means this is for random variable x. But there's no bar on top. This is the actual population distribution, not the sampling distribution of the sample mean."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This part right here is the same thing as the variance of our population distribution. And the x just means this is for random variable x. But there's no bar on top. This is the actual population distribution, not the sampling distribution of the sample mean. So that divided by n. And then if we want the variance of the sampling distribution for y, let me do that in a different color. I'll use blue because that was what we were using for the y random variable. That's going to be equal to this thing over here."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the actual population distribution, not the sampling distribution of the sample mean. So that divided by n. And then if we want the variance of the sampling distribution for y, let me do that in a different color. I'll use blue because that was what we were using for the y random variable. That's going to be equal to this thing over here. And we've done this multiple times. Same exact logic as this. The population distribution for y divided by m. And so once again, I'll just write this out front."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's going to be equal to this thing over here. And we've done this multiple times. Same exact logic as this. The population distribution for y divided by m. And so once again, I'll just write this out front. This is the variance of the differences of the sample means. And if you wanted the standard deviation of the differences of the sample means, you just have to take the square root of both sides of this. If you take the square root of this, you get the standard deviation of the difference of the sample means is equal to the square root of the population distribution of x, or the variance of the population distribution of x divided by n plus the variance of the population distribution of y divided by m. And then the whole reason why I've even done this, and this is just neat because it kind of looks a little bit like a distance formula."}, {"video_title": "Difference of sample means distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "The population distribution for y divided by m. And so once again, I'll just write this out front. This is the variance of the differences of the sample means. And if you wanted the standard deviation of the differences of the sample means, you just have to take the square root of both sides of this. If you take the square root of this, you get the standard deviation of the difference of the sample means is equal to the square root of the population distribution of x, or the variance of the population distribution of x divided by n plus the variance of the population distribution of y divided by m. And then the whole reason why I've even done this, and this is just neat because it kind of looks a little bit like a distance formula. And I'll kind of throw that out there as we get more sophisticated with our statistics and try to visualize what all of this kind of stuff means in more advanced topics. But the whole point of this is now we can make inferences about a difference of means. If we have two samples and we want to say, and we take the means of both of those samples and we find some difference, we can make some conclusions about how likely that difference was just by chance."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And one of these things that you'll find in probability is that you can always do a more interesting problem. So now I'm going to think about, I'm going to take a fair coin, and I'm going to flip it three times. And I want to find the probability of at least one head out of the three flips. So the easiest way to think about this is how many equally likely possibilities there are. In the last video we saw, if we flip a coin three times, there's eight possibilities. For the first flip, there's two possibilities. Second flip, there's two possibilities."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the easiest way to think about this is how many equally likely possibilities there are. In the last video we saw, if we flip a coin three times, there's eight possibilities. For the first flip, there's two possibilities. Second flip, there's two possibilities. And in the third flip, there are two possibilities. So 2 times 2 times 2, there are eight equally likely possibilities if I'm flipping a coin three times. Now how many of those possibilities have at least one head?"}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Second flip, there's two possibilities. And in the third flip, there are two possibilities. So 2 times 2 times 2, there are eight equally likely possibilities if I'm flipping a coin three times. Now how many of those possibilities have at least one head? Well, we drew all of the possibilities over here, so we just have to count how many of these have at least one head. So that's 1, 2, 3, 4, 5, 6, 7. So 7 of these have at least one head in them, and this last one does not."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now how many of those possibilities have at least one head? Well, we drew all of the possibilities over here, so we just have to count how many of these have at least one head. So that's 1, 2, 3, 4, 5, 6, 7. So 7 of these have at least one head in them, and this last one does not. So 7 of the 8 have at least one head. Now you're probably thinking, OK, Sal, you were able to do it by writing out all of the possibilities, but that would be really hard if I said at least one head out of 20 flips. This would work well because I only had three flips."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 7 of these have at least one head in them, and this last one does not. So 7 of the 8 have at least one head. Now you're probably thinking, OK, Sal, you were able to do it by writing out all of the possibilities, but that would be really hard if I said at least one head out of 20 flips. This would work well because I only had three flips. So let me make it clear. This is in three flips. This would have been a lot harder to do or more time-consuming to do if I had 20 flips."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would work well because I only had three flips. So let me make it clear. This is in three flips. This would have been a lot harder to do or more time-consuming to do if I had 20 flips. Is there some shortcut here? Is there some other way to think about it? And you couldn't just do it in some simple way."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would have been a lot harder to do or more time-consuming to do if I had 20 flips. Is there some shortcut here? Is there some other way to think about it? And you couldn't just do it in some simple way. You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore. Or you could get heads again, but you don't have to. So it becomes a little bit more complicated."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you couldn't just do it in some simple way. You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore. Or you could get heads again, but you don't have to. So it becomes a little bit more complicated. But there is an easy way to think about it where you could use this methodology right over here. You'll actually see this on a lot of exams where they make it seem like a harder problem, but if you just think about it the right way, all of a sudden it becomes simpler. One way to think about it is the probability of at least one heads in three flips is the same thing as the probability of not getting all tails."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it becomes a little bit more complicated. But there is an easy way to think about it where you could use this methodology right over here. You'll actually see this on a lot of exams where they make it seem like a harder problem, but if you just think about it the right way, all of a sudden it becomes simpler. One way to think about it is the probability of at least one heads in three flips is the same thing as the probability of not getting all tails. If we got all tails, then we don't have at least one head. So these two things are equivalent. The probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "One way to think about it is the probability of at least one heads in three flips is the same thing as the probability of not getting all tails. If we got all tails, then we don't have at least one head. So these two things are equivalent. The probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips. Let me write in three flips. So what's the probability of not getting all tails? Well, that's going to be 1 minus the probability of getting all tails."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips. Let me write in three flips. So what's the probability of not getting all tails? Well, that's going to be 1 minus the probability of getting all tails. And since it's three flips, it's the probability of tails, tails, and tails. Because any of the other situations are going to have at least one head in them. And that's all of the other possibilities."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that's going to be 1 minus the probability of getting all tails. And since it's three flips, it's the probability of tails, tails, and tails. Because any of the other situations are going to have at least one head in them. And that's all of the other possibilities. And this is the only other leftover possibility. If you add them together, you're going to get one. Let me write it this way."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's all of the other possibilities. And this is the only other leftover possibility. If you add them together, you're going to get one. Let me write it this way. The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive. This is all of the possible circumstances. So your chances of getting either not all tails or all tails, and these are mutual exclusives, so we can add them, so the probability of not all tails or the probability of all tails is going to be equal to 1."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write it this way. The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive. This is all of the possible circumstances. So your chances of getting either not all tails or all tails, and these are mutual exclusives, so we can add them, so the probability of not all tails or the probability of all tails is going to be equal to 1. These are mutual exclusives. You're either going to have not all tails, which means a head shows up, or you're going to have all tails, but you can't have both of these things happening. And since they're mutual exclusives, and you're saying the probability of this or this happening, you can add their probabilities."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So your chances of getting either not all tails or all tails, and these are mutual exclusives, so we can add them, so the probability of not all tails or the probability of all tails is going to be equal to 1. These are mutual exclusives. You're either going to have not all tails, which means a head shows up, or you're going to have all tails, but you can't have both of these things happening. And since they're mutual exclusives, and you're saying the probability of this or this happening, you can add their probabilities. And this is essentially all of the possible events. So this is essentially, if you combine these, this is the probability of any of the events happening, and that's going to be a 1 or 100% chance. So another way to think about it is the probability of not all tails is going to be 1 minus the probability of all tails."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And since they're mutual exclusives, and you're saying the probability of this or this happening, you can add their probabilities. And this is essentially all of the possible events. So this is essentially, if you combine these, this is the probability of any of the events happening, and that's going to be a 1 or 100% chance. So another way to think about it is the probability of not all tails is going to be 1 minus the probability of all tails. So that's what we did right over here. And the probability of all tails is pretty straightforward. That's the probability of it's going to be 1 half, because you have a 1 half chance of getting a tails on the first flip, times, let me write it here so it becomes a little clearer, so this is going to be 1 minus the probability of getting all tails, well, you have a 1 half chance of getting tails on the first flip, and then you're going to have to get another tails on the second flip, and then you're going to have to get another tails on the third flip."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So another way to think about it is the probability of not all tails is going to be 1 minus the probability of all tails. So that's what we did right over here. And the probability of all tails is pretty straightforward. That's the probability of it's going to be 1 half, because you have a 1 half chance of getting a tails on the first flip, times, let me write it here so it becomes a little clearer, so this is going to be 1 minus the probability of getting all tails, well, you have a 1 half chance of getting tails on the first flip, and then you're going to have to get another tails on the second flip, and then you're going to have to get another tails on the third flip. And then 1 half times 1 half times 1 half, this is going to be 1 eighth. And then 1 minus 1 eighth, or 8 eighths minus 1 eighth, is going to be equal to 7 eighths. So we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem, we can say, let's say we have 10 flips, the probability of at least 1 head in 10 flips."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's the probability of it's going to be 1 half, because you have a 1 half chance of getting a tails on the first flip, times, let me write it here so it becomes a little clearer, so this is going to be 1 minus the probability of getting all tails, well, you have a 1 half chance of getting tails on the first flip, and then you're going to have to get another tails on the second flip, and then you're going to have to get another tails on the third flip. And then 1 half times 1 half times 1 half, this is going to be 1 eighth. And then 1 minus 1 eighth, or 8 eighths minus 1 eighth, is going to be equal to 7 eighths. So we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem, we can say, let's say we have 10 flips, the probability of at least 1 head in 10 flips. Well, we use the same idea, this is the same thing, this is going to be equal to the probability of not all tails in 10 flips. So we're just saying the probability of not getting all of the flips going to be tails, all of the flips is tails, not all tails in 10 flips. And this is going to be 1 minus the probability of flipping tails 10 times."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem, we can say, let's say we have 10 flips, the probability of at least 1 head in 10 flips. Well, we use the same idea, this is the same thing, this is going to be equal to the probability of not all tails in 10 flips. So we're just saying the probability of not getting all of the flips going to be tails, all of the flips is tails, not all tails in 10 flips. And this is going to be 1 minus the probability of flipping tails 10 times. So it's 1 minus 10 tails in a row. And so this is going to be equal to, this part right over here, let me write this. So this is going to be this 1, let me just rewrite it, this is equal to 1 minus, and this part is going to be, well, 1 tail, another tail, so it's 1 half times 1 half, and I'm going to do this 10 times."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is going to be 1 minus the probability of flipping tails 10 times. So it's 1 minus 10 tails in a row. And so this is going to be equal to, this part right over here, let me write this. So this is going to be this 1, let me just rewrite it, this is equal to 1 minus, and this part is going to be, well, 1 tail, another tail, so it's 1 half times 1 half, and I'm going to do this 10 times. Let me write this a little neater, because I need a 1 half, so that's 5, 6, 7, 8, 9, and 10. And so we really just have to, the numerator is going to be 1. So this is going to be 1, this is going to be equal to 1, let me do it in that same color of green."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be this 1, let me just rewrite it, this is equal to 1 minus, and this part is going to be, well, 1 tail, another tail, so it's 1 half times 1 half, and I'm going to do this 10 times. Let me write this a little neater, because I need a 1 half, so that's 5, 6, 7, 8, 9, and 10. And so we really just have to, the numerator is going to be 1. So this is going to be 1, this is going to be equal to 1, let me do it in that same color of green. This is going to be equal to 1 minus, our numerator, you just have 1 times itself 10 times, so that's 1. And then on the denominator, you have 2 times 2 is 4, 4 times 2 is 8, 16, 32, 64, 128, 256, 512, 1,024. This is the same exact same thing as 1 is 1024 over 1024 minus 1 over 1024, which is equal to 1023, or 1,023, over 1,024, we have a common denominator here, so 1,000, doing that same blue, over 1,000 and 1,024."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be 1, this is going to be equal to 1, let me do it in that same color of green. This is going to be equal to 1 minus, our numerator, you just have 1 times itself 10 times, so that's 1. And then on the denominator, you have 2 times 2 is 4, 4 times 2 is 8, 16, 32, 64, 128, 256, 512, 1,024. This is the same exact same thing as 1 is 1024 over 1024 minus 1 over 1024, which is equal to 1023, or 1,023, over 1,024, we have a common denominator here, so 1,000, doing that same blue, over 1,000 and 1,024. So if you flip a coin 10 times in a row, a fair coin, you're probability of getting at least one heads in that 10 flips is pretty high. It's 1,023 over 1,024, and you can get a calculator out to figure that out in terms of a percentage. Actually, let me just do that just for fun."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the same exact same thing as 1 is 1024 over 1024 minus 1 over 1024, which is equal to 1023, or 1,023, over 1,024, we have a common denominator here, so 1,000, doing that same blue, over 1,000 and 1,024. So if you flip a coin 10 times in a row, a fair coin, you're probability of getting at least one heads in that 10 flips is pretty high. It's 1,023 over 1,024, and you can get a calculator out to figure that out in terms of a percentage. Actually, let me just do that just for fun. So if we have 1,023 divided by 1,024, that gives us, you have a 99.9% chance that you're going to have at least one heads. So this is, if we round, this is equal to 99.9% chance. And I rounded a little bit."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me just do that just for fun. So if we have 1,023 divided by 1,024, that gives us, you have a 99.9% chance that you're going to have at least one heads. So this is, if we round, this is equal to 99.9% chance. And I rounded a little bit. It's actually even slightly higher than that. And this is a pretty powerful tool, or a pretty powerful way to think about it, because it would have taken you forever to write all of the scenarios down. In fact, there would have been 1,024 scenarios to write down."}, {"video_title": "Coin flipping probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I rounded a little bit. It's actually even slightly higher than that. And this is a pretty powerful tool, or a pretty powerful way to think about it, because it would have taken you forever to write all of the scenarios down. In fact, there would have been 1,024 scenarios to write down. So doing this exercise for 10 flips would have taken up all of our time. But when you think about it in a slightly different way, when you just say, look, the probability of getting at least one heads in 10 flips is the same thing as the probability of not getting all tails. That's 1 minus the probability of getting all tails."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "He placed ping pong balls numbered from zero to 32, so I guess that would be, what, 33 ping pong balls, in a drum and mixed them well. Note that the median of the population is 16, right? The median number, of course, yes, and that population is 16. He then took a random sample of five balls and calculated the median of the sample. So we have this population of balls. He takes a, we know the population parameter. We know that the population median is 16, but then he starts taking a sample of five balls, so n equals five, and he calculates a sample median."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "He then took a random sample of five balls and calculated the median of the sample. So we have this population of balls. He takes a, we know the population parameter. We know that the population median is 16, but then he starts taking a sample of five balls, so n equals five, and he calculates a sample median. Sample median. And then he replaced the balls and repeated this process for a total of 50 trials. His results are summarized in the dot plot below, where each dot represents the sample median from a sample of five balls."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "We know that the population median is 16, but then he starts taking a sample of five balls, so n equals five, and he calculates a sample median. Sample median. And then he replaced the balls and repeated this process for a total of 50 trials. His results are summarized in the dot plot below, where each dot represents the sample median from a sample of five balls. So he does this. He takes these five balls, puts them back in, then he does it again, then he does it again, and every time, he calculates the sample median for that sample, and he plots that on the dot plot. So, and he'll do this for 50 samples."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "His results are summarized in the dot plot below, where each dot represents the sample median from a sample of five balls. So he does this. He takes these five balls, puts them back in, then he does it again, then he does it again, and every time, he calculates the sample median for that sample, and he plots that on the dot plot. So, and he'll do this for 50 samples. And each dot here represents that sample statistic, so it shows that four times we got a sample median. In four of those 50 samples, we got a sample median of 20. In five of those sample medians, we got a sample median of 10."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So, and he'll do this for 50 samples. And each dot here represents that sample statistic, so it shows that four times we got a sample median. In four of those 50 samples, we got a sample median of 20. In five of those sample medians, we got a sample median of 10. And so what he ends up creating with these dots is really an approximation of the sampling distribution of the sample medians. Now, to judge whether it is a biased or unbiased estimator for the population median, well, actually, pause the video. See if you can figure that out."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "In five of those sample medians, we got a sample median of 10. And so what he ends up creating with these dots is really an approximation of the sampling distribution of the sample medians. Now, to judge whether it is a biased or unbiased estimator for the population median, well, actually, pause the video. See if you can figure that out. All right, now let's do this together. Now, to judge it, let's think about where the true population parameter is, the population median. It's 16, we know that."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "See if you can figure that out. All right, now let's do this together. Now, to judge it, let's think about where the true population parameter is, the population median. It's 16, we know that. And so that is right over here, the true population parameter. So if we were dealing with a biased, a biased estimator for the population parameter, then as we get our approximation of the sampling distribution, we would expect it to be somewhat skewed. So for example, if the sampling, if this approximation of the sampling distribution looks something like that, then we say, okay, that looks like a biased estimator."}, {"video_title": "Sample statistic bias worked example   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "It's 16, we know that. And so that is right over here, the true population parameter. So if we were dealing with a biased, a biased estimator for the population parameter, then as we get our approximation of the sampling distribution, we would expect it to be somewhat skewed. So for example, if the sampling, if this approximation of the sampling distribution looks something like that, then we say, okay, that looks like a biased estimator. Or if it was looking something like that, we'd say, okay, that looks like a biased estimator. But if this approximation for our sampling distribution that Alejandro is constructing, where we see that roughly the same proportion of the sample statistics came out below as came out above the true parameter, and it's not, it doesn't have to be exact, but it seems roughly the case, this seems pretty unbiased. And so to answer the question, based on these results, it does appear that the sample median is an unbiased estimator of the population median."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so it has various outcomes of those two free throws and then the corresponding probability, missing both free throws, 0.2, making exactly one free throw, 0.5, and making both free throws, 0.1. Is this a valid probability model? Pause this video and see if you can make a conclusion there. So let's think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that. And also, they would all have to be positive values, or I guess I should say, none of them can be negative values. You could have a scenario that has a 0% probability."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So let's think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that. And also, they would all have to be positive values, or I guess I should say, none of them can be negative values. You could have a scenario that has a 0% probability. And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%? So if I had 0.2 to 0.5, that is 0.7 plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You could have a scenario that has a 0% probability. And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%? So if I had 0.2 to 0.5, that is 0.7 plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model. In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this is not a valid probability model. In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no. Let's do another example. So here, we are told, you are a space alien. You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no. Let's do another example. So here, we are told, you are a space alien. You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then, you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then, you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundredth. So in the last example, we wanted to see whether a probability model was valid, whether it was legitimate. Here, we wanna construct a legitimate probability model."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundredth. So in the last example, we wanted to see whether a probability model was valid, whether it was legitimate. Here, we wanna construct a legitimate probability model. Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from that is, are chickens, because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Here, we wanna construct a legitimate probability model. Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from that is, are chickens, because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens. And so, what is this going to be? It's going to be 97 over 97, 47, and 77. You add them up."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "97 of the 97 plus 47 plus 77 animals are chickens. And so, what is this going to be? It's going to be 97 over 97, 47, and 77. You add them up. Three sevens is 21. And then, let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You add them up. Three sevens is 21. And then, let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens. And so, I'll just write 97, two 21s. They say that we can answer as fractions, so the problem's gonna go that way. What about cows?"}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So 97 of the 221 animals are chickens. And so, I'll just write 97, two 21s. They say that we can answer as fractions, so the problem's gonna go that way. What about cows? Well, 47 of the 221 are cows. So there's a 47, two 21st probability of getting a cow. And then, last but not least, you have 77 of the 221s are human."}, {"video_title": "Valid discrete probability distribution examples   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "What about cows? Well, 47 of the 221 are cows. So there's a 47, two 21st probability of getting a cow. And then, last but not least, you have 77 of the 221s are human. Is this a legitimate probability distribution? Well, add these up. If you add these three fractions up, the denominator's gonna be 221, and we already know that 97 plus 47 plus 77 is 221, so it definitely adds up to one, and none of these are negative."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in that presidential election, there are two candidates. There's candidate A and candidate B. And there's some reality. Let's say I live in a very decisive country, and everyone is going to vote for either, and everyone participates in the election, and everyone is going to vote for either candidate A or candidate B. And so there's some percentage. There's some reality there that P, let me write it over here, maybe 1 minus P percent. Let me do the P first."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I live in a very decisive country, and everyone is going to vote for either, and everyone participates in the election, and everyone is going to vote for either candidate A or candidate B. And so there's some percentage. There's some reality there that P, let me write it over here, maybe 1 minus P percent. Let me do the P first. There's some reality that maybe P percent will vote for B. And I could switch them around if I wanted. So P percent are going to vote for B, and the rest of the people are going to vote for A."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do the P first. There's some reality that maybe P percent will vote for B. And I could switch them around if I wanted. So P percent are going to vote for B, and the rest of the people are going to vote for A. So maybe 1 minus P percent are going to vote for A. 1 minus P. And you might already recognize that this is a Bernoulli distribution. There's one of two values for a sample I can get."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So P percent are going to vote for B, and the rest of the people are going to vote for A. So maybe 1 minus P percent are going to vote for A. 1 minus P. And you might already recognize that this is a Bernoulli distribution. There's one of two values for a sample I can get. And right here, the values I said, you're either voting for candidate A or you're voting for candidate B. It's very hard to deal with those values. You can't calculate a mean between A and B and all of that."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's one of two values for a sample I can get. And right here, the values I said, you're either voting for candidate A or you're voting for candidate B. It's very hard to deal with those values. You can't calculate a mean between A and B and all of that. Those are letters. They're not numbers. So to make it manipulatable mathematically, we're going to say sampling someone who's going to vote for A is equivalent to sampling a 0."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can't calculate a mean between A and B and all of that. Those are letters. They're not numbers. So to make it manipulatable mathematically, we're going to say sampling someone who's going to vote for A is equivalent to sampling a 0. And sampling someone who's going to vote for B is equivalent to sampling a 1. And if you do that with a Bernoulli distribution, we learned in the video on Bernoulli distributions that the mean of this distribution right here is going to be equal to P. And it's a pretty straightforward proof for how we got that. So the mean of this distribution, which will actually be not a value that this distribution can take on, is going to be someplace over here."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So to make it manipulatable mathematically, we're going to say sampling someone who's going to vote for A is equivalent to sampling a 0. And sampling someone who's going to vote for B is equivalent to sampling a 1. And if you do that with a Bernoulli distribution, we learned in the video on Bernoulli distributions that the mean of this distribution right here is going to be equal to P. And it's a pretty straightforward proof for how we got that. So the mean of this distribution, which will actually be not a value that this distribution can take on, is going to be someplace over here. And it is going to be equal to P. Now, my country has 100 million people. It is practically or is definitely impossible for me to be able to go and ask all 100 million people who are they going to vote for. So I won't be able to exactly figure out what these parameters are going to be, what my mean is, what P is going to be."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the mean of this distribution, which will actually be not a value that this distribution can take on, is going to be someplace over here. And it is going to be equal to P. Now, my country has 100 million people. It is practically or is definitely impossible for me to be able to go and ask all 100 million people who are they going to vote for. So I won't be able to exactly figure out what these parameters are going to be, what my mean is, what P is going to be. But instead of doing that, what I'm going to do is do a random survey. I'm going to sample this population, look at that data, and then get an estimate of what P really is. Because this is what I really care about."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I won't be able to exactly figure out what these parameters are going to be, what my mean is, what P is going to be. But instead of doing that, what I'm going to do is do a random survey. I'm going to sample this population, look at that data, and then get an estimate of what P really is. Because this is what I really care about. I really care about P. So I'm going to try to estimate P with a sample. And then we're also going to think about how good of an estimate that is. So let's say I am going to randomly survey 100 people."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because this is what I really care about. I really care about P. So I'm going to try to estimate P with a sample. And then we're also going to think about how good of an estimate that is. So let's say I am going to randomly survey 100 people. And let's say I got the following results. Let's say that 57 people say that they were going to vote for person A. Let me write it this way."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say I am going to randomly survey 100 people. And let's say I got the following results. Let's say that 57 people say that they were going to vote for person A. Let me write it this way. So 57 people say they're going to vote for A. Or that's equivalent to getting 57 samples of 0. And then the rest of the people, once again, very decisive population."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write it this way. So 57 people say they're going to vote for A. Or that's equivalent to getting 57 samples of 0. And then the rest of the people, once again, very decisive population. No one is undecided. The rest of the people, so 43 people, say they're going to vote for B. Or that's the equivalent of sampling 43 ones."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the rest of the people, once again, very decisive population. No one is undecided. The rest of the people, so 43 people, say they're going to vote for B. Or that's the equivalent of sampling 43 ones. Now, given this sample here, what is my sample mean and my sample variance? My sample mean right here, well, that's just going to be the average of these 0's and 1's. So I got 57 0's."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or that's the equivalent of sampling 43 ones. Now, given this sample here, what is my sample mean and my sample variance? My sample mean right here, well, that's just going to be the average of these 0's and 1's. So I got 57 0's. So it's going to be 57 times 0 plus my 43 1's. So the sum of all of my samples. So it's 43 1's plus 43 times 1 over the total number of samples I took, over 100."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I got 57 0's. So it's going to be 57 times 0 plus my 43 1's. So the sum of all of my samples. So it's 43 1's plus 43 times 1 over the total number of samples I took, over 100. So what does this get me? So 57 times 0 is 0. 43 times 1 divided by 100 is 0.43."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 43 1's plus 43 times 1 over the total number of samples I took, over 100. So what does this get me? So 57 times 0 is 0. 43 times 1 divided by 100 is 0.43. That is my sample mean, the mean of just the 100 data points that I actually got. Now, what is my sample variance? Is going to be equal to the sum of my squared distances to the mean divided by my samples minus 1."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "43 times 1 divided by 100 is 0.43. That is my sample mean, the mean of just the 100 data points that I actually got. Now, what is my sample variance? Is going to be equal to the sum of my squared distances to the mean divided by my samples minus 1. Remember, this is a sample variance. And we want to get the best estimator of the real variance of this distribution. And to do that, you don't divide by 100."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is going to be equal to the sum of my squared distances to the mean divided by my samples minus 1. Remember, this is a sample variance. And we want to get the best estimator of the real variance of this distribution. And to do that, you don't divide by 100. You're going to divide by 100 minus 1. We learned that many, many videos ago. So I have 57 samples of 0."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to do that, you don't divide by 100. You're going to divide by 100 minus 1. We learned that many, many videos ago. So I have 57 samples of 0. So I have 57 samples of 0. And so each of those samples are 0 minus 0.43 away from the mean. Each of those samples are 0."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I have 57 samples of 0. So I have 57 samples of 0. And so each of those samples are 0 minus 0.43 away from the mean. Each of those samples are 0. You subtract 0.43. This is the difference between 0 and 0.43. And if I want the squared distance, I square it."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Each of those samples are 0. You subtract 0.43. This is the difference between 0 and 0.43. And if I want the squared distance, I square it. That's how we calculate variance. There's 57 of those. And then there's 43 times that I sampled a 1 in my sample population."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if I want the squared distance, I square it. That's how we calculate variance. There's 57 of those. And then there's 43 times that I sampled a 1 in my sample population. 43 times I sampled a 1. And the 1 is 1 minus 0.43 away from the mean, because that is the mean. And I want to square that distance."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then there's 43 times that I sampled a 1 in my sample population. 43 times I sampled a 1. And the 1 is 1 minus 0.43 away from the mean, because that is the mean. And I want to square that distance. And then I don't want to just divide it by n. I don't want to just divide it by 100. Remember, I'm trying to estimate the true population mean in order for this to be the best estimator of that. And I gave you an intuition of why many, many videos ago."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I want to square that distance. And then I don't want to just divide it by n. I don't want to just divide it by 100. Remember, I'm trying to estimate the true population mean in order for this to be the best estimator of that. And I gave you an intuition of why many, many videos ago. We divide by 100 minus 1. We divide by 100 minus 1, or 99. Let's get the calculator out to actually figure out our sample variance."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I gave you an intuition of why many, many videos ago. We divide by 100 minus 1. We divide by 100 minus 1, or 99. Let's get the calculator out to actually figure out our sample variance. Let me get the calculator out. And we have, so I'll do the numerator first. I have 57 times 0 minus 0.43 squared plus 43 times 1 minus 0.43 squared."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's get the calculator out to actually figure out our sample variance. Let me get the calculator out. And we have, so I'll do the numerator first. I have 57 times 0 minus 0.43 squared plus 43 times 1 minus 0.43 squared. And then all of that divided by 100 minus 1, or 99, is equal to 0.2475. So this is equal to, so my sample variance is equal to 0.2475. And if I want to figure out my sample standard deviation, I just take the square root of that."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I have 57 times 0 minus 0.43 squared plus 43 times 1 minus 0.43 squared. And then all of that divided by 100 minus 1, or 99, is equal to 0.2475. So this is equal to, so my sample variance is equal to 0.2475. And if I want to figure out my sample standard deviation, I just take the square root of that. My sample standard deviation is just going to be the square root of my sample variance. So I take the square root of that value that I just had, which is 0.497. So actually, let me just round that up as 0.50."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if I want to figure out my sample standard deviation, I just take the square root of that. My sample standard deviation is just going to be the square root of my sample variance. So I take the square root of that value that I just had, which is 0.497. So actually, let me just round that up as 0.50. So my sample standard deviation is 0.50. Now, if you just look at this, you say, OK, well your best estimate of the percentage of people voting for A or B is really what you just saw here. Your best estimate of the mean is that 43% of people are going to vote for B and everyone else is going to vote for A."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So actually, let me just round that up as 0.50. So my sample standard deviation is 0.50. Now, if you just look at this, you say, OK, well your best estimate of the percentage of people voting for A or B is really what you just saw here. Your best estimate of the mean is that 43% of people are going to vote for B and everyone else is going to vote for A. But an interesting question is, how good of a sample is that? And let's take it to the next level. Let's try to think of an interval around 43% for which we are 95% we're reasonably confident, or roughly 95% sure, that the real mean is in that interval."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Your best estimate of the mean is that 43% of people are going to vote for B and everyone else is going to vote for A. But an interesting question is, how good of a sample is that? And let's take it to the next level. Let's try to think of an interval around 43% for which we are 95% we're reasonably confident, or roughly 95% sure, that the real mean is in that interval. And let me make it very clear. Let me draw. So when we get our sample mean, we are sampling from the sampling distribution of the sampling mean."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's try to think of an interval around 43% for which we are 95% we're reasonably confident, or roughly 95% sure, that the real mean is in that interval. And let me make it very clear. Let me draw. So when we get our sample mean, we are sampling from the sampling distribution of the sampling mean. Let's let me draw that. The sampling distribution of the sample mean. So since we're sampling from a discrete distribution, it's actually going to be a discrete distribution, but it's going to have 100 possible values."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when we get our sample mean, we are sampling from the sampling distribution of the sampling mean. Let's let me draw that. The sampling distribution of the sample mean. So since we're sampling from a discrete distribution, it's actually going to be a discrete distribution, but it's going to have 100 possible values. This can take on 100 different values here, really anything between 0 and 1. But I'll draw it kind of continuous, because it would be hard for me to draw 100 different bars. If I did, you'd have a bar there, you'd have a bar there."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So since we're sampling from a discrete distribution, it's actually going to be a discrete distribution, but it's going to have 100 possible values. This can take on 100 different values here, really anything between 0 and 1. But I'll draw it kind of continuous, because it would be hard for me to draw 100 different bars. If I did, you'd have a bar there, you'd have a bar there. The odds that your sample mean would be 1 would be very low probability, and then you would have one more bar, a bar like that, a bar like that. But that takes forever to draw, so I'm just going to approximate it with this normal curve right over there. And so the sampling distribution of the sample mean, let me write it over here."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If I did, you'd have a bar there, you'd have a bar there. The odds that your sample mean would be 1 would be very low probability, and then you would have one more bar, a bar like that, a bar like that. But that takes forever to draw, so I'm just going to approximate it with this normal curve right over there. And so the sampling distribution of the sample mean, let me write it over here. So this is the sampling distribution of the sample mean, it has some mean here. And I can denote it with the mu sub x bar. This tells us this is the mean of the sampling of the sample distribution."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so the sampling distribution of the sample mean, let me write it over here. So this is the sampling distribution of the sample mean, it has some mean here. And I can denote it with the mu sub x bar. This tells us this is the mean of the sampling of the sample distribution. But we know from many, many videos that this is going to be the same thing as the population mean that we are sampling from, that each sample comes from, each of these 100 samples come from. So this is going to be equal to mu, which is going to be equal to p. So this is going to be equal to mu, which is equal to p. Now, this variance over here, the variance of this distribution, let me draw it like this. Or even better, let's say the standard deviation of this distribution."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This tells us this is the mean of the sampling of the sample distribution. But we know from many, many videos that this is going to be the same thing as the population mean that we are sampling from, that each sample comes from, each of these 100 samples come from. So this is going to be equal to mu, which is going to be equal to p. So this is going to be equal to mu, which is equal to p. Now, this variance over here, the variance of this distribution, let me draw it like this. Or even better, let's say the standard deviation of this distribution. That distance right over here. The standard deviation of the sampling distribution of the sample mean, we've seen it multiple times already, it's going to be this standard deviation. It's going to be the standard deviation of our population distribution."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or even better, let's say the standard deviation of this distribution. That distance right over here. The standard deviation of the sampling distribution of the sample mean, we've seen it multiple times already, it's going to be this standard deviation. It's going to be the standard deviation of our population distribution. So that standard deviation is going to be that distance over there. So there's some standard deviation associated with this distribution. It's going to be that standard deviation divided by the square root of our sample size."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be the standard deviation of our population distribution. So that standard deviation is going to be that distance over there. So there's some standard deviation associated with this distribution. It's going to be that standard deviation divided by the square root of our sample size. And we saw many videos ago why that at least experimentally makes sense, or why it intuitively makes sense. So it's going to be the square root of 100. So it's going to be this guy divided by 10."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be that standard deviation divided by the square root of our sample size. And we saw many videos ago why that at least experimentally makes sense, or why it intuitively makes sense. So it's going to be the square root of 100. So it's going to be this guy divided by 10. Now, we do not know what this guy is. The only way to figure out what that guy is is to actually survey 100 million people, which would have been impossible. So to estimate the standard deviation of this, we will use our sampling standard deviation as our best estimate for the population standard deviation."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be this guy divided by 10. Now, we do not know what this guy is. The only way to figure out what that guy is is to actually survey 100 million people, which would have been impossible. So to estimate the standard deviation of this, we will use our sampling standard deviation as our best estimate for the population standard deviation. So we could say, now remember, this is an estimate. We cannot come up with the exact number for this just from a sample, but we can estimate it. Because this is our best estimator for this standard deviation, and if we divide it by 10, we will have our best estimator for the standard deviation of the sampling distribution of the sampling mean."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So to estimate the standard deviation of this, we will use our sampling standard deviation as our best estimate for the population standard deviation. So we could say, now remember, this is an estimate. We cannot come up with the exact number for this just from a sample, but we can estimate it. Because this is our best estimator for this standard deviation, and if we divide it by 10, we will have our best estimator for the standard deviation of the sampling distribution of the sampling mean. So remember, this is just an estimate. So you kind of have to take everything after this point with a little bit of a grain of salt. So it's going to be roughly equal to, or an estimate of it is going to be 0.5."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because this is our best estimator for this standard deviation, and if we divide it by 10, we will have our best estimator for the standard deviation of the sampling distribution of the sampling mean. So remember, this is just an estimate. So you kind of have to take everything after this point with a little bit of a grain of salt. So it's going to be roughly equal to, or an estimate of it is going to be 0.5. And remember, every time we take a different sample from here, this number is going to change. So this isn't like something in stone. This is dependent on our sample."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be roughly equal to, or an estimate of it is going to be 0.5. And remember, every time we take a different sample from here, this number is going to change. So this isn't like something in stone. This is dependent on our sample. So it's going to wiggle around a little bit, depending on what numbers we actually get in our sample. But it's going to be 0.50. This is the s right here."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is dependent on our sample. So it's going to wiggle around a little bit, depending on what numbers we actually get in our sample. But it's going to be 0.50. This is the s right here. This is the s right over here. This 0.50 divided by 10, which is equal to 0.05. So our best estimate of this standard deviation is 0.05, or you could even view it as 5%."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the s right here. This is the s right over here. This 0.50 divided by 10, which is equal to 0.05. So our best estimate of this standard deviation is 0.05, or you could even view it as 5%. Now, what I want to do is come up with an interval around this sample mean where I'm reasonably confident, using all of my estimates and all of that, where I'm reasonably confident that there's a 95% chance that the true mean is within two standard deviations. Or let me put it this way. There's a 95% chance that the true mean is in that interval."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our best estimate of this standard deviation is 0.05, or you could even view it as 5%. Now, what I want to do is come up with an interval around this sample mean where I'm reasonably confident, using all of my estimates and all of that, where I'm reasonably confident that there's a 95% chance that the true mean is within two standard deviations. Or let me put it this way. There's a 95% chance that the true mean is in that interval. So let me write this down. I want to find an interval such that I am reasonably confident, and I'm putting this kind of touchy-feely language over here, because it's all around the fact that I don't know for a fact that this standard deviation is 0.05. I'm just estimating."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a 95% chance that the true mean is in that interval. So let me write this down. I want to find an interval such that I am reasonably confident, and I'm putting this kind of touchy-feely language over here, because it's all around the fact that I don't know for a fact that this standard deviation is 0.05. I'm just estimating. But I'm reasonably confident that there is a 95% chance that the true mean of the population, which is the same thing as the proportion of the population who are going to vote for person B, or the proportion of the population that are going to be a 1. So this is also, we just have to remember, that mu is equal to p, that there's a 95% chance that the true p is in that interval. And actually, since I've already gone 14 minutes into this video, I'm going to pause this video."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just estimating. But I'm reasonably confident that there is a 95% chance that the true mean of the population, which is the same thing as the proportion of the population who are going to vote for person B, or the proportion of the population that are going to be a 1. So this is also, we just have to remember, that mu is equal to p, that there's a 95% chance that the true p is in that interval. And actually, since I've already gone 14 minutes into this video, I'm going to pause this video. I'm going to stop this video here. And maybe I'll even let you think about it, just based on what everything we've done so far. We figured out the sample mean right over here."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, since I've already gone 14 minutes into this video, I'm going to pause this video. I'm going to stop this video here. And maybe I'll even let you think about it, just based on what everything we've done so far. We figured out the sample mean right over here. We've figured out an estimate for the, and remember, this is just a sampling mean. We don't know the true, this is the mean of our sample. We don't know the true mean of the sampling distribution."}, {"video_title": "Margin of error 1   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We figured out the sample mean right over here. We've figured out an estimate for the, and remember, this is just a sampling mean. We don't know the true, this is the mean of our sample. We don't know the true mean of the sampling distribution. And we also don't know the true standard deviation of the sampling distribution. But we were able to estimate it with the sample standard deviation. Now, everything that we have so far, and based on what we've seen before on confidence intervals and all of that, how can we find an interval such that roughly, and I'm saying roughly because we had to estimate the standard deviation, that there's a 95% chance that the true mean of our population, or that p, the proportion of the population saying 1, is in that interval."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm using it essentially to get some practice on some statistics problems. So here, number two. The grades on a statistics midterm for a high school are normally distributed with a mean of 81 and a standard deviation of 6.3. All right. Calculate the z-scores for each of the following exam grades, draw and label a sketch for each example. We could probably do it all on the same example, but the first thing we'd have to do is just remember what is a z-score? What is a z-score?"}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right. Calculate the z-scores for each of the following exam grades, draw and label a sketch for each example. We could probably do it all on the same example, but the first thing we'd have to do is just remember what is a z-score? What is a z-score? Z-score is literally just measuring how many standard deviations away from the mean. Just like that. So we literally just have to calculate how many standard deviations each of these guys are from the mean, and that's their z-scores."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is a z-score? Z-score is literally just measuring how many standard deviations away from the mean. Just like that. So we literally just have to calculate how many standard deviations each of these guys are from the mean, and that's their z-scores. So let me do number or part a. So we have 65. So first we can just figure out how far is 65 from the mean, let me just draw one chart here that we could use the entire time."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we literally just have to calculate how many standard deviations each of these guys are from the mean, and that's their z-scores. So let me do number or part a. So we have 65. So first we can just figure out how far is 65 from the mean, let me just draw one chart here that we could use the entire time. So if this is our distribution, let's see, we have a mean of 81. So we have a mean of 81, that's our mean. And then a standard deviation of 6.3."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So first we can just figure out how far is 65 from the mean, let me just draw one chart here that we could use the entire time. So if this is our distribution, let's see, we have a mean of 81. So we have a mean of 81, that's our mean. And then a standard deviation of 6.3. So our distribution, they're telling us that it's normally distributed, so I can draw a nice bell curve here. They're saying that it's normally distributed, so that's as good of a bell curve as I'm capable of drawing. This is the mean right there at 81."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then a standard deviation of 6.3. So our distribution, they're telling us that it's normally distributed, so I can draw a nice bell curve here. They're saying that it's normally distributed, so that's as good of a bell curve as I'm capable of drawing. This is the mean right there at 81. And the standard deviation is 6.3. So one standard deviation above and below is going to be 6.3 away from that mean. So if we go 6.3 in the positive direction, that value right there is going to be 87.3."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the mean right there at 81. And the standard deviation is 6.3. So one standard deviation above and below is going to be 6.3 away from that mean. So if we go 6.3 in the positive direction, that value right there is going to be 87.3. If we go 6.3 in the negative direction, where does that get us, that is close to what, 74.7. If we add 6, it'll get us to 80.7, and then 0.3 will get us to 81. So that's one standard deviation below and above the mean, and then you would add another 6.3 to go to two standard deviations."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if we go 6.3 in the positive direction, that value right there is going to be 87.3. If we go 6.3 in the negative direction, where does that get us, that is close to what, 74.7. If we add 6, it'll get us to 80.7, and then 0.3 will get us to 81. So that's one standard deviation below and above the mean, and then you would add another 6.3 to go to two standard deviations. So on and so forth. So that's at least a drawing of the distribution itself. So let's figure out the z-scores for each of these scores."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's one standard deviation below and above the mean, and then you would add another 6.3 to go to two standard deviations. So on and so forth. So that's at least a drawing of the distribution itself. So let's figure out the z-scores for each of these scores. So 65, or each of these grades. 65 is how far? 65 is, you know, it's maybe going to be here someplace."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's figure out the z-scores for each of these scores. So 65, or each of these grades. 65 is how far? 65 is, you know, it's maybe going to be here someplace. So we first want to say, well, how far is it just from our mean, from our mean? So the distance is, we just want a positive number here. Well, actually, you want a negative number, because you want your z-score to be positive or negative."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "65 is, you know, it's maybe going to be here someplace. So we first want to say, well, how far is it just from our mean, from our mean? So the distance is, we just want a positive number here. Well, actually, you want a negative number, because you want your z-score to be positive or negative. Negative would mean to the left of the mean, and positive would mean to the right of the mean. So we say 65 minus 81. So that's literally how far away we are, but we want that in terms of standard deviations."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, actually, you want a negative number, because you want your z-score to be positive or negative. Negative would mean to the left of the mean, and positive would mean to the right of the mean. So we say 65 minus 81. So that's literally how far away we are, but we want that in terms of standard deviations. So we divide that by the length, or the magnitude, of our standard deviation. So 65 minus 81. Let's see, 81 minus 65 is what?"}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's literally how far away we are, but we want that in terms of standard deviations. So we divide that by the length, or the magnitude, of our standard deviation. So 65 minus 81. Let's see, 81 minus 65 is what? It is 5 plus 11, it's 16. So this is going to be minus 16 over 6.3. And take our calculator out, and let's see, if we have minus 16 divided by 6.3, you get minus 2.54."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see, 81 minus 65 is what? It is 5 plus 11, it's 16. So this is going to be minus 16 over 6.3. And take our calculator out, and let's see, if we have minus 16 divided by 6.3, you get minus 2.54. So approximately equal to minus 2.54. That's the z-score for a grade of 65. Pretty straightforward."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "And take our calculator out, and let's see, if we have minus 16 divided by 6.3, you get minus 2.54. So approximately equal to minus 2.54. That's the z-score for a grade of 65. Pretty straightforward. Let's do a couple more. Let's do all of them. 83."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Pretty straightforward. Let's do a couple more. Let's do all of them. 83. So how far is away from the mean? Well, it's 83 minus 81. It's two grades above the mean."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "83. So how far is away from the mean? Well, it's 83 minus 81. It's two grades above the mean. But we want it in terms of standard deviations. How many standard deviations? So this was part a. a was right here."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's two grades above the mean. But we want it in terms of standard deviations. How many standard deviations? So this was part a. a was right here. We were at 2.5 standard deviations below the mean. So this was part a. 1, 2, and then 0.5, so this was a right there, 65."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this was part a. a was right here. We were at 2.5 standard deviations below the mean. So this was part a. 1, 2, and then 0.5, so this was a right there, 65. And then part b, 83. 83 is going to be right here, a little bit higher. We're right here."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "1, 2, and then 0.5, so this was a right there, 65. And then part b, 83. 83 is going to be right here, a little bit higher. We're right here. And the z-score here, 83 minus 81 divided by 6.3 will get us, let's see, clear the calculator. So we have, well, 83 minus 81 is 2. Divided by 6.3, 0.32, roughly."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're right here. And the z-score here, 83 minus 81 divided by 6.3 will get us, let's see, clear the calculator. So we have, well, 83 minus 81 is 2. Divided by 6.3, 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard deviations above the mean. And so it would be roughly 1 third of the standard deviation along the way, right?"}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Divided by 6.3, 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard deviations above the mean. And so it would be roughly 1 third of the standard deviation along the way, right? Because this was one whole standard deviation. So we're 0.3 of a standard deviation above the mean. Choice number c. Or not choice, part c, I guess I should call it."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so it would be roughly 1 third of the standard deviation along the way, right? Because this was one whole standard deviation. So we're 0.3 of a standard deviation above the mean. Choice number c. Or not choice, part c, I guess I should call it. 93. Well, we do the same exercise. 93 is how much above the mean?"}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Choice number c. Or not choice, part c, I guess I should call it. 93. Well, we do the same exercise. 93 is how much above the mean? Well, 93 minus 81 is 12. But we want it in terms of standard deviations. So 12 is how many standard deviations above the mean?"}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "93 is how much above the mean? Well, 93 minus 81 is 12. But we want it in terms of standard deviations. So 12 is how many standard deviations above the mean? Well, it's going to be almost 2. Let's take the calculator out. So we get 12 divided by 6.3."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 12 is how many standard deviations above the mean? Well, it's going to be almost 2. Let's take the calculator out. So we get 12 divided by 6.3. It's 1.9 standard deviations. It's z-score. It's z-score is 1.9, which means it's 1.9 standard deviations above the mean."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we get 12 divided by 6.3. It's 1.9 standard deviations. It's z-score. It's z-score is 1.9, which means it's 1.9 standard deviations above the mean. So the mean is 81. We go one whole standard deviation, and then 0.9 standard deviations. And that's where a score of 93 would lie right there."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's z-score is 1.9, which means it's 1.9 standard deviations above the mean. So the mean is 81. We go one whole standard deviation, and then 0.9 standard deviations. And that's where a score of 93 would lie right there. It's z-score is 1.9. That all that means is 1.9 standard deviations above the mean. Let's do the last one."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's where a score of 93 would lie right there. It's z-score is 1.9. That all that means is 1.9 standard deviations above the mean. Let's do the last one. I'll do it in magenta. D, part d. Score of 100. We don't even need the problem anymore."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's do the last one. I'll do it in magenta. D, part d. Score of 100. We don't even need the problem anymore. A score of 100. Well, same thing. We figure out how far is 100 above the mean."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't even need the problem anymore. A score of 100. Well, same thing. We figure out how far is 100 above the mean. Remember, the mean was 81. And we divide that by the length or the size or the magnitude of our standard deviation. So 100 minus 81 is equal to 19 over 6.3."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "We figure out how far is 100 above the mean. Remember, the mean was 81. And we divide that by the length or the size or the magnitude of our standard deviation. So 100 minus 81 is equal to 19 over 6.3. So it's going to be a little over 3 standard deviations. And in the next problem, we'll see what does that imply in terms of the probability of that actually occurring. But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 100 minus 81 is equal to 19 over 6.3. So it's going to be a little over 3 standard deviations. And in the next problem, we'll see what does that imply in terms of the probability of that actually occurring. But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, if I were to round. So it's very close to 3.02."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, if I were to round. So it's very close to 3.02. It's z-score is 3.02. Or a grade of 100 is 3.02 standard deviations above the mean. So remember, this was the mean right here."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's very close to 3.02. It's z-score is 3.02. Or a grade of 100 is 3.02 standard deviations above the mean. So remember, this was the mean right here. Right here at 81. We go one standard deviation above the mean. Two standard deviations above the mean."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "So remember, this was the mean right here. Right here at 81. We go one standard deviation above the mean. Two standard deviations above the mean. The third standard deviation above the mean is right there. So we're sitting right there on our chart. A little bit above that."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two standard deviations above the mean. The third standard deviation above the mean is right there. So we're sitting right there on our chart. A little bit above that. 3.02 standard deviations above the mean, that's where a score of 100 would be. And you can see the probability, the height of this, that's what the chart tells us. It's actually a very low probability."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "A little bit above that. 3.02 standard deviations above the mean, that's where a score of 100 would be. And you can see the probability, the height of this, that's what the chart tells us. It's actually a very low probability. And actually, it's not just a very low probability of getting something higher than that. Because we've learned before in a probability density function, the probability, if this is a continuous, not a discrete, the probability of getting exactly that is 0 if this wasn't discrete. But since this is a score, it's not a test, we know that it's actually a discrete probability function."}, {"video_title": "ck12.org normal distribution problems z-score   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually a very low probability. And actually, it's not just a very low probability of getting something higher than that. Because we've learned before in a probability density function, the probability, if this is a continuous, not a discrete, the probability of getting exactly that is 0 if this wasn't discrete. But since this is a score, it's not a test, we know that it's actually a discrete probability function. But the probability is low of getting higher than that. Because you can see where we sit in the bell curve. Well, anyway, hopefully this at least clarified how to solve for z-scores, which is pretty straightforward mathematically."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "At one of its large factories, 2% of the chips produced are defective in some way. A quality check involves randomly selecting and testing 500 chips. What are the mean and standard deviation of the number of defective processing chips in these samples? So like always, try to pause this video and have a go at it on your own, and then we will work through it together. All right, so let me define a random variable that we're gonna find the mean and standard deviation of. And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. So let's let X be equal to the number of defective chips in 500-chip sample."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So like always, try to pause this video and have a go at it on your own, and then we will work through it together. All right, so let me define a random variable that we're gonna find the mean and standard deviation of. And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. So let's let X be equal to the number of defective chips in 500-chip sample. So the first thing to recognize is that this will be a binomial variable. This is binomial. How do we know it's binomial?"}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So let's let X be equal to the number of defective chips in 500-chip sample. So the first thing to recognize is that this will be a binomial variable. This is binomial. How do we know it's binomial? Well, it's made up of 500, it's a finite number of trials right over here. The probability of getting a defective chip, you could view this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "How do we know it's binomial? Well, it's made up of 500, it's a finite number of trials right over here. The probability of getting a defective chip, you could view this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. So we would view the probability of a defect, or I should say a defective chip, it is constant across these 500 trials, and we will assume that they are independent of each other, 0.02. You might be saying, hey, well, are we replacing the chips before or after? But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. So we would view the probability of a defect, or I should say a defective chip, it is constant across these 500 trials, and we will assume that they are independent of each other, 0.02. You might be saying, hey, well, are we replacing the chips before or after? But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. They're not really telling us that right over here. So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here. And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. They're not really telling us that right over here. So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here. And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. So the mean, the mean of x, which is the same thing as the expected value of x, is going to be equal to the number of trials, n, times the probability of success on each trial, times p. So what is this going to be? Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. So the mean, the mean of x, which is the same thing as the expected value of x, is going to be equal to the number of trials, n, times the probability of success on each trial, times p. So what is this going to be? Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02. And what is this going to be? 500 times 2 hundredths is going to be, it's going to be equal to 10. So that is your expected value of the number of defective processing chips, or the mean."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So it's 500 times 0.02. And what is this going to be? 500 times 2 hundredths is going to be, it's going to be equal to 10. So that is your expected value of the number of defective processing chips, or the mean. Now what about the standard deviation? So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x. So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So that is your expected value of the number of defective processing chips, or the mean. Now what about the standard deviation? So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x. So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial. And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial. And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. So times 0.98, and all of this is under the radical sign. I didn't make that radical sign long enough. And so what is this going to be?"}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. So times 0.98, and all of this is under the radical sign. I didn't make that radical sign long enough. And so what is this going to be? Well, let's see, 500 times 0.02, we already said that this is going to be 10, 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable   AP Statistics   Khan Academy.mp3", "Sentence": "And so what is this going to be? Well, let's see, 500 times 0.02, we already said that this is going to be 10, 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value. So I'm gonna take 9.8, and then take the square root of it, and I get three point, if I round to the nearest hundredth, 3.13. So this is approximately 3.13 for the standard deviation. If I wanted the variance, it would be 9.8."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "What we're going to do in this video is talk about the idea of a sampling distribution. Now, just to make things a little bit concrete, let's imagine that we have a population of some kind. Let's say it's a bunch of balls, each of them have a number written on it. For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter. And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter?"}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter. And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in and then we'll pick another ball."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in and then we'll pick another ball. So these are going to be independent events and it's gonna be with replacement. And so let's say we could pick a one and then a one. We could pick a one then a two, a one and a three."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And remember, when we pick one ball, we'll record that number, then we'll put it back in and then we'll pick another ball. So these are going to be independent events and it's gonna be with replacement. And so let's say we could pick a one and then a one. We could pick a one then a two, a one and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or three and a three."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "We could pick a one then a two, a one and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "We could pick a three and a one, a three and a two, or three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. Here it is 1.5. Here it is two. Here it is 1.5."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So for this one, the sample mean is one. Here it is 1.5. Here it is two. Here it is 1.5. Here it is two. Here it is 2.5. Here it is two."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Here it is 1.5. Here it is two. Here it is 2.5. Here it is two. Here it is 2.5. And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Here it is two. Here it is 2.5. And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. So let me make a little chart right over here, a little graph right over here."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5. We could get a two, we could get a 2.5."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5. We could get a two, we could get a 2.5. Or we can get a three. And now let's see the frequency of it. And I will put that over here."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "We could get a two, we could get a 2.5. Or we can get a three. And now let's see the frequency of it. And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5?"}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two."}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5?"}, {"video_title": "Introduction to sampling distributions   Sampling distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5? Well, there's two 2.5s, so two out of the nine times. Another way you could interpret this is, when you take a random sample with replacement of two balls, you have a 2 9th chance of having a sample mean of 2.5. And then last but not least, right over here, there's one scenario out of the nine where you get 2 3, so 1 9th."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "And we're gonna talk about two things. The different conclusions you might make based on the different significance levels that you might set, and also why it's important to set your significance levels ahead of time, before you conduct an experiment and calculate the p-values, for frankly, ethical purposes. So to help us get this, let's look at a scenario right over here, which tells us, Rahim heard that spinning, rather than flipping a penny, raises the probability above 50% that the penny lands showing heads. That's actually quite fascinating, if that's true. He tested this by spinning 10 different pennies, 10 times each, so that would be a total of 100 spins. His hypotheses were, his null hypothesis is that, by spinning, your proportion doesn't change, rather, versus flipping, it's still 50%. And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "That's actually quite fascinating, if that's true. He tested this by spinning 10 different pennies, 10 times each, so that would be a total of 100 spins. His hypotheses were, his null hypothesis is that, by spinning, your proportion doesn't change, rather, versus flipping, it's still 50%. And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. In his 100 spins, the penny landed showing heads in 59 spins. Rahim calculated that the statistic, so this is the sample proportion here, it's 59 out of 100 were heads, so that's 0.59, or 59 hundredths, and he calculated, had an associated p-value of approximately 0.036. So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make?"}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. In his 100 spins, the penny landed showing heads in 59 spins. Rahim calculated that the statistic, so this is the sample proportion here, it's 59 out of 100 were heads, so that's 0.59, or 59 hundredths, and he calculated, had an associated p-value of approximately 0.036. So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? And, while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out. So, let's, first of all, remind ourselves what a p-value even is."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? And, while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out. So, let's, first of all, remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. And if that is low enough, if it is some, if it's below some threshold, which is our significance level, then we will reject the null hypothesis. And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "So, let's, first of all, remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. And if that is low enough, if it is some, if it's below some threshold, which is our significance level, then we will reject the null hypothesis. And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05. And because of that, we would reject the null hypothesis. And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05. And because of that, we would reject the null hypothesis. And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. And if you reject your null hypothesis, you could also say that suggests our alternative hypothesis that the true proportion of spins that a penny would land showing heads is greater than 50%. Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. And if you reject your null hypothesis, you could also say that suggests our alternative hypothesis that the true proportion of spins that a penny would land showing heads is greater than 50%. Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. And so here, we would say that we fail, fail to reject our null hypothesis. So we're failing to reject this right over here, and it will not help us suggest our alternative hypothesis. And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. And so here, we would say that we fail, fail to reject our null hypothesis. So we're failing to reject this right over here, and it will not help us suggest our alternative hypothesis. And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. If you're a researcher of some kind, you want to have an interesting result. You want to discover something. You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. If you're a researcher of some kind, you want to have an interesting result. You want to discover something. You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. We can reject the assumption, the status quo. I found something that actually makes a difference. And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis."}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. We can reject the assumption, the status quo. I found something that actually makes a difference. And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. If you did that, that would be very unethical. In future videos, we'll start thinking about the question of, okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 1 100th?"}, {"video_title": "Comparing P-values to different significance levels   AP Statistics   Khan Academy.mp3", "Sentence": "And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. If you did that, that would be very unethical. In future videos, we'll start thinking about the question of, okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 1 100th? When should it be 5 100ths? When should it be 10 100ths? Or when should it be something else?"}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so let's say we can set that, and let's make that 60% of the gumballs are green. But let's say someone else comes along and they don't actually know the proportion of gumballs that are green, but they can take samples. And so let's say they take samples of 50 at a time. And so they draw a sample. The sample proportion right over here actually just happened to be 0.6. But then they could draw another sample. This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so they draw a sample. The sample proportion right over here actually just happened to be 0.6. But then they could draw another sample. This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. Now you could say, all right, well, these are all different estimates, but for any given estimate, how confident are we that a certain range around that estimate actually contains the true population proportion? And so if we look at this tab right over here, that's what confidence intervals are good for. And in a previous video, we talked about how you calculate the confidence interval."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. Now you could say, all right, well, these are all different estimates, but for any given estimate, how confident are we that a certain range around that estimate actually contains the true population proportion? And so if we look at this tab right over here, that's what confidence intervals are good for. And in a previous video, we talked about how you calculate the confidence interval. What we wanna do is say, well, there's a 95% chance, and we get that from this confidence level, and 95% is the confidence level people typically use. And so there's a 95% chance that whatever our sample proportion is, that it's within two standard deviations of the true proportion, or that the true proportion is going to be contained in an interval that are two standard deviations on either side of our sample proportion. Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And in a previous video, we talked about how you calculate the confidence interval. What we wanna do is say, well, there's a 95% chance, and we get that from this confidence level, and 95% is the confidence level people typically use. And so there's a 95% chance that whatever our sample proportion is, that it's within two standard deviations of the true proportion, or that the true proportion is going to be contained in an interval that are two standard deviations on either side of our sample proportion. Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. And so this is two standard errors to the right and two standard errors to the left of our sample proportion. And our confidence interval is this entire interval going from this left point to this right point. And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. And so this is two standard errors to the right and two standard errors to the left of our sample proportion. And our confidence interval is this entire interval going from this left point to this right point. And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. Now, what's interesting here about this simulation is that we can see what percentage of the time does our confidence interval, does it actually contain the true parameter? So let me just draw 25 samples at a time. And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter?"}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. Now, what's interesting here about this simulation is that we can see what percentage of the time does our confidence interval, does it actually contain the true parameter? So let me just draw 25 samples at a time. And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? And we can keep sampling over here. And we can see the more samples that we take, it really is approaching that close to 95% of the time, our confidence interval does indeed contain the true parameter. And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? And we can keep sampling over here. And we can see the more samples that we take, it really is approaching that close to 95% of the time, our confidence interval does indeed contain the true parameter. And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. That if we calculate a confidence interval based on a confidence level of 95%, that it is indeed the case that roughly 95% of the time, the true parameter, the population proportion, will be contained in that interval. And I could just draw more and more and more samples. And we can actually see that happening."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. That if we calculate a confidence interval based on a confidence level of 95%, that it is indeed the case that roughly 95% of the time, the true parameter, the population proportion, will be contained in that interval. And I could just draw more and more and more samples. And we can actually see that happening. Every now and then for sure, you get a sample where even when you calculate your confidence interval, the true parameter, the true population proportion is not contained. But that is the exception that happens very infrequently. 95% of the time, your true population parameter is contained in that interval."}, {"video_title": "Confidence interval simulation   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And we can actually see that happening. Every now and then for sure, you get a sample where even when you calculate your confidence interval, the true parameter, the true population proportion is not contained. But that is the exception that happens very infrequently. 95% of the time, your true population parameter is contained in that interval. Now another interesting thing to see is if we increase our sample size, our confidence interval is going to get narrower. So if we increase our sample size, we'll just make it 200. Now let's draw some samples."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Aubrey wanted to see if there's a connection between the time a given exam takes place and the average score of this exam. She collected data about exams from the previous year. Plot the data in a scatter plot, and let's see, they give us a couple of rows here. This is the class, then they give us the period of the day that the class happened, and then they give us the average score on an exam. And one could, you know, we have to be a little careful with this study, maybe there's some correlation depending on what subject is taught during what period, but let's just use her data, at least just based on her data see if, see, well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see, on the vertical, on the horizontal axis, we have period, and on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average scores? So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the class, then they give us the period of the day that the class happened, and then they give us the average score on an exam. And one could, you know, we have to be a little careful with this study, maybe there's some correlation depending on what subject is taught during what period, but let's just use her data, at least just based on her data see if, see, well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see, on the vertical, on the horizontal axis, we have period, and on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average scores? So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period one, average score 93. Period one, average score 93, right over there."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period one, average score 93. Period one, average score 93, right over there. Period six, 87. Period six, 87. 80, oh, that's not the right place, and then we can move it if we want."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Period one, average score 93, right over there. Period six, 87. Period six, 87. 80, oh, that's not the right place, and then we can move it if we want. 87, right over there. Period two, 70. Two, period two, 70."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "80, oh, that's not the right place, and then we can move it if we want. 87, right over there. Period two, 70. Two, period two, 70. Period four, 62. Four, four and 62, right over there. Period four and 86."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Two, period two, 70. Period four, 62. Four, four and 62, right over there. Period four and 86. Period four and again an 86, that's right over there. Period one, 73. Period one, 73."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Period four and 86. Period four and again an 86, that's right over there. Period one, 73. Period one, 73. Period three, average score of 73 as well. Period three, 73. Period one, 80, average score of 80."}, {"video_title": "Constructing a scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Period one, 73. Period three, average score of 73 as well. Period three, 73. Period one, 80, average score of 80. So period one, average score of 80. And then period three, average score of 96. Period three, average score of 96."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "The scatter plot below displays a set of bivariate data along with its least squares regression line. Consider removing the outlier at 95 comma one. So 95 comma one, we're talking about that outlier right over there, and calculating a new least squares regression line. What effects would removing the outlier have? Choose all answers that apply. Like always, pause this video and see if you could figure it out. Well let's see, even with this outlier here, we have an upward sloping regression line."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "What effects would removing the outlier have? Choose all answers that apply. Like always, pause this video and see if you could figure it out. Well let's see, even with this outlier here, we have an upward sloping regression line. And so it looks like our r already is going to be greater than zero, and of course it's going to be less than one. So our r is going to be greater than zero and less than one. We know it's not going to be equal one because then we would go perfectly through all of the dots."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "Well let's see, even with this outlier here, we have an upward sloping regression line. And so it looks like our r already is going to be greater than zero, and of course it's going to be less than one. So our r is going to be greater than zero and less than one. We know it's not going to be equal one because then we would go perfectly through all of the dots. And it's clear that this point right over here is indeed an outlier. The residual between this point and the line is quite high. We have a pretty big distance right over here."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "We know it's not going to be equal one because then we would go perfectly through all of the dots. And it's clear that this point right over here is indeed an outlier. The residual between this point and the line is quite high. We have a pretty big distance right over here. It would be a negative residual. And so this point is definitely bringing down the r, and it's definitely bringing down the slope of the regression line. If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "We have a pretty big distance right over here. It would be a negative residual. And so this point is definitely bringing down the r, and it's definitely bringing down the slope of the regression line. If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. The only reason why the line isn't doing that is it's trying to get close to this point right over here. So if we remove this outlier, our r would increase. So r would increase."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. The only reason why the line isn't doing that is it's trying to get close to this point right over here. So if we remove this outlier, our r would increase. So r would increase. And also the slope of our line would increase. And slope would increase. We'd have a better fit to this positively correlated data."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "So r would increase. And also the slope of our line would increase. And slope would increase. We'd have a better fit to this positively correlated data. And we would no longer have this point dragging the slope down anymore. So let's see which choices apply. The coefficient of determination, r squared, would increase."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "We'd have a better fit to this positively correlated data. And we would no longer have this point dragging the slope down anymore. So let's see which choices apply. The coefficient of determination, r squared, would increase. Well, if r would increase, then squaring that value would increase as well, so I will circle that. The coefficient, the correlation coefficient r would get close to zero. No, in fact, it would get closer to one because we would have a better fit here."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "The coefficient of determination, r squared, would increase. Well, if r would increase, then squaring that value would increase as well, so I will circle that. The coefficient, the correlation coefficient r would get close to zero. No, in fact, it would get closer to one because we would have a better fit here. And so I will rule that out. The slope of the least squares regression line would increase. Yes, indeed, this point, this outlier is pulling it down."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "No, in fact, it would get closer to one because we would have a better fit here. And so I will rule that out. The slope of the least squares regression line would increase. Yes, indeed, this point, this outlier is pulling it down. If you take it out, it'll allow the slope to increase. So I will circle that as well. Let's do another example."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "Yes, indeed, this point, this outlier is pulling it down. If you take it out, it'll allow the slope to increase. So I will circle that as well. Let's do another example. The scatter plot below displays a set of bivariate data along with its least squares regression line. Same idea. Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "Let's do another example. The scatter plot below displays a set of bivariate data along with its least squares regression line. Same idea. Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. So what would happen this time? So as is, without removing this outlier, we have a negative slope for the regression line. So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. So what would happen this time? So as is, without removing this outlier, we have a negative slope for the regression line. So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. Without even removing the outlier, we know it's not going to be negative one. If r was exactly negative one, then it would be a downward-sloping line that went exactly through all of the points. But if we remove this point, what's going to happen?"}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. Without even removing the outlier, we know it's not going to be negative one. If r was exactly negative one, then it would be a downward-sloping line that went exactly through all of the points. But if we remove this point, what's going to happen? Well, this least squares regression is being pulled down here by this outlier. So if you were to remove this point, the least squares regression line could move up on the left-hand side, and so you'll probably have a line that looks more like that. And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "But if we remove this point, what's going to happen? Well, this least squares regression is being pulled down here by this outlier. So if you were to remove this point, the least squares regression line could move up on the left-hand side, and so you'll probably have a line that looks more like that. And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. And so clearly, the new line that I drew after removing the outlier, this has a more negative slope. So removing the outlier would decrease r. r would get closer to negative one. It would be closer to being a perfect negative correlation."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. And so clearly, the new line that I drew after removing the outlier, this has a more negative slope. So removing the outlier would decrease r. r would get closer to negative one. It would be closer to being a perfect negative correlation. And also, it would decrease the slope. Decrease the slope, which choices match that? The coefficient of determination r squared would decrease."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "It would be closer to being a perfect negative correlation. And also, it would decrease the slope. Decrease the slope, which choices match that? The coefficient of determination r squared would decrease. So let's be very careful. r was already negative. If we decrease it, it's going to become more negative."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "The coefficient of determination r squared would decrease. So let's be very careful. r was already negative. If we decrease it, it's going to become more negative. If you square something that is more negative, it's not going to become smaller. Let's say before you remove the data point, r was, I'm just gonna make up a value, let's say it was negative 0.4, and then after removing the outlier, r becomes more negative, and it's going to be equal to negative 0.5. Well, if you square this, this would be positive 0.16, while this would be positive 0.25."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "If we decrease it, it's going to become more negative. If you square something that is more negative, it's not going to become smaller. Let's say before you remove the data point, r was, I'm just gonna make up a value, let's say it was negative 0.4, and then after removing the outlier, r becomes more negative, and it's going to be equal to negative 0.5. Well, if you square this, this would be positive 0.16, while this would be positive 0.25. So if r is already negative, and if you make it more negative, it would not decrease r squared. It actually would increase r squared. So I will rule this one out."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "Well, if you square this, this would be positive 0.16, while this would be positive 0.25. So if r is already negative, and if you make it more negative, it would not decrease r squared. It actually would increase r squared. So I will rule this one out. The slope of the least squares regression line would increase. No, it's going to decrease. It's going to be a stronger negative correlation."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "So I will rule this one out. The slope of the least squares regression line would increase. No, it's going to decrease. It's going to be a stronger negative correlation. Rule that one out. The y-intercept of the least squares regression line would increase. Yes, by getting rid of this outlier, you could think of it as the left side of this line is going to increase, or another way to think about it, the slope of this line is going to decrease."}, {"video_title": "Impact of removing outliers on regression lines   AP Statistics   Khan Academy.mp3", "Sentence": "It's going to be a stronger negative correlation. Rule that one out. The y-intercept of the least squares regression line would increase. Yes, by getting rid of this outlier, you could think of it as the left side of this line is going to increase, or another way to think about it, the slope of this line is going to decrease. It's going to become more negative. We know that the least squares regression line will always go through the mean of both variables. So we're just gonna pivot around the mean of both variables, which would mean that the y-intercept will go higher."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "Here is computer output from Elise Square's regression analysis on her sample. So just to be clear what's going on, she took a sample of phones, they're not telling us exactly how many, but she took a number of phones and she found a linear relationship between processor speed and prices. So this is price right over here and this is processor speed right over here. And then she plotted her sample for every phone would be a data point. And so you see that. And then she put those data points into her computer and was able to come up with a line, a regression line for her sample. And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And then she plotted her sample for every phone would be a data point. And so you see that. And then she put those data points into her computer and was able to come up with a line, a regression line for her sample. And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there. And for her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there. And for her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe. If she were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe. If she were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate. And just because she sees this positive linear relationship in her sample doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship. And so that's why she's doing this hypothesis test."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "But since this is a sample, it's just an estimate. And just because she sees this positive linear relationship in her sample doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship. And so that's why she's doing this hypothesis test. And in a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population. So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And so that's why she's doing this hypothesis test. And in a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population. So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. So the regression line might look something like that. Where the equation of the regression line for the population, Y hat, would be alpha plus beta times X. And so our null hypothesis is that beta is equal to zero."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. So the regression line might look something like that. Where the equation of the regression line for the population, Y hat, would be alpha plus beta times X. And so our null hypothesis is that beta is equal to zero. And the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero. Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y?"}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And so our null hypothesis is that beta is equal to zero. And the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero. Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true, assuming this is the actual slope of the population regression line, I guess you could think about it, what is the probability of us getting this result right over here? And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true, assuming this is the actual slope of the population regression line, I guess you could think about it, what is the probability of us getting this result right over here? And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. And for this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this. So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. And for this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this. So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. Maybe they just figured out for us that this probability is 0.004. And we have to be very, very careful here, because here, they're actually giving us, I guess you could call it a two-sided p-value. If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater?"}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. Maybe they just figured out for us that this probability is 0.004. And we have to be very, very careful here, because here, they're actually giving us, I guess you could call it a two-sided p-value. If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? So if this is t equals zero right here in the middle, and this is 2.999, we care about this region. We care about this right tail. This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999?"}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? So if this is t equals zero right here in the middle, and this is 2.999, we care about this region. We care about this right tail. This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? So it's giving us both of these areas. So if you want the p-value for this scenario, we would just look at this. And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? So it's giving us both of these areas. So if you want the p-value for this scenario, we would just look at this. And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. So this is going to be equal to 0.002. And what you do in any significance test is then compare your p-value to your level of significance. And so if you look at 0.002 and compare it to 0.01, which of these is greater?"}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. So this is going to be equal to 0.002. And what you do in any significance test is then compare your p-value to your level of significance. And so if you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first your eyes might say, hey, two is greater than one, but this is 2,000th versus 1,000th. This is 10,000th right over here. So in this situation, our p-value is less than our level of significance."}, {"video_title": "Using a P-value to make conclusions in a test about slope   AP Statistics   Khan Academy.mp3", "Sentence": "And so if you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first your eyes might say, hey, two is greater than one, but this is 2,000th versus 1,000th. This is 10,000th right over here. So in this situation, our p-value is less than our level of significance. And so we're saying, hey, the probability of getting a result this extreme or more extreme is so low if we assume our null hypothesis that in this situation we will reject, we will decide to reject our null hypothesis, which would suggest the alternative. So is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones? Yes."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's imagine ourselves in some type of a strange casino with very strange games. And you walk up to a table, and on that table there is an empty bag. And the guy who runs the table says, look, I've got some marbles here, three green marbles, two orange marbles, and I'm going to stick them in the bag. And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play?"}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game?"}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game."}, {"video_title": "Dependent probability introduction   Probability and Statistics   Khan Academy.mp3", "Sentence": "30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game. Now, one thing I will let you think about is, would you want to play this game if you could replace the green marble, the first pick after the first pick? If you could replace the green marble, would you want to pick? Would you want to play the game in that scenario?"}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "And under which situations does it look skewed right? So does it look something like this? And under which situations does it look skewed left? Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions. So with that in our minds, let's do some examples here. So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions. So with that in our minds, let's do some examples here. So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "All right, so right over here, we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. So the mean is going to be 12%."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. So the mean is going to be 12%. So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So the mean is going to be 12%. So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail. So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "You're gonna have a big, long tail. So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it out."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it out. All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, pause this video and see if you can figure it out. All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88. So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10?"}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So P is 0.88. So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition?"}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "This is actually going to be over 100. So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "I didn't even have to calculate it. I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb. Is it gonna be perfectly normal? No."}, {"video_title": "Normal conditions for sampling distributions of sample proportions   AP Statistics   Khan Academy.mp3", "Sentence": "It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb. Is it gonna be perfectly normal? No. In fact, if we didn't have this rule of thumb to kind of draw the line, some might even argue that, well, we still have a longer tail to the left than we do to the right. Maybe it's skewed to the left. But using this threshold, using this rule of thumb, which is the standard in statistics, this would be viewed as approximately normal."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "My idea is to make the background color of my website yellow. But after making that change, how do I feel good about this actually having the intended consequence? Well, that's where significance tests come into play. What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% or 10%. You might see other ones, but we're gonna set a significance level for this particular case."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% or 10%. You might see other ones, but we're gonna set a significance level for this particular case. Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what is the probability of getting a sample with the statistics that we get? And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "You might see other ones, but we're gonna set a significance level for this particular case. Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what is the probability of getting a sample with the statistics that we get? And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going to be. And then we decide whether we can reject the null hypothesis."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going to be. And then we decide whether we can reject the null hypothesis. So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And then we decide whether we can reject the null hypothesis. So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypothesis."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypothesis. And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "I would just say that we do not, do not reject, reject the null hypothesis. And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which you can interpret as saying that, hey, if we assume the null hypothesis is true, that there's no change due to making the background yellow, I would have a 50% chance of getting this result. And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which you can interpret as saying that, hey, if we assume the null hypothesis is true, that there's no change due to making the background yellow, I would have a 50% chance of getting this result. And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very, very, very clear."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very, very, very clear. Our p-value, the thing that we're using to decide whether or not we reject the null hypothesis, this is the probability of getting your sample statistics given that the null hypothesis is true. Sometimes people confuse this and they say, hey, is this the probability that the null hypothesis is true given the sample, given the sample statistics that we got? And I would say clearly, no, that is not the case."}, {"video_title": "P-values and significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Now, there's one last point of clarification that I wanna make very, very, very clear. Our p-value, the thing that we're using to decide whether or not we reject the null hypothesis, this is the probability of getting your sample statistics given that the null hypothesis is true. Sometimes people confuse this and they say, hey, is this the probability that the null hypothesis is true given the sample, given the sample statistics that we got? And I would say clearly, no, that is not the case. We are not trying to gauge the probability that the null hypothesis is true or not. What we are trying to do is say, hey, if we assume the null hypothesis were true, what is the probability that we got the result that we did for our sample? And if that probability is low, if it's below some threshold that we set ahead of time, then we decide to reject the null hypothesis and say that we have evidence for the alternative."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "Now before I calculate the correlation coefficient, let's just make sure we understand some of these other statistics that they've given us. So we assume that these are samples of the x and the corresponding y from a broader population. And so we have the sample mean for x and the sample standard deviation for x. The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things. All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "Until you realize a few things. All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here. So in this particular situation, r is going to be equal to one over n minus one. We have four pairs, so it's gonna be one over three, and it's gonna be times a sum of the products of the z-scores. So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x sample mean divided by the x sample standard deviation, 0.816, that times one, now we're looking at the y variable, the y z-score, so it's one minus three, one minus three over the y sample standard deviation, 2.160, and we're just going to keep doing that."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here. So in this particular situation, r is going to be equal to one over n minus one. We have four pairs, so it's gonna be one over three, and it's gonna be times a sum of the products of the z-scores. So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x sample mean divided by the x sample standard deviation, 0.816, that times one, now we're looking at the y variable, the y z-score, so it's one minus three, one minus three over the y sample standard deviation, 2.160, and we're just going to keep doing that. I'll do it like this. So the next one, it's going to be two minus two over 0.816, this is where I got the two from, and I'm subtracting from that the sample mean right over here, times, now we're looking at this two, two minus three over 2.160 plus, I'm happy there's only four pairs here, two minus two again, two minus two over 0.816, times, now we're gonna have three minus three, three minus three over 2.160, and then the last pair, you're going to have three minus two over 0.816 times six minus three, six minus three over 2.160. So before I get a calculator out, let's see if there's some simplifications I can do."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x sample mean divided by the x sample standard deviation, 0.816, that times one, now we're looking at the y variable, the y z-score, so it's one minus three, one minus three over the y sample standard deviation, 2.160, and we're just going to keep doing that. I'll do it like this. So the next one, it's going to be two minus two over 0.816, this is where I got the two from, and I'm subtracting from that the sample mean right over here, times, now we're looking at this two, two minus three over 2.160 plus, I'm happy there's only four pairs here, two minus two again, two minus two over 0.816, times, now we're gonna have three minus three, three minus three over 2.160, and then the last pair, you're going to have three minus two over 0.816 times six minus three, six minus three over 2.160. So before I get a calculator out, let's see if there's some simplifications I can do. Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero. Two minus two is zero, three minus three is zero, is gonna actually be zero times zero, so that whole thing is zero. Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so this is going to be R is equal to 1 3rd times, negative times negative is positive, and so this is going to be two over 0.816 times 2.160, and then plus three minus two is one, six minus three is three, so plus three over 0.816 times 2.160, well these are the same denominator, so actually I could rewrite, if I have two over this thing plus three over this thing, that's going to be five over this thing."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "So before I get a calculator out, let's see if there's some simplifications I can do. Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero. Two minus two is zero, three minus three is zero, is gonna actually be zero times zero, so that whole thing is zero. Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so this is going to be R is equal to 1 3rd times, negative times negative is positive, and so this is going to be two over 0.816 times 2.160, and then plus three minus two is one, six minus three is three, so plus three over 0.816 times 2.160, well these are the same denominator, so actually I could rewrite, if I have two over this thing plus three over this thing, that's going to be five over this thing. So I could rewrite this whole thing, five over 0.816 times 2.160, and now I can just get a calculator out to actually calculate this. So we have one divided by three times five divided by 0.816 times 2.160, the zero won't make a difference, but I'll just write it down, and then I will close that parentheses, and let's see what we get. We get an R of, and since everything else goes to the thousandths place, I'll just round to the thousandths place, an R of 0.946, so R is approximately 0.946."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so this is going to be R is equal to 1 3rd times, negative times negative is positive, and so this is going to be two over 0.816 times 2.160, and then plus three minus two is one, six minus three is three, so plus three over 0.816 times 2.160, well these are the same denominator, so actually I could rewrite, if I have two over this thing plus three over this thing, that's going to be five over this thing. So I could rewrite this whole thing, five over 0.816 times 2.160, and now I can just get a calculator out to actually calculate this. So we have one divided by three times five divided by 0.816 times 2.160, the zero won't make a difference, but I'll just write it down, and then I will close that parentheses, and let's see what we get. We get an R of, and since everything else goes to the thousandths place, I'll just round to the thousandths place, an R of 0.946, so R is approximately 0.946. So what does this tell us? The correlation coefficient is a measure of how well a line can describe the relationship between X and Y. R is always going to be greater than or equal to negative one and less than or equal to one. If R is positive one, it means that an upward-sloping line can completely describe the relationship."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "We get an R of, and since everything else goes to the thousandths place, I'll just round to the thousandths place, an R of 0.946, so R is approximately 0.946. So what does this tell us? The correlation coefficient is a measure of how well a line can describe the relationship between X and Y. R is always going to be greater than or equal to negative one and less than or equal to one. If R is positive one, it means that an upward-sloping line can completely describe the relationship. If R is negative one, it means a downward-sloping line can completely describe the relationship. R anywhere in between says, well, it won't be as good. If R is zero, that means that a line isn't describing the relationships well at all."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "If R is positive one, it means that an upward-sloping line can completely describe the relationship. If R is negative one, it means a downward-sloping line can completely describe the relationship. R anywhere in between says, well, it won't be as good. If R is zero, that means that a line isn't describing the relationships well at all. Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys. So for example, I'm just going to try and hand-draw a line here, and it does turn out that our least squares line will always go through the mean of the X and the Y. So the mean of the X is two, mean of the Y is three."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "If R is zero, that means that a line isn't describing the relationships well at all. Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys. So for example, I'm just going to try and hand-draw a line here, and it does turn out that our least squares line will always go through the mean of the X and the Y. So the mean of the X is two, mean of the Y is three. We'll study that in more depth in future videos. But let's see, this actually does look like a pretty good line. So let me just draw it right over there."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "So the mean of the X is two, mean of the Y is three. We'll study that in more depth in future videos. But let's see, this actually does look like a pretty good line. So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-scores, and how does taking products of corresponding Z-scores get us this property that I just talked about, where an R of one will be strong positive correlation, R of negative one would be strong negative correlation? Well, let's draw the sample means here."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-scores, and how does taking products of corresponding Z-scores get us this property that I just talked about, where an R of one will be strong positive correlation, R of negative one would be strong negative correlation? Well, let's draw the sample means here. So the X sample mean is two. This is our X axis here. This is X equals two."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "Well, let's draw the sample means here. So the X sample mean is two. This is our X axis here. This is X equals two. And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "This is X equals two. And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the mean."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y, one standard deviation above the mean, 2.160, so that would be 5.160, so it would put us someplace around there. And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we would go to one, and then we're gonna go take another 0.160, so it's gonna be someplace right around here."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y, one standard deviation above the mean, 2.160, so that would be 5.160, so it would put us someplace around there. And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we would go to one, and then we're gonna go take another 0.160, so it's gonna be someplace right around here. So for example, for this first pair, one comma one, what were we doing? Well, we said, all right, how many standard deviations is this below the mean? And that turns out to be negative one over 0.816."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we would go to one, and then we're gonna go take another 0.160, so it's gonna be someplace right around here. So for example, for this first pair, one comma one, what were we doing? Well, we said, all right, how many standard deviations is this below the mean? And that turns out to be negative one over 0.816. That's what we have right over here. That's what this would have calculated. And then how many standard deviations for in the Y direction?"}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "And that turns out to be negative one over 0.816. That's what we have right over here. That's what this would have calculated. And then how many standard deviations for in the Y direction? And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it, it might be helping us get closer to the one."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "And then how many standard deviations for in the Y direction? And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it, it might be helping us get closer to the one. If both of them have a negative Z score, that means that there is a positive correlation between the variables. When one is below the mean, the other is, you could say, similarly below the mean. Now, if you go to the next data point, two comma two, right over here, what happened?"}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "And so one way to think about it, it might be helping us get closer to the one. If both of them have a negative Z score, that means that there is a positive correlation between the variables. When one is below the mean, the other is, you could say, similarly below the mean. Now, if you go to the next data point, two comma two, right over here, what happened? Well, the X variable was right on the mean. And because of that, that entire term became zero. The X Z score was zero."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "Now, if you go to the next data point, two comma two, right over here, what happened? Well, the X variable was right on the mean. And because of that, that entire term became zero. The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going to be dividing by a slightly higher value by including that extra pair. If you had a data point where, let's say X was below the mean and Y was above the mean, something like this, if this was one of the points, this term would have been negative because the Y Z score would have been positive and the X Z score would have been negative."}, {"video_title": "Calculating correlation coefficient r   AP Statistics   Khan Academy.mp3", "Sentence": "The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going to be dividing by a slightly higher value by including that extra pair. If you had a data point where, let's say X was below the mean and Y was above the mean, something like this, if this was one of the points, this term would have been negative because the Y Z score would have been positive and the X Z score would have been negative. And so when you put it in the sum, it would have actually taken away from the sum. And so it would have made the R score even lower. Similarly, something like this would have done, would have made the R score even lower because you would have a positive Z score for X and a negative Z score for Y."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And in deciding which car they give me, they're first going to randomly select the engine type. So the engine will come in two different varieties. It'll either be a four-cylinder engine or a six-cylinder engine. And they're literally just gonna flip a fair coin to decide whether I get a four-cylinder engine or a six-cylinder engine. Then they're going to pick the color. And there's four different colors that the cars come in. So I'll write color in a neutral color."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And they're literally just gonna flip a fair coin to decide whether I get a four-cylinder engine or a six-cylinder engine. Then they're going to pick the color. And there's four different colors that the cars come in. So I'll write color in a neutral color. So you could get a red car. That's not red. Let me do that in an actual red color, closer to red."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So I'll write color in a neutral color. So you could get a red car. That's not red. Let me do that in an actual red color, closer to red. You could get a red car. You could get a blue car. Blue car."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Let me do that in an actual red color, closer to red. You could get a red car. You could get a blue car. Blue car. You could get a green car. You could get a green car. Or you could get a white car."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Blue car. You could get a green car. You could get a green car. Or you could get a white car. Or you could get a white car. And once again, they're gonna randomly, let's say, just pick, they're gonna have red, blue, green, and white in little slips of paper in a bowl, and they're just gonna pick one of them out. So all of these are equally likely."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Or you could get a white car. Or you could get a white car. And once again, they're gonna randomly, let's say, just pick, they're gonna have red, blue, green, and white in little slips of paper in a bowl, and they're just gonna pick one of them out. So all of these are equally likely. So given this, that they're just gonna flip a coin to pick the engine, and they're also going to, that all of these, the color is all equally likely, I wanna think about the probability of getting a six-cylinder white car. The probability of getting a six-cylinder white car. So I encourage you to pause the video and think about it on your own."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So all of these are equally likely. So given this, that they're just gonna flip a coin to pick the engine, and they're also going to, that all of these, the color is all equally likely, I wanna think about the probability of getting a six-cylinder white car. The probability of getting a six-cylinder white car. So I encourage you to pause the video and think about it on your own. Well, one way to think about this is, well, what are all of the equally likely possible outcomes, and then which of those match six-cylinder white car? Well, first we could think about the engine decision. We're either going to get a four-cylinder engine."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So I encourage you to pause the video and think about it on your own. Well, one way to think about this is, well, what are all of the equally likely possible outcomes, and then which of those match six-cylinder white car? Well, first we could think about the engine decision. We're either going to get a four-cylinder engine. So the first decision is the engine. You could view it that way. You're gonna get a four-cylinder engine, or you're going to get a six-cylinder engine."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "We're either going to get a four-cylinder engine. So the first decision is the engine. You could view it that way. You're gonna get a four-cylinder engine, or you're going to get a six-cylinder engine. Now, if you got a four-cylinder engine, you're either going to get red, blue, blue, green, green, or white, or white. And if you got a six-cylinder engine, once again, you're either going to get red, blue, I think you see where this is going, that's not blue, red, blue, blue, green, green, or white, or white. So how many possible outcomes are there?"}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You're gonna get a four-cylinder engine, or you're going to get a six-cylinder engine. Now, if you got a four-cylinder engine, you're either going to get red, blue, blue, green, green, or white, or white. And if you got a six-cylinder engine, once again, you're either going to get red, blue, I think you see where this is going, that's not blue, red, blue, blue, green, green, or white, or white. So how many possible outcomes are there? Well, you could just count, you could kind of say the leaves of this tree diagram. One, two, three, four, five, six, seven, eight possible outcomes, and that makes sense. You have two possible engines times four possible colors, two times four, and you see that right here, one group of four, two groups of four."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So how many possible outcomes are there? Well, you could just count, you could kind of say the leaves of this tree diagram. One, two, three, four, five, six, seven, eight possible outcomes, and that makes sense. You have two possible engines times four possible colors, two times four, and you see that right here, one group of four, two groups of four. So this outcome right here is a four-cylinder blue car. And this outcome over here is a six-cylinder green car. So there's eight equally possible outcomes, and which outcome matches the one that we, I guess, are hoping for, the white six-cylinder car?"}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You have two possible engines times four possible colors, two times four, and you see that right here, one group of four, two groups of four. So this outcome right here is a four-cylinder blue car. And this outcome over here is a six-cylinder green car. So there's eight equally possible outcomes, and which outcome matches the one that we, I guess, are hoping for, the white six-cylinder car? Well, that's this one right over here. It's one of eight equally likely events, so we have a 1 8th probability. Now, this wasn't the only way that we could have drawn the T diagram."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So there's eight equally possible outcomes, and which outcome matches the one that we, I guess, are hoping for, the white six-cylinder car? Well, that's this one right over here. It's one of eight equally likely events, so we have a 1 8th probability. Now, this wasn't the only way that we could have drawn the T diagram. We could have thought about color as the first row of this tree. So we could have said, look, we're either gonna get a, let me do it down here so I have a little more space. We're either gonna get a red, a blue, a, that's not blue, changing colors is the hard part, a blue, a green, man, a green, let me get, a green or a white car, or a white car, and then for each of those colors, I'm either gonna get a four-cylinder or a six-cylinder engine."}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now, this wasn't the only way that we could have drawn the T diagram. We could have thought about color as the first row of this tree. So we could have said, look, we're either gonna get a, let me do it down here so I have a little more space. We're either gonna get a red, a blue, a, that's not blue, changing colors is the hard part, a blue, a green, man, a green, let me get, a green or a white car, or a white car, and then for each of those colors, I'm either gonna get a four-cylinder or a six-cylinder engine. So it's either gonna be four or a six, either gonna be four or a six, either gonna be four or a six, either going to be four or a six. This would be another way of drawing a T, a tree diagram to represent all of the outcomes. So what is this outcome right over here?"}, {"video_title": "Count outcomes using tree diagram   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "We're either gonna get a red, a blue, a, that's not blue, changing colors is the hard part, a blue, a green, man, a green, let me get, a green or a white car, or a white car, and then for each of those colors, I'm either gonna get a four-cylinder or a six-cylinder engine. So it's either gonna be four or a six, either gonna be four or a six, either gonna be four or a six, either going to be four or a six. This would be another way of drawing a T, a tree diagram to represent all of the outcomes. So what is this outcome right over here? This is a six-cylinder red car. This is a four-cylinder blue car right over here, which is the one that we care about, a white six-cylinder car, that's this outcome right over here. And once again, you see you have eight equally likely outcomes, and that happens because you have four possible colors, and then for each of those four possible colors, you have two different engine types."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Which of the following are accurate descriptions of the distribution below? Select all that apply. So the first statement is the distribution has an outlier. So an outlier is a data point that's way off of where the other data points are. It's way larger or way smaller than where all of the other data points seem to be clustered. And if we look over here, we have a lot of data points between zero and six. And just think about what they're measuring."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So an outlier is a data point that's way off of where the other data points are. It's way larger or way smaller than where all of the other data points seem to be clustered. And if we look over here, we have a lot of data points between zero and six. And just think about what they're measuring. This is shelf time for each apple at Gorge's Grocer. So for example, we see there's one, two, three, four, five, six, seven apples that have a shelf life of zero days. So they're about to go bad."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "And just think about what they're measuring. This is shelf time for each apple at Gorge's Grocer. So for example, we see there's one, two, three, four, five, six, seven apples that have a shelf life of zero days. So they're about to go bad. You see you have one, two, three, four, five, six, seven, eight apples that are gonna be good for another day. You have two apples that are gonna be good for another six days. And you have one apple that's gonna be good for 10 days."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So they're about to go bad. You see you have one, two, three, four, five, six, seven, eight apples that are gonna be good for another day. You have two apples that are gonna be good for another six days. And you have one apple that's gonna be good for 10 days. And this is unusual. This is an outlier here. It has a way larger shelf life than all of the other data."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "And you have one apple that's gonna be good for 10 days. And this is unusual. This is an outlier here. It has a way larger shelf life than all of the other data. So I would say this definitely does have an outlier. We just have this one data point sitting all the way to the right. Way larger, way more shelf life than everything else."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "It has a way larger shelf life than all of the other data. So I would say this definitely does have an outlier. We just have this one data point sitting all the way to the right. Way larger, way more shelf life than everything else. So it definitely has an outlier. And this one would be the outlier. The distribution has a cluster from four to six days."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Way larger, way more shelf life than everything else. So it definitely has an outlier. And this one would be the outlier. The distribution has a cluster from four to six days. And we indeed do see a cluster from four to six days. A cluster, you can imagine it's a grouping of data that's sitting there, or you have a grouping of apples that have a shelf life between four and six days. And you definitely do see that cluster there."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "The distribution has a cluster from four to six days. And we indeed do see a cluster from four to six days. A cluster, you can imagine it's a grouping of data that's sitting there, or you have a grouping of apples that have a shelf life between four and six days. And you definitely do see that cluster there. And since I already selected two things, I'm definitely not gonna select none of the above. So let me check my answer. Let me do a few more of these."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "And you definitely do see that cluster there. And since I already selected two things, I'm definitely not gonna select none of the above. So let me check my answer. Let me do a few more of these. Which of the following are accurate descriptions of the distribution below? And once again, we're going to select all that apply. So the distribution has an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Let me do a few more of these. Which of the following are accurate descriptions of the distribution below? And once again, we're going to select all that apply. So the distribution has an outlier. So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here that's at the low end. But I don't have any data points that are sitting far above or far below the bulk of the data. If I had a data point that was out here, then yeah, I would say that was an outlier to the right, or a positive outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So the distribution has an outlier. So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here that's at the low end. But I don't have any data points that are sitting far above or far below the bulk of the data. If I had a data point that was out here, then yeah, I would say that was an outlier to the right, or a positive outlier. If I had a data point way to the left off the screen over here, maybe that would be an outlier. But I don't really see any obvious outliers. All of the data, it's pretty clustered together."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "If I had a data point that was out here, then yeah, I would say that was an outlier to the right, or a positive outlier. If I had a data point way to the left off the screen over here, maybe that would be an outlier. But I don't really see any obvious outliers. All of the data, it's pretty clustered together. So I would not say that the distribution has an outlier. The distribution has a peak at 22 degrees. Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "All of the data, it's pretty clustered together. So I would not say that the distribution has an outlier. The distribution has a peak at 22 degrees. Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July. So it does indeed look like we have the most number of days that had a high temperature at 22, most number of days in July had a high temperature at 22 degrees Celsius. So that is a peak, and you can see it. If you imagine this is kind of a mountain, this is a peak right here."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July. So it does indeed look like we have the most number of days that had a high temperature at 22, most number of days in July had a high temperature at 22 degrees Celsius. So that is a peak, and you can see it. If you imagine this is kind of a mountain, this is a peak right here. This is a high point. You have, at least locally, you have the most number of days at 22 degrees Celsius. So I would say it definitely has a peak there."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "If you imagine this is kind of a mountain, this is a peak right here. This is a high point. You have, at least locally, you have the most number of days at 22 degrees Celsius. So I would say it definitely has a peak there. Since I selected something, I'm not gonna select none of the above. Let's do a couple more of these. Which of the following are accurate descriptions of the distribution below?"}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So I would say it definitely has a peak there. Since I selected something, I'm not gonna select none of the above. Let's do a couple more of these. Which of the following are accurate descriptions of the distribution below? So the first one, the distribution has an outlier. So let's see, this number of guests by day at Seth's Sandwich Shop. So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Which of the following are accurate descriptions of the distribution below? So the first one, the distribution has an outlier. So let's see, this number of guests by day at Seth's Sandwich Shop. So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests. No days where he had between 20 and 39 guests. Looks like there's about nine days where he had between 40 and 59 guests. Looks like 20 days where he had between 60 and 79 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests. No days where he had between 20 and 39 guests. Looks like there's about nine days where he had between 40 and 59 guests. Looks like 20 days where he had between 60 and 79 guests. All the way, it looks like this is maybe eight days that he had between 180 and 199 guests. But the question of outliers, there doesn't seem to be any day where he had an unusual number of guests. There's not a day that's way out here where he had like 500 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Looks like 20 days where he had between 60 and 79 guests. All the way, it looks like this is maybe eight days that he had between 180 and 199 guests. But the question of outliers, there doesn't seem to be any day where he had an unusual number of guests. There's not a day that's way out here where he had like 500 guests. So I would say this distribution does not have an outlier. The distribution has a cluster from zero to 39 guests. So zero to 39 guests is right over here, zero to 39 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "There's not a day that's way out here where he had like 500 guests. So I would say this distribution does not have an outlier. The distribution has a cluster from zero to 39 guests. So zero to 39 guests is right over here, zero to 39 guests. And there's no days where he had between zero and 39 guests, either zero to 19 or 20 to 39. So there's definitely not a cluster there. I would say that the cluster would be between, were days that had between 40 and 199 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So zero to 39 guests is right over here, zero to 39 guests. And there's no days where he had between zero and 39 guests, either zero to 19 or 20 to 39. So there's definitely not a cluster there. I would say that the cluster would be between, were days that had between 40 and 199 guests. Definitely not zero and 39. There was no days that were between zero and 39 guests. So I would say none of the above, very confidently."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "I would say that the cluster would be between, were days that had between 40 and 199 guests. Definitely not zero and 39. There was no days that were between zero and 39 guests. So I would say none of the above, very confidently. Let's do one more of these. Which of the following are accurate descriptions of the distribution below? All right."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So I would say none of the above, very confidently. Let's do one more of these. Which of the following are accurate descriptions of the distribution below? All right. The distribution has a peak from 12 to 13 points. Let me see what this is measuring or what this data is about. Test scores by student in Ms."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "All right. The distribution has a peak from 12 to 13 points. Let me see what this is measuring or what this data is about. Test scores by student in Ms. Friend's class. So you had one student who got between a zero and a one on the 20 point scale. So got between, I guess you may be out of 20 questions, got between zero and one points."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Test scores by student in Ms. Friend's class. So you had one student who got between a zero and a one on the 20 point scale. So got between, I guess you may be out of 20 questions, got between zero and one points. And then you see that those students got between two and three or four and five or six and seven. Then we have another student who got between eight and nine. Looks like three students got between 10 and 11."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So got between, I guess you may be out of 20 questions, got between zero and one points. And then you see that those students got between two and three or four and five or six and seven. Then we have another student who got between eight and nine. Looks like three students got between 10 and 11. And then we keep increasing. This looks like it's about 12 students got either a 16 or a 17 or something in between, maybe if you could get decimal points on that test. And then it looks like 10 students got from 18 to 19."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "Looks like three students got between 10 and 11. And then we keep increasing. This looks like it's about 12 students got either a 16 or a 17 or something in between, maybe if you could get decimal points on that test. And then it looks like 10 students got from 18 to 19. All right, so this says the distribution has a peak from 12 to 13 points. 12 to 13 points. There were five students, but this isn't a peak."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "And then it looks like 10 students got from 18 to 19. All right, so this says the distribution has a peak from 12 to 13 points. 12 to 13 points. There were five students, but this isn't a peak. If you just go to 14 to 15 points, you have more students. So this is definitely not a peak. If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "There were five students, but this isn't a peak. If you just go to 14 to 15 points, you have more students. So this is definitely not a peak. If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak. You would say this distribution has a peak, has the most number of students who got between 16 and 17 points. So that's the peak right there, not 12 to 13 points. So I would not select that first choice."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak. You would say this distribution has a peak, has the most number of students who got between 16 and 17 points. So that's the peak right there, not 12 to 13 points. So I would not select that first choice. The distribution has an outlier. Well, yeah, look at this. You have this outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "So I would not select that first choice. The distribution has an outlier. Well, yeah, look at this. You have this outlier. Most of the students scored between eight and 19 points. And then you have this one student who got between zero and one. It's really an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "You have this outlier. Most of the students scored between eight and 19 points. And then you have this one student who got between zero and one. It's really an outlier. You even see this, when you look at it visually, it's not even connected to the rest of the distribution. It's way to the left. If something's way to the left or way to the right, that's an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions   6h grade   Khan Academy.mp3", "Sentence": "It's really an outlier. You even see this, when you look at it visually, it's not even connected to the rest of the distribution. It's way to the left. If something's way to the left or way to the right, that's an outlier. If it's unusually low or unusually high. So I would say this distribution definitely does have an outlier. And I'm not gonna pick none of the above since I found a choice."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "Here, we're gonna think about how we can make inferences from a regression line. And so the idea of statistical inference is new to you, or hypothesis testing, once again, watch those videos as well. But let's say we think there's a positive association between shoe size and height. And so what we might wanna do is, we could, here on the horizontal axis, that is shoe size, our sizes could go size one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, and it could keep going up from there. And then on this height, on this axis, our y-axis, this would be height. So one foot, two feet, three feet, four feet, five feet, six feet, seven feet. And then you could, to see if there's an association, you might take a sample, let's say you take a random sample of 20 people from the population, and in future videos, we'll talk about the conditions necessary for making appropriate inferences."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "And so what we might wanna do is, we could, here on the horizontal axis, that is shoe size, our sizes could go size one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, and it could keep going up from there. And then on this height, on this axis, our y-axis, this would be height. So one foot, two feet, three feet, four feet, five feet, six feet, seven feet. And then you could, to see if there's an association, you might take a sample, let's say you take a random sample of 20 people from the population, and in future videos, we'll talk about the conditions necessary for making appropriate inferences. Well, let's say those 20 people are these 20 data points. So there's a young child, then maybe there's a grown adult with bigger feet, and who's taller, and then three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and so you have these 20 data points, and then what you're likely to do is input them into a computer. You could do it by hand, but we have computers now to do that for us usually."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "And then you could, to see if there's an association, you might take a sample, let's say you take a random sample of 20 people from the population, and in future videos, we'll talk about the conditions necessary for making appropriate inferences. Well, let's say those 20 people are these 20 data points. So there's a young child, then maybe there's a grown adult with bigger feet, and who's taller, and then three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and so you have these 20 data points, and then what you're likely to do is input them into a computer. You could do it by hand, but we have computers now to do that for us usually. And the computer could try to fit a regression line. And there's many techniques for doing it, but one typical technique is to try to overall minimize the square distance between these points and that line. And this regression line will have an equation as any line would have, and we tend to show that as saying y hat, this hat tells us that this is a regression line, is equal to the y-intercept, a, plus the slope times our x variable."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "You could do it by hand, but we have computers now to do that for us usually. And the computer could try to fit a regression line. And there's many techniques for doing it, but one typical technique is to try to overall minimize the square distance between these points and that line. And this regression line will have an equation as any line would have, and we tend to show that as saying y hat, this hat tells us that this is a regression line, is equal to the y-intercept, a, plus the slope times our x variable. So this right over here would be a. Now to be clear, if you took another sample, you might get different results here. In fact, let's call this y sub one for our first sample, a sub one, b sub one, and this is a sub one."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "And this regression line will have an equation as any line would have, and we tend to show that as saying y hat, this hat tells us that this is a regression line, is equal to the y-intercept, a, plus the slope times our x variable. So this right over here would be a. Now to be clear, if you took another sample, you might get different results here. In fact, let's call this y sub one for our first sample, a sub one, b sub one, and this is a sub one. If you were to take another sample of 20 folks, so let's do that, maybe you get one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and then you tried to fit a line to that, that line might look something like this. It might have a slightly different y-intercept and a slightly different slope, so we could call that for the second sample, y sub two or y hat sub two is equal to a sub two plus b sub two times x. And so every time you take a sample, you are likely to get different results for these values, which are essentially statistics."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "In fact, let's call this y sub one for our first sample, a sub one, b sub one, and this is a sub one. If you were to take another sample of 20 folks, so let's do that, maybe you get one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and then you tried to fit a line to that, that line might look something like this. It might have a slightly different y-intercept and a slightly different slope, so we could call that for the second sample, y sub two or y hat sub two is equal to a sub two plus b sub two times x. And so every time you take a sample, you are likely to get different results for these values, which are essentially statistics. Remember, statistics are things that we can get from samples and we're trying to estimate true population parameters. Well, what would be the true population parameters we're trying to estimate? Well, imagine a world, imagine a world here that you're able to find out the true linear relationship, or maybe there is some true linear relationship between shoe size and height."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "And so every time you take a sample, you are likely to get different results for these values, which are essentially statistics. Remember, statistics are things that we can get from samples and we're trying to estimate true population parameters. Well, what would be the true population parameters we're trying to estimate? Well, imagine a world, imagine a world here that you're able to find out the true linear relationship, or maybe there is some true linear relationship between shoe size and height. You could get it if, theoretically, you could measure every human being on the planet and depending what you define as the population, it could be all living people or all people who'll ever live. This isn't practical, but let's just say that you actually could. You would have billions of data points here for the true population, and then if you were to fit a regression line to that, you could view this as the true population regression line."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "Well, imagine a world, imagine a world here that you're able to find out the true linear relationship, or maybe there is some true linear relationship between shoe size and height. You could get it if, theoretically, you could measure every human being on the planet and depending what you define as the population, it could be all living people or all people who'll ever live. This isn't practical, but let's just say that you actually could. You would have billions of data points here for the true population, and then if you were to fit a regression line to that, you could view this as the true population regression line. And so that would be y hat is equal to, and to make it clear that here, the y-intercept and the slope, this would be the true population parameters. Instead of saying a, we say alpha, and instead of saying b, we say beta times x. But it's very hard to come up exactly with what alpha and beta are, and so that's why we estimate it with a's and b's based on a sample."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "You would have billions of data points here for the true population, and then if you were to fit a regression line to that, you could view this as the true population regression line. And so that would be y hat is equal to, and to make it clear that here, the y-intercept and the slope, this would be the true population parameters. Instead of saying a, we say alpha, and instead of saying b, we say beta times x. But it's very hard to come up exactly with what alpha and beta are, and so that's why we estimate it with a's and b's based on a sample. Now, what's interesting with this in mind is we can start to make inferences based on our sample. So we know that, for example, b sub two is unlikely to be exactly beta, but how confident can we be that there is at least a positive linear relationship or a non-zero linear relationship? Or can we create a confidence interval around this statistic in order to have a good sense of where the true parameter might actually be?"}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "But it's very hard to come up exactly with what alpha and beta are, and so that's why we estimate it with a's and b's based on a sample. Now, what's interesting with this in mind is we can start to make inferences based on our sample. So we know that, for example, b sub two is unlikely to be exactly beta, but how confident can we be that there is at least a positive linear relationship or a non-zero linear relationship? Or can we create a confidence interval around this statistic in order to have a good sense of where the true parameter might actually be? And the simple answer is yes. And to do so, we'll use the same exact ideas that we did when we made inferences based on proportions or based on means. The way that you can make an inference, for example, for your true population slope of your regression line, you say, okay, I took a sample."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "Or can we create a confidence interval around this statistic in order to have a good sense of where the true parameter might actually be? And the simple answer is yes. And to do so, we'll use the same exact ideas that we did when we made inferences based on proportions or based on means. The way that you can make an inference, for example, for your true population slope of your regression line, you say, okay, I took a sample. I got this slope right over here, so I'll just call that b two, and then I could create a confidence interval around that. And so that confidence interval is going to be based on some critical value times ideally the standard deviation of the sampling distribution of your sample statistic. In this case, it would be the sample regression line slope."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "The way that you can make an inference, for example, for your true population slope of your regression line, you say, okay, I took a sample. I got this slope right over here, so I'll just call that b two, and then I could create a confidence interval around that. And so that confidence interval is going to be based on some critical value times ideally the standard deviation of the sampling distribution of your sample statistic. In this case, it would be the sample regression line slope. But because we don't know exactly what this is, we can't figure out precisely what this is going to be from a sample, we are going to estimate it with what's known as the standard error of the statistic, and we'll go into more depth in this in future videos. And since we're estimating here, we're going to use a critical t value here, which we have studied before. And so based on your confidence level you wanna have, let's say it's 95%, based on the degrees of freedom, which we'll see will come out of how many data points we have, we can figure this out."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "In this case, it would be the sample regression line slope. But because we don't know exactly what this is, we can't figure out precisely what this is going to be from a sample, we are going to estimate it with what's known as the standard error of the statistic, and we'll go into more depth in this in future videos. And since we're estimating here, we're going to use a critical t value here, which we have studied before. And so based on your confidence level you wanna have, let's say it's 95%, based on the degrees of freedom, which we'll see will come out of how many data points we have, we can figure this out. And from our sample, we can figure this out, and we can figure this out, and then we would have constructed a confidence interval. We'll also see that you could do hypothesis testing here. You could say, hey, let's set up a null hypothesis, and the null hypothesis is going to be that there's no non-zero linear relationship, or that the true population slope of the regression line, or slope of the population regression line, is equal to zero, and that the alternative hypothesis is that the true relationship could either be greater than zero, it's a positive linear relationship, or that it's just non-zero."}, {"video_title": "Introduction to inference about slope in linear regression   AP Statistics   Khan Academy.mp3", "Sentence": "And so based on your confidence level you wanna have, let's say it's 95%, based on the degrees of freedom, which we'll see will come out of how many data points we have, we can figure this out. And from our sample, we can figure this out, and we can figure this out, and then we would have constructed a confidence interval. We'll also see that you could do hypothesis testing here. You could say, hey, let's set up a null hypothesis, and the null hypothesis is going to be that there's no non-zero linear relationship, or that the true population slope of the regression line, or slope of the population regression line, is equal to zero, and that the alternative hypothesis is that the true relationship could either be greater than zero, it's a positive linear relationship, or that it's just non-zero. And then what you could do is, assuming this, you could see what's the probability of getting a statistic that is at least this extreme, or more extreme. And if that's below some threshold, you might reject the null hypothesis, which would suggest the alternative. So this and this are things that we have done before, where you're creating a confidence interval around a statistic, or you're doing hypothesis testing, making assumptions about a true parameter."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Each time one of you catches a fish, you release it back into the water. Jeremy offers you the choice of two different bets. Bet number one, bet number one. We don't encourage betting, but I guess Jeremy wants to bet. If the next three fish he catches are all sunfish, you will pay him $100. Otherwise, he will pay you $20. Bet two, if you catch at least two sunfish of the next three fish that you catch, he will pay you $50."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't encourage betting, but I guess Jeremy wants to bet. If the next three fish he catches are all sunfish, you will pay him $100. Otherwise, he will pay you $20. Bet two, if you catch at least two sunfish of the next three fish that you catch, he will pay you $50. Otherwise, you will pay him $25. What is your expected value from problem, from, what is the expected value from bet one? Round your answer to the nearest cent."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Bet two, if you catch at least two sunfish of the next three fish that you catch, he will pay you $50. Otherwise, you will pay him $25. What is your expected value from problem, from, what is the expected value from bet one? Round your answer to the nearest cent. And I encourage you to pause this video and try to think about it on your own. So let's see, the expected value of bet one. So the expected value of, let's just say bet one, or we'll say bet one is the, let's just define a random variable here just to be a little bit better about this."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Round your answer to the nearest cent. And I encourage you to pause this video and try to think about it on your own. So let's see, the expected value of bet one. So the expected value of, let's just say bet one, or we'll say bet one is the, let's just define a random variable here just to be a little bit better about this. So let's say X is equal to, is equal to what you pay, what you, or I guess you could say, because you might get something, what your profit is. Your profit is from bet one. From bet one."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the expected value of, let's just say bet one, or we'll say bet one is the, let's just define a random variable here just to be a little bit better about this. So let's say X is equal to, is equal to what you pay, what you, or I guess you could say, because you might get something, what your profit is. Your profit is from bet one. From bet one. And it's a random variable. And so the expected value, the expected value of X is going to be equal to, well let's see, what's the probability, it's going to be 100, it's going to be negative $100 times the probability that he catches three fish. So the probability that Jeremy, Jeremy catches three sunfish, the next three, three, the next three, the next three fishy catches are going to be sunfish, times $100, times, or I should say, well you're going to pay that, so since you're paying it, we'll put it as negative 100, because we're saying that this is your expected profit."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "From bet one. And it's a random variable. And so the expected value, the expected value of X is going to be equal to, well let's see, what's the probability, it's going to be 100, it's going to be negative $100 times the probability that he catches three fish. So the probability that Jeremy, Jeremy catches three sunfish, the next three, three, the next three, the next three fishy catches are going to be sunfish, times $100, times, or I should say, well you're going to pay that, so since you're paying it, we'll put it as negative 100, because we're saying that this is your expected profit. So you're going to lose money there. And otherwise, so that's going to be one minus this probability, the probability that Jeremy catches, catches three sunfish. In that situation, he'll pay you $20."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability that Jeremy, Jeremy catches three sunfish, the next three, three, the next three, the next three fishy catches are going to be sunfish, times $100, times, or I should say, well you're going to pay that, so since you're paying it, we'll put it as negative 100, because we're saying that this is your expected profit. So you're going to lose money there. And otherwise, so that's going to be one minus this probability, the probability that Jeremy catches, catches three sunfish. In that situation, he'll pay you $20. You get $20 there. So the important thing is, is to figure out the probability that Jeremy catches three sunfish. Well the sunfish are 10 out of the 20 fish, so at any given time he's trying to catch fish, there's a 10 in 20 chance, or you could say a 1 1 2 probability that's going to be a sunfish."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "In that situation, he'll pay you $20. You get $20 there. So the important thing is, is to figure out the probability that Jeremy catches three sunfish. Well the sunfish are 10 out of the 20 fish, so at any given time he's trying to catch fish, there's a 10 in 20 chance, or you could say a 1 1 2 probability that's going to be a sunfish. So the probability that you get three sunfish in a row is going to be 1 1 2, times 1 1 2, times 1 1 2. And they put the fish back in, so that's why it stays 10 out of the 20 fish. If he wasn't putting the fish back in, then the second sunfish, you would have a nine out of 20 chance of the second one being a sunfish."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well the sunfish are 10 out of the 20 fish, so at any given time he's trying to catch fish, there's a 10 in 20 chance, or you could say a 1 1 2 probability that's going to be a sunfish. So the probability that you get three sunfish in a row is going to be 1 1 2, times 1 1 2, times 1 1 2. And they put the fish back in, so that's why it stays 10 out of the 20 fish. If he wasn't putting the fish back in, then the second sunfish, you would have a nine out of 20 chance of the second one being a sunfish. But in this case, they keep replacing the fish every time they catch it. So there's a 1 8th chance that Jeremy catches three sunfish. So this right over here is 1 8th."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "If he wasn't putting the fish back in, then the second sunfish, you would have a nine out of 20 chance of the second one being a sunfish. But in this case, they keep replacing the fish every time they catch it. So there's a 1 8th chance that Jeremy catches three sunfish. So this right over here is 1 8th. And one minus 1 8th, this is 7 8ths. 7 8ths. So you have a 1 8th chance of paying $100, and a 7 8ths chance of getting $20."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right over here is 1 8th. And one minus 1 8th, this is 7 8ths. 7 8ths. So you have a 1 8th chance of paying $100, and a 7 8ths chance of getting $20. And so this gets us two. This gets us two. So you're expected, I guess you could say profit here."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have a 1 8th chance of paying $100, and a 7 8ths chance of getting $20. And so this gets us two. This gets us two. So you're expected, I guess you could say profit here. There's a 1 8th chance, 1 8th probability, that you lose $100 here. So times negative 100. But then there is a 7 8th chance that you get, I'll just put parentheses here to make it a little clearer."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you're expected, I guess you could say profit here. There's a 1 8th chance, 1 8th probability, that you lose $100 here. So times negative 100. But then there is a 7 8th chance that you get, I'll just put parentheses here to make it a little clearer. I think the order of operations on the calculator would have taken care of it, but I'll just do it just so that it looks the same. 7 8ths, there's a 7 8th chance that you get $20. And so your expected payoff here is positive $5."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "But then there is a 7 8th chance that you get, I'll just put parentheses here to make it a little clearer. I think the order of operations on the calculator would have taken care of it, but I'll just do it just so that it looks the same. 7 8ths, there's a 7 8th chance that you get $20. And so your expected payoff here is positive $5. So your expected payoff here is equal to $5. So this is your expected value from bet one. Now let's think about bet two."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so your expected payoff here is positive $5. So your expected payoff here is equal to $5. So this is your expected value from bet one. Now let's think about bet two. Bet two. If you catch at least two sunfish of the next three fish you catch, he will pay you 50. Otherwise, you will pay him 25."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's think about bet two. Bet two. If you catch at least two sunfish of the next three fish you catch, he will pay you 50. Otherwise, you will pay him 25. So let's think about the probability of catching at least two sunfish of the next three fish that you catch. Now there's a bunch of ways to think about this, but since there's only kind of three times that you're trying to catch the fish, and there's only one of two outcomes, you could actually write all the possible outcomes that are possible here. You could get sunfish, sunfish, sunfish."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Otherwise, you will pay him 25. So let's think about the probability of catching at least two sunfish of the next three fish that you catch. Now there's a bunch of ways to think about this, but since there's only kind of three times that you're trying to catch the fish, and there's only one of two outcomes, you could actually write all the possible outcomes that are possible here. You could get sunfish, sunfish, sunfish. You could get, well, what else the other type of fish that you have, or the trout. You could have sunfish, sunfish, trout. You can have sunfish, trout, sunfish."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could get sunfish, sunfish, sunfish. You could get, well, what else the other type of fish that you have, or the trout. You could have sunfish, sunfish, trout. You can have sunfish, trout, sunfish. You could have sunfish, trout, trout. You could have trout, sunfish, sunfish. You could have trout, sunfish, trout."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can have sunfish, trout, sunfish. You could have sunfish, trout, trout. You could have trout, sunfish, sunfish. You could have trout, sunfish, trout. You could have trout, trout, sunfish. or you could have all trout. And you see here that each of these, each time you go, there's two possibilities, and so when, or each time you try to catch a fish, there's two possibilities."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could have trout, sunfish, trout. You could have trout, trout, sunfish. or you could have all trout. And you see here that each of these, each time you go, there's two possibilities, and so when, or each time you try to catch a fish, there's two possibilities. So if you're doing it three times, there's two times two times two possibilities. One, two, three, four, five, six, seven, eight possibilities here. Now out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish?"}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you see here that each of these, each time you go, there's two possibilities, and so when, or each time you try to catch a fish, there's two possibilities. So if you're doing it three times, there's two times two times two possibilities. One, two, three, four, five, six, seven, eight possibilities here. Now out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish? Well you catch at least two sunfish in this one, in this one, in that one, in this one, and I think that is it. That is, yep, this is only one sunfish, one sunfish, one sunfish, and no sunfish. So in four out of the eight equally likely outcomes, you catch at least two sunfish."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish? Well you catch at least two sunfish in this one, in this one, in that one, in this one, and I think that is it. That is, yep, this is only one sunfish, one sunfish, one sunfish, and no sunfish. So in four out of the eight equally likely outcomes, you catch at least two sunfish. So your probability of catching at least two sunfish, probability of at least two sunfish, sunfish, is equal to 4 8ths or 1 1.5. So let's see, what's the expected value? Let's say Y is the expected profit from bet."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So in four out of the eight equally likely outcomes, you catch at least two sunfish. So your probability of catching at least two sunfish, probability of at least two sunfish, sunfish, is equal to 4 8ths or 1 1.5. So let's see, what's the expected value? Let's say Y is the expected profit from bet. So let's let Y equals, another random variable is equal to expected profit from bet two. So the expected value of our random variable Y, you have a 1 1.5 chance that you win. So you have a 1 1.5 chance of getting $50, and then you have the 1 1.5 chance, the rest of the probability."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say Y is the expected profit from bet. So let's let Y equals, another random variable is equal to expected profit from bet two. So the expected value of our random variable Y, you have a 1 1.5 chance that you win. So you have a 1 1.5 chance of getting $50, and then you have the 1 1.5 chance, the rest of the probability. If there's a 1 1.5 chance you win, there's gonna be a 1 minus 1 1.5, or essentially a 1 1.5 chance that you lose. And so this is going, so you have a 1 1.5 chance of having to pay $25. So let's see what this is."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you have a 1 1.5 chance of getting $50, and then you have the 1 1.5 chance, the rest of the probability. If there's a 1 1.5 chance you win, there's gonna be a 1 minus 1 1.5, or essentially a 1 1.5 chance that you lose. And so this is going, so you have a 1 1.5 chance of having to pay $25. So let's see what this is. This is 1 1.5 times 50 plus 1 1.5 times negative 25. This is going to be 25 minus 12.50, minus 12.50, which is equal to 12.50. So your expected value from bet two is 12.50."}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's see what this is. This is 1 1.5 times 50 plus 1 1.5 times negative 25. This is going to be 25 minus 12.50, minus 12.50, which is equal to 12.50. So your expected value from bet two is 12.50. Your friend says he's willing to take both bets. He's willing to take both bets, a combined total of 50 times. If you want to maximize your expected value, what should you do?"}, {"video_title": "Expected value while fishing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So your expected value from bet two is 12.50. Your friend says he's willing to take both bets. He's willing to take both bets, a combined total of 50 times. If you want to maximize your expected value, what should you do? Well, bet number two, actually both of them are good bets, I guess your friend isn't that sophisticated, but bet number two has a higher expected payoff. So I would take bet two all of the time. So I would take bet two all of the time."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And he says, well we sell four types of flowers. We sell roses, tulips, sunflowers, and lilies. And you say, what type of pots could I put them in? And he says, well, you could pick any flower, and then you could pick any of our three pots. We have brown pots, we have yellow pots, and we have green pots. So the question that I ask to you is, how many types of, I guess, flower and pots put together can you walk out of this florist store with? For example, you could get a rose in a brown pot."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And he says, well, you could pick any flower, and then you could pick any of our three pots. We have brown pots, we have yellow pots, and we have green pots. So the question that I ask to you is, how many types of, I guess, flower and pots put together can you walk out of this florist store with? For example, you could get a rose in a brown pot. You could get a rose in a green pot. Or you could get a yellow pot that has a sunflower in it. Or a yellow pot that has a lily in it."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "For example, you could get a rose in a brown pot. You could get a rose in a green pot. Or you could get a yellow pot that has a sunflower in it. Or a yellow pot that has a lily in it. So how many scenarios could you walk out of that store with? And like always, I'll encourage you to pause the video and try to figure it out on your own. So let's think through it."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Or a yellow pot that has a lily in it. So how many scenarios could you walk out of that store with? And like always, I'll encourage you to pause the video and try to figure it out on your own. So let's think through it. So I'll just write the first letters, just to visualize, or just so I don't have to write down everything. So you could have a brown pot. You could have a yellow pot."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So let's think through it. So I'll just write the first letters, just to visualize, or just so I don't have to write down everything. So you could have a brown pot. You could have a yellow pot. Or you could have a green pot. You definitely have to pick a pot, so you're gonna have one of those. And then for each of these three, there's four possible flowers you could have."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You could have a yellow pot. Or you could have a green pot. You definitely have to pick a pot, so you're gonna have one of those. And then for each of these three, there's four possible flowers you could have. You could have a rose with a brown pot. You could have a rose with a yellow pot. You could have a rose with a green pot."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And then for each of these three, there's four possible flowers you could have. You could have a rose with a brown pot. You could have a rose with a yellow pot. You could have a rose with a green pot. You could have a Tulip with a brown pot. A Tulip with a yellow pot. A Tulip with a green pot."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You could have a rose with a green pot. You could have a Tulip with a brown pot. A Tulip with a yellow pot. A Tulip with a green pot. You could have a Sunflower with each of the three pots. And you could have, or you could have a Lily with a brown pot. A Lily with a yellow pot."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "A Tulip with a green pot. You could have a Sunflower with each of the three pots. And you could have, or you could have a Lily with a brown pot. A Lily with a yellow pot. And a Lily with a green pot. So how many scenarios are we talking about? Well, we had three pots, so we have three pots right over here, and we have four possible flowers to put in the pots."}, {"video_title": "Counting pot and flower scenarios   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "A Lily with a yellow pot. And a Lily with a green pot. So how many scenarios are we talking about? Well, we had three pots, so we have three pots right over here, and we have four possible flowers to put in the pots. And so we see that we have four possible flowers for each of the three pots, so it's going to be three times four, three times four possibilities, or 12. And you see them right over here. This is brown with red, brown with, or brown with rose, brown with tulip, brown with sunflower, brown with lily, yellow with rose, yellow with tulip, yellow with sunflower, yellow with lily."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "In other videos, we introduce ourselves to the idea of a density curve, which is a summary of a distribution, a distribution of data, and in the future, we'll also look at things like probability density, but what I want to talk about in this video is think about what we can glean from them, the properties, how we can describe density curves and the distributions they represent. And we have four of them right over here, and the first thing I want to think about is if we can approximate what value would be the middle value, or the median for the data set described by these density curves. So just to remind ourselves, if we have a set of numbers and we order them from least to greatest, the median would be the middle value, or the midway between the middle two values. In a case like this, we want to find the value for which half of the values are above that value and half of the values are below. So when you're looking at a density curve, you'd want to look at the area, and you'd want to say, okay, at what value do we have equal area above and below that value? And so for this one, just eyeballing it, this value right over here would be the median. And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "In a case like this, we want to find the value for which half of the values are above that value and half of the values are below. So when you're looking at a density curve, you'd want to look at the area, and you'd want to say, okay, at what value do we have equal area above and below that value? And so for this one, just eyeballing it, this value right over here would be the median. And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. Here we have a slightly more unusual distribution. This would be called a bimodal distribution, but you have two major lumps right over here. But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. Here we have a slightly more unusual distribution. This would be called a bimodal distribution, but you have two major lumps right over here. But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median. What if we're dealing with non-symmetric distributions? Well, we'd want to do the same principle."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median. What if we're dealing with non-symmetric distributions? Well, we'd want to do the same principle. We'd want to think at what value is the area on the right and the area on the left equal? And once again, this isn't going to be super exact, but I'm going to try to approximate it. You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we'd want to do the same principle. We'd want to think at what value is the area on the right and the area on the left equal? And once again, this isn't going to be super exact, but I'm going to try to approximate it. You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. So that would not be the median. If I move the median a little bit over to the right, this may be right around here, this looks a lot closer. Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. So that would not be the median. If I move the median a little bit over to the right, this may be right around here, this looks a lot closer. Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. Similarly, on this one right over here, maybe right over here. And once again, I'm just approximating it, but that seems reasonable, that this area is equal to that one. Even though this is longer, it's much lower."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. Similarly, on this one right over here, maybe right over here. And once again, I'm just approximating it, but that seems reasonable, that this area is equal to that one. Even though this is longer, it's much lower. This part of the curve is much higher, even though it goes on less to the right. So that's the median. For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Even though this is longer, it's much lower. This part of the curve is much higher, even though it goes on less to the right. So that's the median. For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. But what about the mean? Well, the mean is you take each of the possible values and you weight it by their frequencies, you weight it by their frequencies, and you add all of that up. And so for symmetric distributions, your mean and your median are actually going to be the same."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. But what about the mean? Well, the mean is you take each of the possible values and you weight it by their frequencies, you weight it by their frequencies, and you add all of that up. And so for symmetric distributions, your mean and your median are actually going to be the same. So this is going to be your mean as well, this is going to be your mean as well. If you wanna think about it in terms of physics, the mean would be your balancing point, the point at which you would wanna put a little fulcrum and you would wanna balance the distribution. And so you could put a little fulcrum here and you could imagine that this thing would balance."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And so for symmetric distributions, your mean and your median are actually going to be the same. So this is going to be your mean as well, this is going to be your mean as well. If you wanna think about it in terms of physics, the mean would be your balancing point, the point at which you would wanna put a little fulcrum and you would wanna balance the distribution. And so you could put a little fulcrum here and you could imagine that this thing would balance. This thing would balance. And that all comes out of this idea of the weighted average of all of these possible values. What about for these less symmetric distributions?"}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And so you could put a little fulcrum here and you could imagine that this thing would balance. This thing would balance. And that all comes out of this idea of the weighted average of all of these possible values. What about for these less symmetric distributions? Well, let's think about it over here. Where would I have to put the fulcrum? Or what does our intuition say if we wanted to balance this?"}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "What about for these less symmetric distributions? Well, let's think about it over here. Where would I have to put the fulcrum? Or what does our intuition say if we wanted to balance this? Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case. And so our balance point is probably going to be something closer to that. And once again, this is me approximating it, but this would roughly be our mean."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Or what does our intuition say if we wanted to balance this? Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case. And so our balance point is probably going to be something closer to that. And once again, this is me approximating it, but this would roughly be our mean. It would sit in this case to the right of our median. Let me make it clear. This median is referring to that, the mean is referring to this."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, this is me approximating it, but this would roughly be our mean. It would sit in this case to the right of our median. Let me make it clear. This median is referring to that, the mean is referring to this. In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here. So the mean would be this value right over there. And there's actually a term for these non-symmetric distributions where the mean is varying from the median."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "This median is referring to that, the mean is referring to this. In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here. So the mean would be this value right over there. And there's actually a term for these non-symmetric distributions where the mean is varying from the median. Distributions like this are referred to as being skewed. And this distribution, where you have the mean to the right of the median, where you have this long tail to the right, this is called right skewed. Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And there's actually a term for these non-symmetric distributions where the mean is varying from the median. Distributions like this are referred to as being skewed. And this distribution, where you have the mean to the right of the median, where you have this long tail to the right, this is called right skewed. Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. That's the direction in which it will be skewed. Or if the mean is to that direction of the median. So the mean is to the right of the median."}, {"video_title": "Median, mean and skew from density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. That's the direction in which it will be skewed. Or if the mean is to that direction of the median. So the mean is to the right of the median. So generally speaking, that's going to be a right skewed distribution. So the opposite of that, here the mean is to the left of the median, and we have this long tail on the left of our distribution. So generally speaking, we will describe these as left skewed distributions."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just to visualize that, in this video, we will actually plot these, and we'll get a sense of this random variable's probability distribution. So let's do that. So let's see, maybe, actually, maybe I'll do it like this, so that we can see the probabilities. And actually, I can, let me erase this business right over here. Whoops, that's not working. Let me, here, that might work. Let me erase this real fast, these little scribbles that I had offscreen, and then we can actually plot the distribution."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, I can, let me erase this business right over here. Whoops, that's not working. Let me, here, that might work. Let me erase this real fast, these little scribbles that I had offscreen, and then we can actually plot the distribution. All right, so the one axis, I'm gonna put all of the different outcomes. So let me, that looks like a pretty straight line. And then this axis, I'm going to plot the probabilities."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me erase this real fast, these little scribbles that I had offscreen, and then we can actually plot the distribution. All right, so the one axis, I'm gonna put all of the different outcomes. So let me, that looks like a pretty straight line. And then this axis, I'm going to plot the probabilities. And that looks like a pretty straight line. And let's see what the probabilities are. We have, let's see, it's all gonna be in terms of 30 seconds, and we get as high as 10 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then this axis, I'm going to plot the probabilities. And that looks like a pretty straight line. And let's see what the probabilities are. We have, let's see, it's all gonna be in terms of 30 seconds, and we get as high as 10 30 seconds. So let's say this right over here is, that right over there is 10 30 seconds. 10 30 seconds. Halfway up there, we have two 5 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have, let's see, it's all gonna be in terms of 30 seconds, and we get as high as 10 30 seconds. So let's say this right over here is, that right over there is 10 30 seconds. 10 30 seconds. Halfway up there, we have two 5 30 seconds. So let's see, that looks like about half. That right over there is 5 30 seconds. And then one 30 second would be about this."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Halfway up there, we have two 5 30 seconds. So let's see, that looks like about half. That right over there is 5 30 seconds. And then one 30 second would be about this. It's one, two, let's see, how to split it up. One, two, three, actually let me do it a little bit. One, two, three, four, five."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then one 30 second would be about this. It's one, two, let's see, how to split it up. One, two, three, actually let me do it a little bit. One, two, three, four, five. All right, so let's say this is one 30 second right over here. And our probabilities. So this right over here, probability, so this is what the values of the random variable could take on."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "One, two, three, four, five. All right, so let's say this is one 30 second right over here. And our probabilities. So this right over here, probability, so this is what the values of the random variable could take on. So I'll just make a little histogram here. So x equals zero. And then, and the probability there, and actually, since I want to do a histogram, it will look like this."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right over here, probability, so this is what the values of the random variable could take on. So I'll just make a little histogram here. So x equals zero. And then, and the probability there, and actually, since I want to do a histogram, it will look like this. Actually, let me do it a little bit different. So, put it right here. So x is equal to zero."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then, and the probability there, and actually, since I want to do a histogram, it will look like this. Actually, let me do it a little bit different. So, put it right here. So x is equal to zero. Right there, the probability is one 30 second. And I can shade that in. Now, I have the probability that x equals one."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So x is equal to zero. Right there, the probability is one 30 second. And I can shade that in. Now, I have the probability that x equals one. X equals one is five 30 seconds. So let me draw that. So five 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, I have the probability that x equals one. X equals one is five 30 seconds. So let me draw that. So five 30 seconds. So, put the bar there. So we shade that in. So this right over here is the probability that x is equal to one, that we get one."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So five 30 seconds. So, put the bar there. So we shade that in. So this right over here is the probability that x is equal to one, that we get one. That one, exactly one out of the five flips result in heads. Now we have the probability x equals two. X equals two is 10 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right over here is the probability that x is equal to one, that we get one. That one, exactly one out of the five flips result in heads. Now we have the probability x equals two. X equals two is 10 30 seconds. So that's going to look like this. That's going to look like this. Alright."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "X equals two is 10 30 seconds. So that's going to look like this. That's going to look like this. Alright. My best attempt at hand drawing it. So, somehow I like the aesthetics of hand drawn things more. Sometimes if you just get a computer to graph it, I don't know, sometimes it loses a little bit of its personality."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Alright. My best attempt at hand drawing it. So, somehow I like the aesthetics of hand drawn things more. Sometimes if you just get a computer to graph it, I don't know, sometimes it loses a little bit of its personality. Alright. So that right over there is the probability that we get that x, that our random variable x is equal to two. Then we have the probability that x equals three, which is also 10 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Sometimes if you just get a computer to graph it, I don't know, sometimes it loses a little bit of its personality. Alright. So that right over there is the probability that we get that x, that our random variable x is equal to two. Then we have the probability that x equals three, which is also 10 30 seconds. So, that is also 10 30 seconds. So let me draw that. This is also 10 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we have the probability that x equals three, which is also 10 30 seconds. So, that is also 10 30 seconds. So let me draw that. This is also 10 30 seconds. Shade this in. Dum da dum da dum. Alright."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is also 10 30 seconds. Shade this in. Dum da dum da dum. Alright. I find this strangely therapeutic. Alright. So this is the probability that x is equal to three."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Alright. I find this strangely therapeutic. Alright. So this is the probability that x is equal to three. Now x equals four, that's five 30 seconds. So we go back right over here. That's five 30 seconds."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the probability that x is equal to three. Now x equals four, that's five 30 seconds. So we go back right over here. That's five 30 seconds. So, shade that one in. So this right over here is x is equal to four. And then finally the probability that x equals five is one 30 second again."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's five 30 seconds. So, shade that one in. So this right over here is x is equal to four. And then finally the probability that x equals five is one 30 second again. Same level as this right over here. Shade it in. So this right over here is our random variable x equaling five."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally the probability that x equals five is one 30 second again. Same level as this right over here. Shade it in. So this right over here is our random variable x equaling five. And so when you visually show this probability distribution, it's important to realize this is a discrete probability distribution. This is a discrete random variable. It can only take on a finite number of values."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right over here is our random variable x equaling five. And so when you visually show this probability distribution, it's important to realize this is a discrete probability distribution. This is a discrete random variable. It can only take on a finite number of values. Actually, I should say it's a finite discrete random variable. You could have something that takes on discrete values, but in theory it could take on an infinite number of discrete values. You could just keep counting higher and higher and higher."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "It can only take on a finite number of values. Actually, I should say it's a finite discrete random variable. You could have something that takes on discrete values, but in theory it could take on an infinite number of discrete values. You could just keep counting higher and higher and higher. But this is discrete in that it's kind of these particular values. It can't take on any value in between. And it's also finite."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could just keep counting higher and higher and higher. But this is discrete in that it's kind of these particular values. It can't take on any value in between. And it's also finite. It can only take on x equals zero, x equals one, x equals two, x equals three, x equals four, or x equals five. And you see when you plot its probability distribution, this discrete probability distribution, you have it, you know, it starts at one 30 second, it goes up, and then it comes back down. And it has this symmetry."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it's also finite. It can only take on x equals zero, x equals one, x equals two, x equals three, x equals four, or x equals five. And you see when you plot its probability distribution, this discrete probability distribution, you have it, you know, it starts at one 30 second, it goes up, and then it comes back down. And it has this symmetry. And a distribution like this, this right over here, a discrete distribution like this, we call this a binomial distribution. And we'll talk in the future about why it's called a binomial distribution. But a big clue, actually I'll tell you why it's called a binomial distribution, is that these probabilities, you can get them using binomial coefficients, using combinatorics."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it has this symmetry. And a distribution like this, this right over here, a discrete distribution like this, we call this a binomial distribution. And we'll talk in the future about why it's called a binomial distribution. But a big clue, actually I'll tell you why it's called a binomial distribution, is that these probabilities, you can get them using binomial coefficients, using combinatorics. In another video we'll talk about, especially when we talk about the binomial theorem, why we even call those things binomial coefficients. But it's really based on taking powers of binomials in algebra. But this is a very, very, very, very important distribution."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "But a big clue, actually I'll tell you why it's called a binomial distribution, is that these probabilities, you can get them using binomial coefficients, using combinatorics. In another video we'll talk about, especially when we talk about the binomial theorem, why we even call those things binomial coefficients. But it's really based on taking powers of binomials in algebra. But this is a very, very, very, very important distribution. It's very important in statistics because for a lot of discrete processes, one might assume that the underlying distribution is a binomial distribution. And when we get further into statistics, we will talk why people do that. Now if you were to, if you have much more than five cases here, if instead of saying that the number of heads for flipping a coin five times, you said x is equal to the number of heads of flipping a coin five million times, then you can imagine you'd have much, much, you know, the bars would get narrower and narrower relative to the whole hump."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "But this is a very, very, very, very important distribution. It's very important in statistics because for a lot of discrete processes, one might assume that the underlying distribution is a binomial distribution. And when we get further into statistics, we will talk why people do that. Now if you were to, if you have much more than five cases here, if instead of saying that the number of heads for flipping a coin five times, you said x is equal to the number of heads of flipping a coin five million times, then you can imagine you'd have much, much, you know, the bars would get narrower and narrower relative to the whole hump. And what it would start to do, it would start to approach something that looks really, something that looks really like a bell curve. Let me do that in a color that you can see better that I haven't used yet. So it would start to look, so if you had more and more of these, if you had more and more of these possibilities, it's going to start approaching what looks like a bell curve."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now if you were to, if you have much more than five cases here, if instead of saying that the number of heads for flipping a coin five times, you said x is equal to the number of heads of flipping a coin five million times, then you can imagine you'd have much, much, you know, the bars would get narrower and narrower relative to the whole hump. And what it would start to do, it would start to approach something that looks really, something that looks really like a bell curve. Let me do that in a color that you can see better that I haven't used yet. So it would start to look, so if you had more and more of these, if you had more and more of these possibilities, it's going to start approaching what looks like a bell curve. And you've probably heard the notion of a bell curve. And the bell curve is a normal distribution. So if you, one way to think about it is the normal distribution is a probability density function."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it would start to look, so if you had more and more of these, if you had more and more of these possibilities, it's going to start approaching what looks like a bell curve. And you've probably heard the notion of a bell curve. And the bell curve is a normal distribution. So if you, one way to think about it is the normal distribution is a probability density function. It's a continuous case. So the yellow one, that we're approaching a normal distribution, and a normal distribution in kind of the classical sense is going to keep going on and on. Normal distribution."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you, one way to think about it is the normal distribution is a probability density function. It's a continuous case. So the yellow one, that we're approaching a normal distribution, and a normal distribution in kind of the classical sense is going to keep going on and on. Normal distribution. And it's related to the binomial, you know, a lot of times in statistics, people will assume a normal distribution because they say, okay, it's the product of kind of an almost an infinite number of random processes happening. Here we're taking a coin, we're flipping it five times, but if you imagine kind of molecules interacting or humans interacting, you're saying, oh, there's almost an infinite number of interactions, and then that's going to result in a normal distribution, which is very, very important in science and statistics. Binomial distribution is the discrete version of that."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Normal distribution. And it's related to the binomial, you know, a lot of times in statistics, people will assume a normal distribution because they say, okay, it's the product of kind of an almost an infinite number of random processes happening. Here we're taking a coin, we're flipping it five times, but if you imagine kind of molecules interacting or humans interacting, you're saying, oh, there's almost an infinite number of interactions, and then that's going to result in a normal distribution, which is very, very important in science and statistics. Binomial distribution is the discrete version of that. And one little point of notion, you know, these are where the distributions are, this is where they come from, this is how they're related. If you kind of think about, as you get more and more trials, the binomial distribution is going to really approach the normal distribution. But it's really important to think about where these things come from, and we'll talk about it much more in the statistics, because it is reasonable to assume an underlying binomial distribution or a normal distribution for a lot of different types of processes, but sometimes it's not."}, {"video_title": "Visualizing a binomial distribution   Probability and Statistics   Khan Academy.mp3", "Sentence": "Binomial distribution is the discrete version of that. And one little point of notion, you know, these are where the distributions are, this is where they come from, this is how they're related. If you kind of think about, as you get more and more trials, the binomial distribution is going to really approach the normal distribution. But it's really important to think about where these things come from, and we'll talk about it much more in the statistics, because it is reasonable to assume an underlying binomial distribution or a normal distribution for a lot of different types of processes, but sometimes it's not. And, you know, even in things like economics, sometimes people assume a normal distribution when it's actually much more likely that the things on the ends are going to happen, which might lead to things like economic crises or whatever else. But anyway, I don't want to get off topic. The whole point here is just to appreciate, hey, you know, we started with this random variable, the number of heads from flipping a coin five times, and we plotted it and we were able to see, we were able to visualize this binomial distribution, and I'm kind of telling you, I haven't really shown you, that if you were to have many, many more flips and you defined the random variable in a similar way, then this histogram is going to look a lot, this bar chart is going to look a lot more like a bell curve, and if you had essentially an infinite number of them, you would start having a continuous probability distribution, or I should say, probability density function, and that would be, that would get us closer to a normal distribution."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "One is when we are dealing with proportions. So I'll write that on the left side right over here. And the other is when we are dealing with means. In the proportion case, when we're doing our significance test, we will set up some null hypothesis that usually deals with the population proportion. We might say it is equal to some value. Let's just call that P sub one. And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "In the proportion case, when we're doing our significance test, we will set up some null hypothesis that usually deals with the population proportion. We might say it is equal to some value. Let's just call that P sub one. And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. So let me just go with that one. It's not equal to P sub one. And then what we do to actually test, to actually do the significance test is we take a sample from the population."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. So let me just go with that one. It's not equal to P sub one. And then what we do to actually test, to actually do the significance test is we take a sample from the population. It's going to have a sample size of N. We need to make sure that we feel good about making the inference. We've talked about the conditions for inference in previous videos multiple times. But from this, we calculate the sample proportion."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And then what we do to actually test, to actually do the significance test is we take a sample from the population. It's going to have a sample size of N. We need to make sure that we feel good about making the inference. We've talked about the conditions for inference in previous videos multiple times. But from this, we calculate the sample proportion. And then from this, we calculate the P value. And the way that we do the P value, remember, the P value is the probability of getting a sample proportion at least this extreme. And if it's below some threshold, we reject the null hypothesis and it suggests the alternative."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "But from this, we calculate the sample proportion. And then from this, we calculate the P value. And the way that we do the P value, remember, the P value is the probability of getting a sample proportion at least this extreme. And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. And over here, the way we do that is we find an associated Z value for that P, for that sample proportion. And the way that we calculate it, we say, okay, look, our Z is going to be how many of the sampling distributions, standard deviations, are we away from the mean? And remember, the mean of the sampling distribution is going to be the population proportion."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. And over here, the way we do that is we find an associated Z value for that P, for that sample proportion. And the way that we calculate it, we say, okay, look, our Z is going to be how many of the sampling distributions, standard deviations, are we away from the mean? And remember, the mean of the sampling distribution is going to be the population proportion. So here, we got this sample statistic, this sample proportion. The difference between that and the assumed proportion, remember, when we do these significance tests, we try to figure out the probability assuming the null hypothesis is true. And so when we see this P sub zero, this is the assumed proportion from the null hypothesis."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And remember, the mean of the sampling distribution is going to be the population proportion. So here, we got this sample statistic, this sample proportion. The difference between that and the assumed proportion, remember, when we do these significance tests, we try to figure out the probability assuming the null hypothesis is true. And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. So that's the difference between these two, the sample proportion and the assumed proportion. And then you'd wanna divide it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution of the sample proportion. And this works out well for proportions because in proportions, I can figure out what this is."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. So that's the difference between these two, the sample proportion and the assumed proportion. And then you'd wanna divide it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution of the sample proportion. And this works out well for proportions because in proportions, I can figure out what this is. This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value. And in this case, I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion. Now with means, there's definitely some similarities here."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And this works out well for proportions because in proportions, I can figure out what this is. This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value. And in this case, I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion. Now with means, there's definitely some similarities here. You will make a null hypothesis. Maybe you assume the population mean is equal to mu one. And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Now with means, there's definitely some similarities here. You will make a null hypothesis. Maybe you assume the population mean is equal to mu one. And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. And you're gonna do something very simple. You take your population, take a sample of size N. Instead of calculating a sample proportion, you calculate a sample mean. And actually, you can calculate other things like a sample standard deviation."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. And you're gonna do something very simple. You take your population, take a sample of size N. Instead of calculating a sample proportion, you calculate a sample mean. And actually, you can calculate other things like a sample standard deviation. But now you have an issue. You say, well, ideally, I would use a Z statistic. And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And actually, you can calculate other things like a sample standard deviation. But now you have an issue. You say, well, ideally, I would use a Z statistic. And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. So that would be this right over here. That's what that zero means, the assumed mean from the null hypothesis. And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. So that would be this right over here. That's what that zero means, the assumed mean from the null hypothesis. And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. But this is not so easy to figure out. In order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of N. We know what N is going to be if we conducted the sample, but we don't know what the standard deviation is. So instead, what we do is we estimate this."}, {"video_title": "When to use z or t statistics in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. But this is not so easy to figure out. In order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of N. We know what N is going to be if we conducted the sample, but we don't know what the standard deviation is. So instead, what we do is we estimate this. And so we'll take the sample mean. We subtract from that the assumed population mean from the null hypothesis. And we divide by an estimate of this, which is going to be our sample standard deviation divided by the square root of N. But because this is an estimate, we actually get a better result."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Liz's math test included a survey question asking how many hours students spent studying for the test. The scatter plot and trend line below show the relationship between how many hours students spent studying and their score on the test. The line fitted to model the data has a slope of 15. So the line that they're talking about is right here. So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So the line that they're talking about is right here. So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope?"}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that the student who scored zero studied for an average of 15 hours."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "The model predicts that the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time."}, {"video_title": "Interpreting slope of regression line   AP Statistics   Khan Academy.mp3", "Sentence": "No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. Yes, that is exactly what we were thinking about when we were looking at the model. That's what a slope of 15 tells you. You increase studying time by an hour, it increases score by 15 points."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to start with a fair coin, and I'm going to flip it 4 times. Flip it 4 times. And the first question I want to ask is, what is the probability that I get exactly 1 head? Or heads. Actually, this is one of those confusing things. When you're talking about what side of the coin, even though I've been not doing this consistently, I'm tempted to say if you're saying 1, it feels like you should do the singular, which would be head, but I read up a little bit of it on the internet, and it seems like when you're talking about coins, you really should say 1 heads, which seems a little bit difficult for me, but I'll try to go with that. So what is the probability of getting exactly 1 heads?"}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or heads. Actually, this is one of those confusing things. When you're talking about what side of the coin, even though I've been not doing this consistently, I'm tempted to say if you're saying 1, it feels like you should do the singular, which would be head, but I read up a little bit of it on the internet, and it seems like when you're talking about coins, you really should say 1 heads, which seems a little bit difficult for me, but I'll try to go with that. So what is the probability of getting exactly 1 heads? And I put that in quotes to say, well, we're just talking about 1 head there, but it's called heads when you're dealing with coins. Anyway, I think you get what I'm talking about. Now to think about this, let's think about how many different possible ways we can get 4 flips of a coin."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is the probability of getting exactly 1 heads? And I put that in quotes to say, well, we're just talking about 1 head there, but it's called heads when you're dealing with coins. Anyway, I think you get what I'm talking about. Now to think about this, let's think about how many different possible ways we can get 4 flips of a coin. We're going to have 1 flip, then another flip, then another flip, then another flip. And this first flip has 2 possibilities, it could be heads or tails. The second flip has 2 possibilities."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now to think about this, let's think about how many different possible ways we can get 4 flips of a coin. We're going to have 1 flip, then another flip, then another flip, then another flip. And this first flip has 2 possibilities, it could be heads or tails. The second flip has 2 possibilities. It could be heads or tails. The third flip has 2 possibilities, it could be heads or tails. And the fourth flip has 2 possibilities."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "The second flip has 2 possibilities. It could be heads or tails. The third flip has 2 possibilities, it could be heads or tails. And the fourth flip has 2 possibilities. It could be heads or tails. So you have 2 times 2 times 2 times 2, which is equal to 16 possibilities. 16 possible outcomes when you flip a coin 4 times."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the fourth flip has 2 possibilities. It could be heads or tails. So you have 2 times 2 times 2 times 2, which is equal to 16 possibilities. 16 possible outcomes when you flip a coin 4 times. 16 possible outcomes. And any one of the possible outcomes would be 1 of 16. So if I wanted to say, so if I were to just say the probability, and I'm just going to not talk about this one head."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "16 possible outcomes when you flip a coin 4 times. 16 possible outcomes. And any one of the possible outcomes would be 1 of 16. So if I wanted to say, so if I were to just say the probability, and I'm just going to not talk about this one head. If I just take a, just maybe this thing that has three heads right here. This exact sequence of events, this is the first flip, second flip, third flip, fourth flip. Getting exactly this, this is exactly 1 out of a possible of 16 events."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if I wanted to say, so if I were to just say the probability, and I'm just going to not talk about this one head. If I just take a, just maybe this thing that has three heads right here. This exact sequence of events, this is the first flip, second flip, third flip, fourth flip. Getting exactly this, this is exactly 1 out of a possible of 16 events. Now with that out of the way, let's think about how many possibilities, how many of those 16 possibilities involve getting exactly 1 heads. Well, we could list them. You could get your heads."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Getting exactly this, this is exactly 1 out of a possible of 16 events. Now with that out of the way, let's think about how many possibilities, how many of those 16 possibilities involve getting exactly 1 heads. Well, we could list them. You could get your heads. So this is equal to the probability of getting the heads in the first flip plus the probability of getting the heads in the second flip plus the probability of getting the heads in the third flip. Remember, exactly 1 heads. We're not saying at least 1, exactly 1 heads."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could get your heads. So this is equal to the probability of getting the heads in the first flip plus the probability of getting the heads in the second flip plus the probability of getting the heads in the third flip. Remember, exactly 1 heads. We're not saying at least 1, exactly 1 heads. So probability in the third flip. And then, or the possibility that you get heads in the fourth flip. Tails, heads, and tails."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not saying at least 1, exactly 1 heads. So probability in the third flip. And then, or the possibility that you get heads in the fourth flip. Tails, heads, and tails. And we know already what the probability of each of these things are. There are 16 possible events. And each of these are one of those 16 possible events."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Tails, heads, and tails. And we know already what the probability of each of these things are. There are 16 possible events. And each of these are one of those 16 possible events. So this is going to be 1 over 16. And so we're really saying the probability of getting exactly 1 heads is the same thing as the probability of getting heads in the first flip or the probability of getting heads in the first flip or heads in the second flip or heads in the third flip or heads in the fourth flip. And we can add the probabilities of these different things because they are mutually exclusive."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And each of these are one of those 16 possible events. So this is going to be 1 over 16. And so we're really saying the probability of getting exactly 1 heads is the same thing as the probability of getting heads in the first flip or the probability of getting heads in the first flip or heads in the second flip or heads in the third flip or heads in the fourth flip. And we can add the probabilities of these different things because they are mutually exclusive. Any two of these things cannot happen at the same time. You have to pick one of these scenarios. And so we can add the probabilities."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we can add the probabilities of these different things because they are mutually exclusive. Any two of these things cannot happen at the same time. You have to pick one of these scenarios. And so we can add the probabilities. 1 16th plus 1 16th plus 1 16th plus 1 16th. Did I say that four times? Well, assume that I did."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so we can add the probabilities. 1 16th plus 1 16th plus 1 16th plus 1 16th. Did I say that four times? Well, assume that I did. And so you would get 4 16ths, which is equal to 1 4th. Fair enough. Now let's ask a slightly more interesting question."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, assume that I did. And so you would get 4 16ths, which is equal to 1 4th. Fair enough. Now let's ask a slightly more interesting question. Let's ask ourselves the probability of getting exactly 2 heads. And there's a couple of ways we can think about it. One is just in the traditional way."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's ask a slightly more interesting question. Let's ask ourselves the probability of getting exactly 2 heads. And there's a couple of ways we can think about it. One is just in the traditional way. And let's just look for the number of possibilities and of those equally likely possibilities. And we can only use this methodology because it's a fair coin. So how many of the total possibilities have 2 heads of the total of equally likely possibilities?"}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "One is just in the traditional way. And let's just look for the number of possibilities and of those equally likely possibilities. And we can only use this methodology because it's a fair coin. So how many of the total possibilities have 2 heads of the total of equally likely possibilities? So we know there are 16 equally likely possibilities. How many of those have 2 heads? So I'm actually ahead of time."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many of the total possibilities have 2 heads of the total of equally likely possibilities? So we know there are 16 equally likely possibilities. How many of those have 2 heads? So I'm actually ahead of time. So we save time. I've drawn all of the 16 equally likely possibilities. And how many of these involve 2 heads?"}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm actually ahead of time. So we save time. I've drawn all of the 16 equally likely possibilities. And how many of these involve 2 heads? Well, let's see. This one over here has 2 heads. This one over here has 2 heads."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And how many of these involve 2 heads? Well, let's see. This one over here has 2 heads. This one over here has 2 heads. This one over here has 2 heads. Let's see. That's this one over here has 2 heads."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "This one over here has 2 heads. This one over here has 2 heads. Let's see. That's this one over here has 2 heads. And this one over here has 2 heads. And then this one over here has 2 heads. And I believe we are done after that."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's this one over here has 2 heads. And this one over here has 2 heads. And then this one over here has 2 heads. And I believe we are done after that. So if we count them, 1, 2, 3, 4, 5, 6 of the possibilities have exactly 2 heads. So 6 of the 16 equally likely possibilities have 2 heads. So we have a 3 8th chance of getting exactly 2 heads."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I believe we are done after that. So if we count them, 1, 2, 3, 4, 5, 6 of the possibilities have exactly 2 heads. So 6 of the 16 equally likely possibilities have 2 heads. So we have a 3 8th chance of getting exactly 2 heads. Now, that's kind of what we've been doing in the past. But what I want to do is think about a way so we wouldn't have to write out all the possibilities. And the reason why that's useful is we're only dealing with 4 flips now."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have a 3 8th chance of getting exactly 2 heads. Now, that's kind of what we've been doing in the past. But what I want to do is think about a way so we wouldn't have to write out all the possibilities. And the reason why that's useful is we're only dealing with 4 flips now. But if we were dealing with 10 flips, there's no way that we could write out all the possibilities like this. So I really want a different way of thinking about it. And the different way of thinking about it is if we're saying exactly 2 heads, you can imagine we're having the 4 flips."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the reason why that's useful is we're only dealing with 4 flips now. But if we were dealing with 10 flips, there's no way that we could write out all the possibilities like this. So I really want a different way of thinking about it. And the different way of thinking about it is if we're saying exactly 2 heads, you can imagine we're having the 4 flips. Flip 1, flip 2, flip 3, flip 4. So these are the flips, or you could say the outcome of the flips. And if you're going to have exactly 2 heads, you could say, well, look, I'm going to have 1 head in one of these positions and then 1 head in the other position."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the different way of thinking about it is if we're saying exactly 2 heads, you can imagine we're having the 4 flips. Flip 1, flip 2, flip 3, flip 4. So these are the flips, or you could say the outcome of the flips. And if you're going to have exactly 2 heads, you could say, well, look, I'm going to have 1 head in one of these positions and then 1 head in the other position. So if I'm picking the first, so you could say, and I have kind of a heads 1 and I have a heads 2. And I don't want you to think that these are somehow the heads in the first flip or the heads in the second flip. What I'm saying is we need 2 heads."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if you're going to have exactly 2 heads, you could say, well, look, I'm going to have 1 head in one of these positions and then 1 head in the other position. So if I'm picking the first, so you could say, and I have kind of a heads 1 and I have a heads 2. And I don't want you to think that these are somehow the heads in the first flip or the heads in the second flip. What I'm saying is we need 2 heads. We need a total of 2 heads in all of our flips. And I'm just giving one of the heads a name, and I'm giving the other head a name. And what we're going to see in a few seconds is that we actually don't want to double count."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I'm saying is we need 2 heads. We need a total of 2 heads in all of our flips. And I'm just giving one of the heads a name, and I'm giving the other head a name. And what we're going to see in a few seconds is that we actually don't want to double count. We don't want to count the situation. We don't want to double count this situation. Heads 1, heads 2, tails, tails, and heads 2, heads 1, tails, tails."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what we're going to see in a few seconds is that we actually don't want to double count. We don't want to count the situation. We don't want to double count this situation. Heads 1, heads 2, tails, tails, and heads 2, heads 1, tails, tails. For our purposes, these are the exact same outcomes. So we don't want to double count that, and we're going to have to account for that. But if we just think about it generally, how many different spots, how many of different flips can that first head show up in?"}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Heads 1, heads 2, tails, tails, and heads 2, heads 1, tails, tails. For our purposes, these are the exact same outcomes. So we don't want to double count that, and we're going to have to account for that. But if we just think about it generally, how many different spots, how many of different flips can that first head show up in? Well, there's 4 different flips that that first head could show up in. So there's 4 possibilities, 4 flips, or 4 places that it could show up in. Well, if that first head takes up one of these 4 places, let's just say that first head shows up on the third flip, then how many different places can that second head show up in?"}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But if we just think about it generally, how many different spots, how many of different flips can that first head show up in? Well, there's 4 different flips that that first head could show up in. So there's 4 possibilities, 4 flips, or 4 places that it could show up in. Well, if that first head takes up one of these 4 places, let's just say that first head shows up on the third flip, then how many different places can that second head show up in? Well, if that first head is in one of the 4 places, then that second head can only be in 3 different places. So that second head can only be in 3 different places. And so it could be in any one of these."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, if that first head takes up one of these 4 places, let's just say that first head shows up on the third flip, then how many different places can that second head show up in? Well, if that first head is in one of the 4 places, then that second head can only be in 3 different places. So that second head can only be in 3 different places. And so it could be in any one of these. It could maybe be right over there, any one of those 3 places. And so when you think about it in terms of the first, and I don't want to say the first head, head 1. Actually, let me call it this way."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so it could be in any one of these. It could maybe be right over there, any one of those 3 places. And so when you think about it in terms of the first, and I don't want to say the first head, head 1. Actually, let me call it this way. Head A and head B. That way you won't think that I'm talking about the first flip or the second flip. So this is head A, and this right over there is head B."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me call it this way. Head A and head B. That way you won't think that I'm talking about the first flip or the second flip. So this is head A, and this right over there is head B. So if you had a particular, I mean, these heads are identical. These outcomes aren't different. But the way we talk about it right now, it looks like there's 4 places that we could get this head in, and there's 3 places where we could get this head in."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is head A, and this right over there is head B. So if you had a particular, I mean, these heads are identical. These outcomes aren't different. But the way we talk about it right now, it looks like there's 4 places that we could get this head in, and there's 3 places where we could get this head in. And so if you multiply all of the different ways that you could get all of the different scenarios where this is in 4 different places, and then this is in one of the 3 leftover places, you get 12 different scenarios. But there would only be 12 different scenarios if you viewed this as being different than this. And let me rewrite it with our new."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the way we talk about it right now, it looks like there's 4 places that we could get this head in, and there's 3 places where we could get this head in. And so if you multiply all of the different ways that you could get all of the different scenarios where this is in 4 different places, and then this is in one of the 3 leftover places, you get 12 different scenarios. But there would only be 12 different scenarios if you viewed this as being different than this. And let me rewrite it with our new. So this is head A, this is head B, this is head B, this is head A. There would only be 12 different scenarios if you viewed these 2 things as fundamentally different. But we don't."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let me rewrite it with our new. So this is head A, this is head B, this is head B, this is head A. There would only be 12 different scenarios if you viewed these 2 things as fundamentally different. But we don't. We're actually double counting, because we can always swap these 2 heads and have the exact same outcome. So what you want to do is actually divide it by 2. So you want to divide it by all of the different ways that you can swap 2 different things."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But we don't. We're actually double counting, because we can always swap these 2 heads and have the exact same outcome. So what you want to do is actually divide it by 2. So you want to divide it by all of the different ways that you can swap 2 different things. If we had 3 heads here, you would think about all of the different ways that you could swap 3 different things. If we had 4 heads here, it would be all of the different ways you could swap 4 different things. So there's 12 different scenarios if you couldn't swap them, but you want to divide it by all of the different ways that you can swap 2 things."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you want to divide it by all of the different ways that you can swap 2 different things. If we had 3 heads here, you would think about all of the different ways that you could swap 3 different things. If we had 4 heads here, it would be all of the different ways you could swap 4 different things. So there's 12 different scenarios if you couldn't swap them, but you want to divide it by all of the different ways that you can swap 2 things. So 12 divided by 2 is equal to 6 fundamentally different scenarios, considering that you can swap them. If you assume that head A and head B can be interchangeable, that it's a completely identical outcome for us, because they're really just heads. So there's 6 different scenarios, and we know that there's a total of 16 equally likely scenarios."}, {"video_title": "Getting exactly two heads (combinatorics)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's 12 different scenarios if you couldn't swap them, but you want to divide it by all of the different ways that you can swap 2 things. So 12 divided by 2 is equal to 6 fundamentally different scenarios, considering that you can swap them. If you assume that head A and head B can be interchangeable, that it's a completely identical outcome for us, because they're really just heads. So there's 6 different scenarios, and we know that there's a total of 16 equally likely scenarios. So we could say that the probability of getting exactly 2 heads is 6 times 6 scenarios. And there's a couple of ways. You could say there's 6 scenarios that give us 2 heads of a possible 16, or you could say there are 6 possible scenarios, and the probability of each of those scenarios is 1 16th."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The results are shown in the two plots below. And so the first statement that we have to complete is the mean number of fruits is greater for, and actually let me go to the actual screen, is greater for, we have to pick between freshmen and seniors and then they say the mean is a good measure for the center of distribution of, and we pick either freshmen or seniors. So let me go back to my scratch pad here and let's think about this. So let's first think about the first part. So let's just calculate the mean for each of these distributions. And I encourage you to pause the video and try to calculate it out on your own. So let's first think about the mean number of fruit for freshmen."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's first think about the first part. So let's just calculate the mean for each of these distributions. And I encourage you to pause the video and try to calculate it out on your own. So let's first think about the mean number of fruit for freshmen. So essentially we're just gonna take each of these data points, add them all together, and then divide by the number of data points that we have. So we have one data point at zero. So we have one data point at zero, so I'll write zero."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's first think about the mean number of fruit for freshmen. So essentially we're just gonna take each of these data points, add them all together, and then divide by the number of data points that we have. So we have one data point at zero. So we have one data point at zero, so I'll write zero. And then we have two data points at one. So we could say plus two times one. And then we have two data points at two."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have one data point at zero, so I'll write zero. And then we have two data points at one. So we could say plus two times one. And then we have two data points at two. So we write plus two times two. And then let's see, we have a bunch of data. We have four data points at three."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have two data points at two. So we write plus two times two. And then let's see, we have a bunch of data. We have four data points at three. So we could say we have four threes. So let me circle that. So we have four threes plus four times three."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have four data points at three. So we could say we have four threes. So let me circle that. So we have four threes plus four times three. And then we have three fours, so plus three times four. And then we have a five, so plus five. And then we have a six."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have four threes plus four times three. And then we have three fours, so plus three times four. And then we have a five, so plus five. And then we have a six. Let me do this in a color that you can see. And then we have a six right over here, plus six. And how many total points did we have?"}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have a six. Let me do this in a color that you can see. And then we have a six right over here, plus six. And how many total points did we have? Well, we had one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14. Oh, actually, be careful, we had 15 points. And I didn't put that one in there."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And how many total points did we have? Well, we had one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14. Oh, actually, be careful, we had 15 points. And I didn't put that one in there. So actually, let me just. So we have 15 points. And I can't forget this one over here."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I didn't put that one in there. So actually, let me just. So we have 15 points. And I can't forget this one over here. So plus, my pen is acting a little funny right now, but we'll power through that. Plus 19. Plus 19."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I can't forget this one over here. So plus, my pen is acting a little funny right now, but we'll power through that. Plus 19. Plus 19. So what is this going to be? So this is just going to be zero. This is going to be two."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 19. So what is this going to be? So this is just going to be zero. This is going to be two. This is going to be four. This is going to be 12. My pen is really acting up."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is going to be two. This is going to be four. This is going to be 12. My pen is really acting up. It's almost like it's running out of digital ink or something. And this is going to be another 12. And then we have five, six, and 19."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "My pen is really acting up. It's almost like it's running out of digital ink or something. And this is going to be another 12. And then we have five, six, and 19. So what is this going to be? Two plus four is six. Plus 24 is 30."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have five, six, and 19. So what is this going to be? Two plus four is six. Plus 24 is 30. Plus 11 is 41. Plus 19 gets us to 60. 60 divided by 15 is four."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 24 is 30. Plus 11 is 41. Plus 19 gets us to 60. 60 divided by 15 is four. So the mean number of fruit per day for the freshman is four pieces of fruit per day. So this right over here, that right over there, is our mean for the, let me put that in a color that you can actually see. Now let's do the same calculation for the seniors."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "60 divided by 15 is four. So the mean number of fruit per day for the freshman is four pieces of fruit per day. So this right over here, that right over there, is our mean for the, let me put that in a color that you can actually see. Now let's do the same calculation for the seniors. So we have one data point where they didn't eat any fruit at all each day, not too healthy. Then you have one one. So I'll just write that as, we could write that as one times one, but I'll just write that as one."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's do the same calculation for the seniors. So we have one data point where they didn't eat any fruit at all each day, not too healthy. Then you have one one. So I'll just write that as, we could write that as one times one, but I'll just write that as one. Then we have two twos. So plus two times two. Then we have one, two, three, four, five threes."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll just write that as, we could write that as one times one, but I'll just write that as one. Then we have two twos. So plus two times two. Then we have one, two, three, four, five threes. Five threes. So plus five times three. And then we have, we have three three fours."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then we have one, two, three, four, five threes. Five threes. So plus five times three. And then we have, we have three three fours. So plus three times four. And then we have two fives. Plus two times five."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have, we have three three fours. So plus three times four. And then we have two fives. Plus two times five. And then we have a six. We have a six plus six. And we have a seven."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus two times five. And then we have a six. We have a six plus six. And we have a seven. Someone eats seven pieces of fruit each day. A lot of fiber. Plus seven."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we have a seven. Someone eats seven pieces of fruit each day. A lot of fiber. Plus seven. And now, how many data points did we have? Well we have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16 data points. So we're gonna divide this by 16."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus seven. And now, how many data points did we have? Well we have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16 data points. So we're gonna divide this by 16. So what is this going to be? This is just zero. Let's see, this is just right over, that's zero."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're gonna divide this by 16. So what is this going to be? This is just zero. Let's see, this is just right over, that's zero. This is four. This is 15. This is 12."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's see, this is just right over, that's zero. This is four. This is 15. This is 12. This is 10. So we have one plus four is five. Plus 15 is 20."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 12. This is 10. So we have one plus four is five. Plus 15 is 20. Plus 12 is 32. Plus 10 is 42. 42 plus six is 48."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 15 is 20. Plus 12 is 32. Plus 10 is 42. 42 plus six is 48. 48. Am I doing, 42 plus six is 48. Plus seven, 48 plus seven is 55."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "42 plus six is 48. 48. Am I doing, 42 plus six is 48. Plus seven, 48 plus seven is 55. Did I do that right? Let me do that one more time. One plus four is five."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus seven, 48 plus seven is 55. Did I do that right? Let me do that one more time. One plus four is five. Plus 15 is 20. 32, 42. 42 plus 13 is 55."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "One plus four is five. Plus 15 is 20. 32, 42. 42 plus 13 is 55. So this is equal to 55 over 16, which is the same thing as, let's see, that's the same thing as three. And three times 16 is 48. So 3 7 16ths."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "42 plus 13 is 55. So this is equal to 55 over 16, which is the same thing as, let's see, that's the same thing as three. And three times 16 is 48. So 3 7 16ths. So the mean for the seniors, the mean for the seniors, 3 7 16ths, is right around, let's see, this is three, that's four. So 7 16ths, it's a little less than a half. It's right around there."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 3 7 16ths. So the mean for the seniors, the mean for the seniors, 3 7 16ths, is right around, let's see, this is three, that's four. So 7 16ths, it's a little less than a half. It's right around there. So the mean number of fruits is definitely greater for the freshmen. They have four, their mean number of fruit eaten per day is four versus three and 7 16ths. The mean is a good measure for the center of the distribution of."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's right around there. So the mean number of fruits is definitely greater for the freshmen. They have four, their mean number of fruit eaten per day is four versus three and 7 16ths. The mean is a good measure for the center of the distribution of. So when we think about whether it's freshmen or seniors, so the mean is fairly sensitive to when you have outliers here. For example, someone here was eating 19 pieces of fruit per day. That's an enormous amount of fruit."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mean is a good measure for the center of the distribution of. So when we think about whether it's freshmen or seniors, so the mean is fairly sensitive to when you have outliers here. For example, someone here was eating 19 pieces of fruit per day. That's an enormous amount of fruit. They must be only eating fruit. You can imagine if it was even a bigger outlier, someone was eating 20 or 30 pieces of fruit, just that one data point will skew the entire mean upwards. That wouldn't be the effect on the mode because the mode is the middle number."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's an enormous amount of fruit. They must be only eating fruit. You can imagine if it was even a bigger outlier, someone was eating 20 or 30 pieces of fruit, just that one data point will skew the entire mean upwards. That wouldn't be the effect on the mode because the mode is the middle number. Even if you change this one point all the way out here, it's not going to change what the middle number is. So the mean is more sensitive to these outliers, to these points that are really, really high, really, really low. So and because the seniors don't seem to have any outliers like that, I would say that the mean is a good measure for the center of distribution for the seniors or a better measure for the center of distribution for the seniors."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "That wouldn't be the effect on the mode because the mode is the middle number. Even if you change this one point all the way out here, it's not going to change what the middle number is. So the mean is more sensitive to these outliers, to these points that are really, really high, really, really low. So and because the seniors don't seem to have any outliers like that, I would say that the mean is a good measure for the center of distribution for the seniors or a better measure for the center of distribution for the seniors. So let's fill both of those out. So the mean number of fruit is greater for the freshmen and the mean is a good measure for the center of distribution for the seniors. And you actually even see it here."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So and because the seniors don't seem to have any outliers like that, I would say that the mean is a good measure for the center of distribution for the seniors or a better measure for the center of distribution for the seniors. So let's fill both of those out. So the mean number of fruit is greater for the freshmen and the mean is a good measure for the center of distribution for the seniors. And you actually even see it here. We saw that the mean number for freshmen was at four. But if you just ignored this person right over here and just you kind of thought about the bulk of this distribution right over here, four really doesn't look like the center of it. The center of it looks closer to three here."}, {"video_title": "Comparing means of distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you actually even see it here. We saw that the mean number for freshmen was at four. But if you just ignored this person right over here and just you kind of thought about the bulk of this distribution right over here, four really doesn't look like the center of it. The center of it looks closer to three here. And what happened is this one person eating 19 pieces of fruit per day skewed the mean upwards. While here, that three and 716s really did look closer to the actual distribution, closer to the, actually I shouldn't say, I mean, in both times, we actually did calculate the mean of the actual distribution. But here, since there's no outliers, it does seem the mean seemed much closer to, I guess you could say the middle of this pile right over here."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "This is the scores on midterm and final exams. So this axis, the vertical axis, is the scores. And then it's by student. And the blue bar is the midterm. And the yellow bar is the final. And the question they ask us is, by how many points did Nadia's score improve from the midterm to the final exam? So let's look at Nadia."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "And the blue bar is the midterm. And the yellow bar is the final. And the question they ask us is, by how many points did Nadia's score improve from the midterm to the final exam? So let's look at Nadia. So this is who we're talking about, Nadia. And we care about how many points did she improve from the midterm to the final. Midterm is blue, final is yellow."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "So let's look at Nadia. So this is who we're talking about, Nadia. And we care about how many points did she improve from the midterm to the final. Midterm is blue, final is yellow. So in the midterm, it looks like she scored, and if I were to eyeball it, it looks like 75 points. And on the final, it looks like she scored 80. Looks like she scored 85 points."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "Midterm is blue, final is yellow. So in the midterm, it looks like she scored, and if I were to eyeball it, it looks like 75 points. And on the final, it looks like she scored 80. Looks like she scored 85 points. So it looks like her score improved by 10 points. 10 points. Let's try one more."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "Looks like she scored 85 points. So it looks like her score improved by 10 points. 10 points. Let's try one more. How many students improve their scores from the midterm to the final exam? So to improve from the midterm to the final, that means that the yellow bar for a given student, which is the final, is going to be higher than the midterm bar. That's the only way you can improve from the midterm to the final."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "Let's try one more. How many students improve their scores from the midterm to the final exam? So to improve from the midterm to the final, that means that the yellow bar for a given student, which is the final, is going to be higher than the midterm bar. That's the only way you can improve from the midterm to the final. So Brandon improved from the midterm to the final. Vanessa improved from the midterm to the final. Daniel improved from the midterm to the final."}, {"video_title": "Reading bar charts comparing two sets of data   Pre-Algebra   Khan Academy.mp3", "Sentence": "That's the only way you can improve from the midterm to the final. So Brandon improved from the midterm to the final. Vanessa improved from the midterm to the final. Daniel improved from the midterm to the final. Kevin improved from the midterm to the final. William got a lower score on the final than the midterm. So he did not improve."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "Paul has the option between a high deductible or a low deductible health insurance plan. And when we talk about the deductible and health insurance, if someone says that they have a plan with a $1,000 deductible, that means that the insurance company only pays the medical costs after the first $1,000. So if you have a $1,000 deductible and you say incur medical costs of $800, you're going to pay that $800. The insurance company won't pay anything. If you have a deductible of $1,000 and your total medical costs are $1,200, you're going to pay the first $1,000 and then the insurance company will kick in after that. So with that out of the way, let's think about his two plans. If Paul chooses the low deductible plan, he will have to pay the first $1,000 of any, let me do that in purple, the first $1,000 of any medical costs."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "The insurance company won't pay anything. If you have a deductible of $1,000 and your total medical costs are $1,200, you're going to pay the first $1,000 and then the insurance company will kick in after that. So with that out of the way, let's think about his two plans. If Paul chooses the low deductible plan, he will have to pay the first $1,000 of any, let me do that in purple, the first $1,000 of any medical costs. The low deductible plan costs $8,000 for a year. So in this situation, he's going to pay $8,000 to get the insurance. If he has $900 of medical expenses, the insurance company still pays nothing."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "If Paul chooses the low deductible plan, he will have to pay the first $1,000 of any, let me do that in purple, the first $1,000 of any medical costs. The low deductible plan costs $8,000 for a year. So in this situation, he's going to pay $8,000 to get the insurance. If he has $900 of medical expenses, the insurance company still pays nothing. If he has $2,000 of medical expenses, then he pays the first $1,000 and then the insurance company would pay the next $1,000. If he has $10,000 in medical expenses, he would pay the first $1,000 and then the insurance company would pay the next $9,000. If Paul chooses the high deductible plan, he will have to pay the first $2,500 of any medical costs."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "If he has $900 of medical expenses, the insurance company still pays nothing. If he has $2,000 of medical expenses, then he pays the first $1,000 and then the insurance company would pay the next $1,000. If he has $10,000 in medical expenses, he would pay the first $1,000 and then the insurance company would pay the next $9,000. If Paul chooses the high deductible plan, he will have to pay the first $2,500 of any medical costs. The high deductible plan costs $7,500 a year. And it makes sense that the high deductible plan costs less than the low deductible plan because here the insurance doesn't kick in until he has over $2,500 of medical expenses while here it was only 1,000. To help himself choose a plan, Paul found some statistics about common health problems for people similar to him."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "If Paul chooses the high deductible plan, he will have to pay the first $2,500 of any medical costs. The high deductible plan costs $7,500 a year. And it makes sense that the high deductible plan costs less than the low deductible plan because here the insurance doesn't kick in until he has over $2,500 of medical expenses while here it was only 1,000. To help himself choose a plan, Paul found some statistics about common health problems for people similar to him. Assume that the table below, and I put it up here on the right, correctly shows the probabilities and costs of total medical incidents within the next year. So this right over here, it's saying what's the probability he has zero in medical costs? What's the probability, he has a 25% probability of $1,000, 20% probability $4,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "To help himself choose a plan, Paul found some statistics about common health problems for people similar to him. Assume that the table below, and I put it up here on the right, correctly shows the probabilities and costs of total medical incidents within the next year. So this right over here, it's saying what's the probability he has zero in medical costs? What's the probability, he has a 25% probability of $1,000, 20% probability $4,000. And this is a simplification, a pretty dramatic simplification from the real world. In the real world, the way this makes it sound is there's only five possible medical costs that someone might have, zero, 1,000, 4,000, 7,000, 15,000. In the real world, you could have $20 in medical costs, you could have 20,000, you could have $999."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's the probability, he has a 25% probability of $1,000, 20% probability $4,000. And this is a simplification, a pretty dramatic simplification from the real world. In the real world, the way this makes it sound is there's only five possible medical costs that someone might have, zero, 1,000, 4,000, 7,000, 15,000. In the real world, you could have $20 in medical costs, you could have 20,000, you could have $999. So in the real world, there's many, many more situations here that you would have to kind of redistribute the probabilities accordingly. But with that said, this isn't a bad approximation. It's just saying, okay, roughly, if you wanted to construct this so it's easier to do the math, which the problem writers have done, say, look, okay, 30%, zero, 25%, $1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the real world, you could have $20 in medical costs, you could have 20,000, you could have $999. So in the real world, there's many, many more situations here that you would have to kind of redistribute the probabilities accordingly. But with that said, this isn't a bad approximation. It's just saying, okay, roughly, if you wanted to construct this so it's easier to do the math, which the problem writers have done, say, look, okay, 30%, zero, 25%, $1,000. This is pretty indicative if you had to kind of group all of the possible costs into some major bucket. So it's probably at least a pretty good method for figuring out which insurance policy someone should use. So they say, including the cost of insurance, what are Paul's expected total medical costs with the low deductible plan?"}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's just saying, okay, roughly, if you wanted to construct this so it's easier to do the math, which the problem writers have done, say, look, okay, 30%, zero, 25%, $1,000. This is pretty indicative if you had to kind of group all of the possible costs into some major bucket. So it's probably at least a pretty good method for figuring out which insurance policy someone should use. So they say, including the cost of insurance, what are Paul's expected total medical costs with the low deductible plan? Round your answer to the nearest cent. So actually, I'll get the calculator out for this. So at the low, whoops, with the low deductible plan here, low deductible plan, he's going to have to spend, his total cost, he's gonna spend 8,000 no matter what."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they say, including the cost of insurance, what are Paul's expected total medical costs with the low deductible plan? Round your answer to the nearest cent. So actually, I'll get the calculator out for this. So at the low, whoops, with the low deductible plan here, low deductible plan, he's going to have to spend, his total cost, he's gonna spend 8,000 no matter what. Eight, whoops, what happened to my calculator? He's gonna spend 8,000, my God, I'm having issues. He's gonna spend 8,000 no matter what."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So at the low, whoops, with the low deductible plan here, low deductible plan, he's going to have to spend, his total cost, he's gonna spend 8,000 no matter what. Eight, whoops, what happened to my calculator? He's gonna spend 8,000, my God, I'm having issues. He's gonna spend 8,000 no matter what. So that is $8,000. And then let's see. There's a 30% probability he spends nothing."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "He's gonna spend 8,000 no matter what. So that is $8,000. And then let's see. There's a 30% probability he spends nothing. I could just write that as plus.3 times zero, and I will write it just so that you see I'm taking that into account. Then there's a 25% chance that he has $1,000 in medical costs. And in the low deductible plan, he still has to pay that $1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a 30% probability he spends nothing. I could just write that as plus.3 times zero, and I will write it just so that you see I'm taking that into account. Then there's a 25% chance that he has $1,000 in medical costs. And in the low deductible plan, he still has to pay that $1,000. So plus.25% chance that he pays $1,000. $1,000. And then you might say, okay, plus.2 times 4,000, but remember, if his medical costs are 4,000, he's not going to pay that 4,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in the low deductible plan, he still has to pay that $1,000. So plus.25% chance that he pays $1,000. $1,000. And then you might say, okay, plus.2 times 4,000, but remember, if his medical costs are 4,000, he's not going to pay that 4,000. He's only going to pay the first 1,000. So it's really plus.2. In this situation, his out-of-pocket costs are only $1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you might say, okay, plus.2 times 4,000, but remember, if his medical costs are 4,000, he's not going to pay that 4,000. He's only going to pay the first 1,000. So it's really plus.2. In this situation, his out-of-pocket costs are only $1,000. The insurance company will pay the next 3,000. So.2 times 1,000. And then plus.2, 20% chance, even if he has 7,000 in medical costs, he's only going to have to pay the first 1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "In this situation, his out-of-pocket costs are only $1,000. The insurance company will pay the next 3,000. So.2 times 1,000. And then plus.2, 20% chance, even if he has 7,000 in medical costs, he's only going to have to pay the first 1,000. So.2 times 1,000 again. And then plus.05 times, once again, 1,000. If he has 15,000 in expenses, he's only going to have to pay the first 1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then plus.2, 20% chance, even if he has 7,000 in medical costs, he's only going to have to pay the first 1,000. So.2 times 1,000 again. And then plus.05 times, once again, 1,000. If he has 15,000 in expenses, he's only going to have to pay the first 1,000. Times 1,000. And we get $8,700. And one way you could have thought about it is, okay, he's going to have to pay $8,000 no matter what."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "If he has 15,000 in expenses, he's only going to have to pay the first 1,000. Times 1,000. And we get $8,700. And one way you could have thought about it is, okay, he's going to have to pay $8,000 no matter what. And all of the situations where he ends up paying, that's these four situations right over here, there's a 70% probability of falling into one of these four situations. And in any one of these, he only has to pay $1,000. The insurance company pays everything after that."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "And one way you could have thought about it is, okay, he's going to have to pay $8,000 no matter what. And all of the situations where he ends up paying, that's these four situations right over here, there's a 70% probability of falling into one of these four situations. And in any one of these, he only has to pay $1,000. The insurance company pays everything after that. So you could say 8,000 plus, there's a 70% chance that he's going to pay 1,000. And once again, this table is a pretty big simplification from the real world. There's probably a lot of scenarios where you'd have to pay $500 or $600 or whatever it might be."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "The insurance company pays everything after that. So you could say 8,000 plus, there's a 70% chance that he's going to pay 1,000. And once again, this table is a pretty big simplification from the real world. There's probably a lot of scenarios where you'd have to pay $500 or $600 or whatever it might be. But let's just go with this. And so that's essentially a simplification. There's a 70% chance that he's going to have to pay $1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's probably a lot of scenarios where you'd have to pay $500 or $600 or whatever it might be. But let's just go with this. And so that's essentially a simplification. There's a 70% chance that he's going to have to pay $1,000. And so that's $700 expected cost from that plus the 8,000 from the insurance gets us to $8,700. So let's write that down. So 8,700, my pen is really acting up."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a 70% chance that he's going to have to pay $1,000. And so that's $700 expected cost from that plus the 8,000 from the insurance gets us to $8,700. So let's write that down. So 8,700, my pen is really acting up. I don't know what's going on here. I think I have to get a new tablet. You can't even read that."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 8,700, my pen is really acting up. I don't know what's going on here. I think I have to get a new tablet. You can't even read that. Let me write this. 87, whoops. $8,700, including the cost of insurance, what are Paul's expected total medical costs with the high deductible plan?"}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can't even read that. Let me write this. 87, whoops. $8,700, including the cost of insurance, what are Paul's expected total medical costs with the high deductible plan? Round your answer to the nearest cent. So let's look at the high deductible plan. So he's going to pay 7,500 no matter what."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "$8,700, including the cost of insurance, what are Paul's expected total medical costs with the high deductible plan? Round your answer to the nearest cent. So let's look at the high deductible plan. So he's going to pay 7,500 no matter what. 7,500 no matter what. And then, so there's a zero, we could write $0 times 30% or 30% times $0, but that's just going to be zero. There's a 25% chance he spends $1,000."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So he's going to pay 7,500 no matter what. 7,500 no matter what. And then, so there's a zero, we could write $0 times 30% or 30% times $0, but that's just going to be zero. There's a 25% chance he spends $1,000. So plus.25 times 1,000. There's a 20% chance plus.2. He's not going to spend 4,000 here."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a 25% chance he spends $1,000. So plus.25 times 1,000. There's a 20% chance plus.2. He's not going to spend 4,000 here. He's going to have to spend the first 2,500. So it's a 20% chance he spends 2,500. So times 2,500, insurance company will pay the next 1,500, plus another 20% chance."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "He's not going to spend 4,000 here. He's going to have to spend the first 2,500. So it's a 20% chance he spends 2,500. So times 2,500, insurance company will pay the next 1,500, plus another 20% chance. Even in this situation, he only has to pay 2,500. So times 2,500. And then finally, plus, there's a 5% chance."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So times 2,500, insurance company will pay the next 1,500, plus another 20% chance. Even in this situation, he only has to pay 2,500. So times 2,500. And then finally, plus, there's a 5% chance. Even in this situation, he only has to pay the first 2,500. Times 2,500 gets us to $8,875. $8,875."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally, plus, there's a 5% chance. Even in this situation, he only has to pay the first 2,500. Times 2,500 gets us to $8,875. $8,875. $8,875. And once again, you could think about it as, okay, there's a 25% chance that he pays 1,000, and then there is a 45% chance that he pays $2,500. All of these situations, he's paying $2,500."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "$8,875. $8,875. And once again, you could think about it as, okay, there's a 25% chance that he pays 1,000, and then there is a 45% chance that he pays $2,500. All of these situations, he's paying $2,500. But either way, you would get $8,875. If Paul wants the best payoff in the long run and must buy one of the two insurance plans, he should purchase the, well, his expected total cost of medical, cost of insurance, including medical costs, is lower with the low deductible plan. So this one, he should go with the low, low deductible."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "All of these situations, he's paying $2,500. But either way, you would get $8,875. If Paul wants the best payoff in the long run and must buy one of the two insurance plans, he should purchase the, well, his expected total cost of medical, cost of insurance, including medical costs, is lower with the low deductible plan. So this one, he should go with the low, low deductible. Low deductible. Which, once again, you shouldn't use these videos as kind of insurance advice. This is actually, but also, it's an interesting way to think about it."}, {"video_title": "Comparing insurance with expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this one, he should go with the low, low deductible. Low deductible. Which, once again, you shouldn't use these videos as kind of insurance advice. This is actually, but also, it's an interesting way to think about it. It's typically, well, it's not always typically the case that the low deductible plan is going to have a higher long-term, or the low deductible plan is going to be a better deal. It's usually the, well, I won't make any actuarial statements but at least in this situation, the low deductible plan seems like the better deal. It's lower expected total cost given these probabilities."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So over this axis right over here, the horizontal axis, they have month by month, and we move forward in time, July, August, September, October. And in this axis, the vertical axis, we have the price. So for example, in July, the price of this stock was a little over $10. Then in August, it moved up to, it looks like, around $11. And then we could keep going month by month. And this type of graph right over here is called a line graph because you have the data points for each month. And then we connected them with a line."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Then in August, it moved up to, it looks like, around $11. And then we could keep going month by month. And this type of graph right over here is called a line graph because you have the data points for each month. And then we connected them with a line. And the reason why we connect them with a line is to really see if there's some kind of a trend here, to really show that you have something that's moving from one price to another. And so line graphs tend to be used when you have something that's changing over time. Now with that out of the way, let's actually answer their question."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And then we connected them with a line. And the reason why we connect them with a line is to really see if there's some kind of a trend here, to really show that you have something that's moving from one price to another. And so line graphs tend to be used when you have something that's changing over time. Now with that out of the way, let's actually answer their question. Over the course of the year, is the price of the stock rising, falling, or staying the same? So on a month-to-month basis, you have, for example, from July to August, the price went up. Then from August to September, the price went down."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Now with that out of the way, let's actually answer their question. Over the course of the year, is the price of the stock rising, falling, or staying the same? So on a month-to-month basis, you have, for example, from July to August, the price went up. Then from August to September, the price went down. Then it went up for two months. Then it went down for a month. Then it went up for a couple of more months."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Then from August to September, the price went down. Then it went up for two months. Then it went down for a month. Then it went up for a couple of more months. Then it went really up from February to March, went all the way up to almost $17. Then it went down again. And then it kept going up again."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Then it went up for a couple of more months. Then it went really up from February to March, went all the way up to almost $17. Then it went down again. And then it kept going up again. But they're asking us not, you know, did it go up every month? They're saying over the course of the year. Over the course of the year, is the price of the stock rising, falling, or staying the same?"}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And then it kept going up again. But they're asking us not, you know, did it go up every month? They're saying over the course of the year. Over the course of the year, is the price of the stock rising, falling, or staying the same? And if you go from July, which is where our data starts right over here, our price was around $10. And even though there were a few months where it went down, the overall trend is that the price is going up. The overall trend is that the price is going up."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Over the course of the year, is the price of the stock rising, falling, or staying the same? And if you go from July, which is where our data starts right over here, our price was around $10. And even though there were a few months where it went down, the overall trend is that the price is going up. The overall trend is that the price is going up. And you can even see that. In July, it was $10. And then by June of the next year, it was approaching, I don't know, it looks like it's about almost a little over $16, maybe almost $17."}, {"video_title": "Reading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "The overall trend is that the price is going up. And you can even see that. In July, it was $10. And then by June of the next year, it was approaching, I don't know, it looks like it's about almost a little over $16, maybe almost $17. So it actually had gone up a lot. They don't give us July of the next year, but the overall trend is definitely the upwards direction right over here. And you can see it just visually by looking at this line graph."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And statistics is really a broad category of things that you might do with data. So it generally deals with data, collecting data. So actually let me write these down. It's involving collecting data. Collecting data. You could present data in tables or charts or just as lists of numbers or however you might do it. It is analyzing the data."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's involving collecting data. Collecting data. You could present data in tables or charts or just as lists of numbers or however you might do it. It is analyzing the data. Analyzing, analyzing, presenting and analyzing data. So this whole class of just, you know, all the stuff that you might do with data to answer a question or try to figure out what's going on or just to learn about the world, that whole class of things is called statistics. Now an idea that will come up very frequently in statistics is the notion of variability."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It is analyzing the data. Analyzing, analyzing, presenting and analyzing data. So this whole class of just, you know, all the stuff that you might do with data to answer a question or try to figure out what's going on or just to learn about the world, that whole class of things is called statistics. Now an idea that will come up very frequently in statistics is the notion of variability. Variability. In everyday language, variability, it's how much something is, how much does it vary? How much does it change?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Now an idea that will come up very frequently in statistics is the notion of variability. Variability. In everyday language, variability, it's how much something is, how much does it vary? How much does it change? And it's the same notion in statistics. In statistics, variability is the degree to which data points are different from each other, the degree to which they vary. And just as an example of that, to just make it a little bit more concrete, let's say you were to go to five people and you were to ask them, how many bricks did you eat yesterday?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How much does it change? And it's the same notion in statistics. In statistics, variability is the degree to which data points are different from each other, the degree to which they vary. And just as an example of that, to just make it a little bit more concrete, let's say you were to go to five people and you were to ask them, how many bricks did you eat yesterday? And each of the people say, well, you know, a person once said, I don't eat bricks at all. I don't even know how to do that. I ate zero bricks."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And just as an example of that, to just make it a little bit more concrete, let's say you were to go to five people and you were to ask them, how many bricks did you eat yesterday? And each of the people say, well, you know, a person once said, I don't eat bricks at all. I don't even know how to do that. I ate zero bricks. And then the next person says zero, the next person says zero, fourth person says zero, and the fifth person says zero. Fair enough, so that was our data point on the different data points on, and I'm already doing statistics just by going out there and asking them how many bricks they ate. Then I ask them, how many grapes did you eat yesterday?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I ate zero bricks. And then the next person says zero, the next person says zero, fourth person says zero, and the fifth person says zero. Fair enough, so that was our data point on the different data points on, and I'm already doing statistics just by going out there and asking them how many bricks they ate. Then I ask them, how many grapes did you eat yesterday? And the first person says, I ate zero grapes, but the next person says, I survive on grapes. I ate 235 grapes. The next person says, yeah, I like grapes."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Then I ask them, how many grapes did you eat yesterday? And the first person says, I ate zero grapes, but the next person says, I survive on grapes. I ate 235 grapes. The next person says, yeah, I like grapes. I ate 17 grapes. Then the person after that says that they ate five grapes. And then the next person also survives on grapes, even to a larger degree."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The next person says, yeah, I like grapes. I ate 17 grapes. Then the person after that says that they ate five grapes. And then the next person also survives on grapes, even to a larger degree. They ate 318 grapes. So if you look at these two data sets, one is the number of bricks someone ate yesterday, the other one is how many grapes they ate yesterday, you immediately see that there's more variability here. All of these data points are zero, while these, they change a good bit from data point to data point."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then the next person also survives on grapes, even to a larger degree. They ate 318 grapes. So if you look at these two data sets, one is the number of bricks someone ate yesterday, the other one is how many grapes they ate yesterday, you immediately see that there's more variability here. All of these data points are zero, while these, they change a good bit from data point to data point. So we have a sense that there is more variability in this data set. Now one of the things we will start doing a lot in statistics is trying to measure how much more, how much variability is, how can we quantify that? How can we put a number on it?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "All of these data points are zero, while these, they change a good bit from data point to data point. So we have a sense that there is more variability in this data set. Now one of the things we will start doing a lot in statistics is trying to measure how much more, how much variability is, how can we quantify that? How can we put a number on it? How can we measure variability? This is a big aspect of statistics, but we won't do that in this video. There are future videos for doing that."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How can we put a number on it? How can we measure variability? This is a big aspect of statistics, but we won't do that in this video. There are future videos for doing that. But just as we go into the world of statistics, we should think about, well, when do we even, when should our brain even start getting into statistics mode? Thinking about the tools that we have at our disposal, about collecting data and measuring variability and measuring and finding numbers that somehow represent a data, a pool of data that has variability. And so the question we should ask ourselves is, what questions in the world are statistical questions?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There are future videos for doing that. But just as we go into the world of statistics, we should think about, well, when do we even, when should our brain even start getting into statistics mode? Thinking about the tools that we have at our disposal, about collecting data and measuring variability and measuring and finding numbers that somehow represent a data, a pool of data that has variability. And so the question we should ask ourselves is, what questions in the world are statistical questions? So statistical, statistical questions. So let's come up with a definition for statistical question, the type of question where we would want to start bringing out our statistical, our statistical toolkit. Well, one possible way to think about when you need to bring out your statistical toolkit is these are questions that to answer them, to answer, you need to collect data with variability."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And so the question we should ask ourselves is, what questions in the world are statistical questions? So statistical, statistical questions. So let's come up with a definition for statistical question, the type of question where we would want to start bringing out our statistical, our statistical toolkit. Well, one possible way to think about when you need to bring out your statistical toolkit is these are questions that to answer them, to answer, you need to collect data with variability. To answer, you need to collect data with variability. And apologize for my handwriting. Data with variability."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, one possible way to think about when you need to bring out your statistical toolkit is these are questions that to answer them, to answer, you need to collect data with variability. To answer, you need to collect data with variability. And apologize for my handwriting. Data with variability. That's W-I-T-H, data with variability. Variability. So you're saying, okay, that kind of makes sense, but I need to see some tangible questions, or tangible examples of things that are statistical questions and things that are not statistical questions."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Data with variability. That's W-I-T-H, data with variability. Variability. So you're saying, okay, that kind of makes sense, but I need to see some tangible questions, or tangible examples of things that are statistical questions and things that are not statistical questions. And I would say, fair enough, let's look at some examples. So here I have six questions, and I encourage you to pause this video right now, and before I work through it, think about it. Based on this definition of a statistical question, which of these questions are statistical, would require your kind of statistical toolkit, and which of these are not statistical?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So you're saying, okay, that kind of makes sense, but I need to see some tangible questions, or tangible examples of things that are statistical questions and things that are not statistical questions. And I would say, fair enough, let's look at some examples. So here I have six questions, and I encourage you to pause this video right now, and before I work through it, think about it. Based on this definition of a statistical question, which of these questions are statistical, would require your kind of statistical toolkit, and which of these are not statistical? So I'm assuming you've had a go at it. Let's go through these one by one. So the first question, how much does my pet grapefruit weigh?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Based on this definition of a statistical question, which of these questions are statistical, would require your kind of statistical toolkit, and which of these are not statistical? So I'm assuming you've had a go at it. Let's go through these one by one. So the first question, how much does my pet grapefruit weigh? Now, it's bizarre to begin with to have a pet grapefruit, but is this a statistical question? Well, what do I need to do to answer it? I have to take my pet grapefruit out, I have to weigh it, and then I have to just write that down."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So the first question, how much does my pet grapefruit weigh? Now, it's bizarre to begin with to have a pet grapefruit, but is this a statistical question? Well, what do I need to do to answer it? I have to take my pet grapefruit out, I have to weigh it, and then I have to just write that down. And just doing that, I am collecting data, so you could argue that maybe I'm kinda starting to mess with statistics a little bit, but I'm just getting one data point. So I might weigh it, and I might see my grapefruit weighs one pound, but that's not data with variability, that's just one data point. In order to have variability, I have to have multiple data points, and it should be at least possible that they could vary."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I have to take my pet grapefruit out, I have to weigh it, and then I have to just write that down. And just doing that, I am collecting data, so you could argue that maybe I'm kinda starting to mess with statistics a little bit, but I'm just getting one data point. So I might weigh it, and I might see my grapefruit weighs one pound, but that's not data with variability, that's just one data point. In order to have variability, I have to have multiple data points, and it should be at least possible that they could vary. So for example, all of these folks ate zero bricks, and maybe it was possible that someone actually ate a brick. But here I just have one data point. With one data point, you can't have variability."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "In order to have variability, I have to have multiple data points, and it should be at least possible that they could vary. So for example, all of these folks ate zero bricks, and maybe it was possible that someone actually ate a brick. But here I just have one data point. With one data point, you can't have variability. So this is not a statistical question. I just collect a data point. Next question, what is the average number of cars in a parking lot on Monday mornings?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "With one data point, you can't have variability. So this is not a statistical question. I just collect a data point. Next question, what is the average number of cars in a parking lot on Monday mornings? Well, to think about whether it is a statistical question, we just have to think about what do I have to do to answer that question? Well, I would have to go out to the parking lot on multiple Monday mornings and measure the number of cars. So on the first Monday morning, I might see there are 50 cars."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Next question, what is the average number of cars in a parking lot on Monday mornings? Well, to think about whether it is a statistical question, we just have to think about what do I have to do to answer that question? Well, I would have to go out to the parking lot on multiple Monday mornings and measure the number of cars. So on the first Monday morning, I might see there are 50 cars. The next Monday morning, I might go out there and count, there's 49 cars. The next Monday morning, I might see 50 cars again. The next Monday morning, I might see 63 cars."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So on the first Monday morning, I might see there are 50 cars. The next Monday morning, I might go out there and count, there's 49 cars. The next Monday morning, I might see 50 cars again. The next Monday morning, I might see 63 cars. So I'm collecting multiple data points to answer this question. Then I'm gonna take the average of all of these, but I'm collecting multiple data points to answer this question. And it's definitely possible that there could be variation here, that there could be variability."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "The next Monday morning, I might see 63 cars. So I'm collecting multiple data points to answer this question. Then I'm gonna take the average of all of these, but I'm collecting multiple data points to answer this question. And it's definitely possible that there could be variation here, that there could be variability. So this is a statistical question. Next question, am I hungry? It's an important question."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And it's definitely possible that there could be variation here, that there could be variability. So this is a statistical question. Next question, am I hungry? It's an important question. We need to ask it to ourselves multiple times. In fact, sometimes our bodies just tell it to us. But I am definitely not collecting, I guess you could say I'm collecting some type of feelings from my stomach or how weak I feel or not."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "It's an important question. We need to ask it to ourselves multiple times. In fact, sometimes our bodies just tell it to us. But I am definitely not collecting, I guess you could say I'm collecting some type of feelings from my stomach or how weak I feel or not. But it's definitely not data with variability. I'm either hungry or not hungry on a given day. I mean, if you said broader, how does my hunger change from day to day, and you came up with some type of a scale for rating your hunger, okay, maybe that's more statistical."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But I am definitely not collecting, I guess you could say I'm collecting some type of feelings from my stomach or how weak I feel or not. But it's definitely not data with variability. I'm either hungry or not hungry on a given day. I mean, if you said broader, how does my hunger change from day to day, and you came up with some type of a scale for rating your hunger, okay, maybe that's more statistical. But just am I hungry, a yes, no question, this is not, to answer this, I do not have to collect data with variability. So this is not a statistical question. How many teeth does my mother have?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I mean, if you said broader, how does my hunger change from day to day, and you came up with some type of a scale for rating your hunger, okay, maybe that's more statistical. But just am I hungry, a yes, no question, this is not, to answer this, I do not have to collect data with variability. So this is not a statistical question. How many teeth does my mother have? To do this, I would have to go find my mother, and then I would have to ask her to open her mouth and count the teeth in her mouth. And maybe I get a number like 30. So it's kind of like how much does my pet grapefruit weigh?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many teeth does my mother have? To do this, I would have to go find my mother, and then I would have to ask her to open her mouth and count the teeth in her mouth. And maybe I get a number like 30. So it's kind of like how much does my pet grapefruit weigh? I do have to collect one data point, but one data point is not going to have variability. So I am not collecting data with variability. So this is not a statistical question."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it's kind of like how much does my pet grapefruit weigh? I do have to collect one data point, but one data point is not going to have variability. So I am not collecting data with variability. So this is not a statistical question. If I said how many teeth do all of the mothers that I know have on average, or what's the range of number of teeth that the mothers I know have, well, that would start getting, or that would be statistical. But this is just one data point, so not statistical. How much time do the members of my family spend eating per year?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So this is not a statistical question. If I said how many teeth do all of the mothers that I know have on average, or what's the range of number of teeth that the mothers I know have, well, that would start getting, or that would be statistical. But this is just one data point, so not statistical. How much time do the members of my family spend eating per year? Well, once again, what do I need to do to answer this question? Well, I would have to go either observe or survey my family members, maybe my mom, my wife, my children, and my uncles, aunts, whoever else. I would say, well, how much do you eat each day?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How much time do the members of my family spend eating per year? Well, once again, what do I need to do to answer this question? Well, I would have to go either observe or survey my family members, maybe my mom, my wife, my children, and my uncles, aunts, whoever else. I would say, well, how much do you eat each day? I would add them all up to figure out how much they eat in the year. And maybe family member A eats 813 hours in a year. Family member B ate, I don't know, 732 hours in the year."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I would say, well, how much do you eat each day? I would add them all up to figure out how much they eat in the year. And maybe family member A eats 813 hours in a year. Family member B ate, I don't know, 732 hours in the year. And so you see the general notion that I will be collecting multiple data points from the different family members, and they're very well, and in fact, there's very likely to be variation in that. In fact, I might even see variation from year to year. Person A is probably going to need a different number of hours in the next year."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Family member B ate, I don't know, 732 hours in the year. And so you see the general notion that I will be collecting multiple data points from the different family members, and they're very well, and in fact, there's very likely to be variation in that. In fact, I might even see variation from year to year. Person A is probably going to need a different number of hours in the next year. So I'm definitely going to collect data with variability in order to answer this question. So that is a statistical question. And then finally, I have the question, how many times have I watched Star Wars?"}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Person A is probably going to need a different number of hours in the next year. So I'm definitely going to collect data with variability in order to answer this question. So that is a statistical question. And then finally, I have the question, how many times have I watched Star Wars? Well, this is very similar to how many teeth does my mother have, or how much does my pet grapefruit weigh. I just have to count the number of times that I watched Star Wars, and maybe, you know, I watched it seven times, just one data point. No variability here."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then finally, I have the question, how many times have I watched Star Wars? Well, this is very similar to how many teeth does my mother have, or how much does my pet grapefruit weigh. I just have to count the number of times that I watched Star Wars, and maybe, you know, I watched it seven times, just one data point. No variability here. If I said, on average, how many times have my coworkers watched Star Wars? Well, then I'm going to have to collect data with variability. I'm gonna collect multiple data points, and it's definitely possible that my coworkers have watched it different numbers of times."}, {"video_title": "Statistical questions   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "No variability here. If I said, on average, how many times have my coworkers watched Star Wars? Well, then I'm going to have to collect data with variability. I'm gonna collect multiple data points, and it's definitely possible that my coworkers have watched it different numbers of times. But for this question in particular, where it's just one data point to answer it, how many times have I watched Star Wars? And my answer in this case, actually, I think is seven. Well, then not a statistical question."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "You can see there's seven equally likely possibilities. Let's see, there's one, two, three, four, five, six, seven equally likely possibilities. It looks like in four of them you've spun an elephant. Two of them you have this mouse running away from something. And then in one of them you have this monkey doing some type of acrobatics. Fair enough. Now let's pose ourselves an interesting question."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Two of them you have this mouse running away from something. And then in one of them you have this monkey doing some type of acrobatics. Fair enough. Now let's pose ourselves an interesting question. At least I think it's an interesting question. Let's say we were to spin this spinner 210 times. So we're gonna spin 210, we're gonna spin that spinner 210 times."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now let's pose ourselves an interesting question. At least I think it's an interesting question. Let's say we were to spin this spinner 210 times. So we're gonna spin 210, we're gonna spin that spinner 210 times. And I want you to make a prediction. I want you to predict, I want you to predict the number of times, the number of times we get an elephant. We get an elephant."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So we're gonna spin 210, we're gonna spin that spinner 210 times. And I want you to make a prediction. I want you to predict, I want you to predict the number of times, the number of times we get an elephant. We get an elephant. Number of times we get an elephant out of the 210 times. So why don't you have a go at it. Alright, so I'm assuming like always pause the video and had a try."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "We get an elephant. Number of times we get an elephant out of the 210 times. So why don't you have a go at it. Alright, so I'm assuming like always pause the video and had a try. So one way to think about it is, well for one spin what is the probability of getting an elephant? So let's do this. One spin, so for one spin what is the probability of getting an elephant?"}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Alright, so I'm assuming like always pause the video and had a try. So one way to think about it is, well for one spin what is the probability of getting an elephant? So let's do this. One spin, so for one spin what is the probability of getting an elephant? Well let's see, we have already talked about this is a fair spinner. There's seven equally likely possibilities. And then how many involve getting an elephant?"}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "One spin, so for one spin what is the probability of getting an elephant? Well let's see, we have already talked about this is a fair spinner. There's seven equally likely possibilities. And then how many involve getting an elephant? Well so we have one, two, three, four. Four out of the seven equally likely possibilities involve us getting an elephant. So one reasonable thing to do, and this is actually what I would do, is let's go look."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And then how many involve getting an elephant? Well so we have one, two, three, four. Four out of the seven equally likely possibilities involve us getting an elephant. So one reasonable thing to do, and this is actually what I would do, is let's go look. Four sevenths probability means I should expect that four sevenths of the time, especially if I'm doing a lot of these, if I'm doing it over and over and over again, it's a reasonable expectation that, hey, four sevenths of the time I will get an elephant. I've just calculated the theoretical probability here based on this being a fair spinner. And that should inform that if I were to do a bunch of experiments that four sevenths of the time that I should see me getting the elephant."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So one reasonable thing to do, and this is actually what I would do, is let's go look. Four sevenths probability means I should expect that four sevenths of the time, especially if I'm doing a lot of these, if I'm doing it over and over and over again, it's a reasonable expectation that, hey, four sevenths of the time I will get an elephant. I've just calculated the theoretical probability here based on this being a fair spinner. And that should inform that if I were to do a bunch of experiments that four sevenths of the time that I should see me getting the elephant. So it would be a reasonable prediction to say, look, I'm gonna spin this thing 210 times, and I would expect that four sevenths of those 210 times I would get an elephant. And so let's think about what this is. 210 times four sevenths, 210 divided by seven is 30, 30 times 40 is 120."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And that should inform that if I were to do a bunch of experiments that four sevenths of the time that I should see me getting the elephant. So it would be a reasonable prediction to say, look, I'm gonna spin this thing 210 times, and I would expect that four sevenths of those 210 times I would get an elephant. And so let's think about what this is. 210 times four sevenths, 210 divided by seven is 30, 30 times 40 is 120. So 120 times. My prediction, or maybe your prediction was this as well, I think it's a reasonable prediction, is that if I spin it 210 times, that I'm going to get an elephant 120 times. Now it's very important to think about what this is saying and this is not saying."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "210 times four sevenths, 210 divided by seven is 30, 30 times 40 is 120. So 120 times. My prediction, or maybe your prediction was this as well, I think it's a reasonable prediction, is that if I spin it 210 times, that I'm going to get an elephant 120 times. Now it's very important to think about what this is saying and this is not saying. Is it possible that I get an elephant 121 times, or maybe 119 times? Sure, sure, it's completely reasonable that you might get something different than this. In fact, there's some probability that you get no elephant."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now it's very important to think about what this is saying and this is not saying. Is it possible that I get an elephant 121 times, or maybe 119 times? Sure, sure, it's completely reasonable that you might get something different than this. In fact, there's some probability that you get no elephant. If you consider getting an elephant lucky, that you just happen to keep landing on the monkey or one of the mice. Now it's a very low probability that that would happen if you spun it 210 times, but it is possible. So it's important to realize that this is just a pretty, there's actually a possibility that you might get an elephant on all 210 spins."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "In fact, there's some probability that you get no elephant. If you consider getting an elephant lucky, that you just happen to keep landing on the monkey or one of the mice. Now it's a very low probability that that would happen if you spun it 210 times, but it is possible. So it's important to realize that this is just a pretty, there's actually a possibility that you might get an elephant on all 210 spins. Once again, that's a low probability, but it is possible. So this isn't saying that you're definitely going to get the elephant 120 times. In fact, it's very reasonable that you might get the elephant 123 times, or 128 times, or 110 times, or even 90 times."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So it's important to realize that this is just a pretty, there's actually a possibility that you might get an elephant on all 210 spins. Once again, that's a low probability, but it is possible. So this isn't saying that you're definitely going to get the elephant 120 times. In fact, it's very reasonable that you might get the elephant 123 times, or 128 times, or 110 times, or even 90 times. These are all completely reasonable things to happen. All you would say is that, look, if I had to predict it, this, out of all of the different, I can get the elephant anywhere between zero and 210 times. Out of all of those possibilities, before I even start spinning, I'll say, okay, I think that this is the most reasonable one, that I'm going to get it 4 7ths of the time."}, {"video_title": "Making predictions with probability   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "In fact, it's very reasonable that you might get the elephant 123 times, or 128 times, or 110 times, or even 90 times. These are all completely reasonable things to happen. All you would say is that, look, if I had to predict it, this, out of all of the different, I can get the elephant anywhere between zero and 210 times. Out of all of those possibilities, before I even start spinning, I'll say, okay, I think that this is the most reasonable one, that I'm going to get it 4 7ths of the time. But it's not saying that, hey, yeah, it's not saying that I'm definitely going to get it 120 times. It's not saying that 118 times, or 129 times aren't reasonably possible as well. It's just saying, look, this is a reasonable prediction."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "What I have here is the list of ages of the students in a class. And what I want to explore in this video is different ways of representing this data and then see if we can answer questions about the data. So the first way we can think about it is as a frequency table. Frequency table. Frequency table. And what we're going to do is we're going to look at each, for each age, for each possible age that we've measured here to see how many students in the class are of that age. So we could say the age is one column."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Frequency table. Frequency table. And what we're going to do is we're going to look at each, for each age, for each possible age that we've measured here to see how many students in the class are of that age. So we could say the age is one column. And then the number. The number of students of that age. Or we could even say the frequency."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So we could say the age is one column. And then the number. The number of students of that age. Or we could even say the frequency. Frequency. When people say, how frequent do you do something, they're saying, how often does it happen? How often do you do that thing?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Or we could even say the frequency. Frequency. When people say, how frequent do you do something, they're saying, how often does it happen? How often do you do that thing? Frequency. Or we could also say, actually, let me just write number. I'm always a fan of the simpler number at age, which we could also consider the frequency at that age."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How often do you do that thing? Frequency. Or we could also say, actually, let me just write number. I'm always a fan of the simpler number at age, which we could also consider the frequency at that age. Frequency of students. All right. So what's the lowest age that we have here?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "I'm always a fan of the simpler number at age, which we could also consider the frequency at that age. Frequency of students. All right. So what's the lowest age that we have here? Well, the lowest age is five. So I'll start with five. And how many students in the class are age five?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So what's the lowest age that we have here? Well, the lowest age is five. So I'll start with five. And how many students in the class are age five? How frequent is the number five? Let's see. There is one, two."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And how many students in the class are age five? How frequent is the number five? Let's see. There is one, two. Let me keep scanning. Looks like there's only two fives. So I could write a two here."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There is one, two. Let me keep scanning. Looks like there's only two fives. So I could write a two here. There are two fives. And now let's go to six. How many sixes are there?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So I could write a two here. There are two fives. And now let's go to six. How many sixes are there? Let's see. There is one sixth. There's only one six-year-old in the class."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many sixes are there? Let's see. There is one sixth. There's only one six-year-old in the class. All right. Seven-year-olds. See, there's one, two, three, four seven-year-olds."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There's only one six-year-old in the class. All right. Seven-year-olds. See, there's one, two, three, four seven-year-olds. Now what about eight-year-olds? I'm going to do this in a color that I have not used yet. Eight-year-olds, we have no eight-year-olds."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "See, there's one, two, three, four seven-year-olds. Now what about eight-year-olds? I'm going to do this in a color that I have not used yet. Eight-year-olds, we have no eight-year-olds. Zero eight-year-olds. And then we have nine-year-olds. Let's see."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Eight-year-olds, we have no eight-year-olds. Zero eight-year-olds. And then we have nine-year-olds. Let's see. Nine-year-olds, we have one, two, three, four nine-year-olds. 10-year-olds, what do we have? We have one 10-year-old right over there."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Let's see. Nine-year-olds, we have one, two, three, four nine-year-olds. 10-year-olds, what do we have? We have one 10-year-old right over there. And then 11-year-olds. There are no 11-year-olds. And then let me scroll up a little bit."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have one 10-year-old right over there. And then 11-year-olds. There are no 11-year-olds. And then let me scroll up a little bit. And then finally 12-year-olds. 12-year-olds, there are one, two 12-year-olds. So what we have just constructed is a frequency table."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then let me scroll up a little bit. And then finally 12-year-olds. 12-year-olds, there are one, two 12-year-olds. So what we have just constructed is a frequency table. It's a frequency table. You can see for each age how many students are at that age. So it's giving you the same information as we have up here."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So what we have just constructed is a frequency table. It's a frequency table. You can see for each age how many students are at that age. So it's giving you the same information as we have up here. You could take this table and construct what we have up here. You would just write down two fives, one sixth, four sevens, no eights, four nines, one 10, no 11s, and two 12s. And then you would just have this list of numbers."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So it's giving you the same information as we have up here. You could take this table and construct what we have up here. You would just write down two fives, one sixth, four sevens, no eights, four nines, one 10, no 11s, and two 12s. And then you would just have this list of numbers. Now a way to visually look at a frequency table is a dot plot. So let me draw a dot plot right over here. So a dot plot."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And then you would just have this list of numbers. Now a way to visually look at a frequency table is a dot plot. So let me draw a dot plot right over here. So a dot plot. And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually. So in a dot plot, what we would have of, actually let me just not draw an even arrow there. We have the different age groups."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So a dot plot. And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually. So in a dot plot, what we would have of, actually let me just not draw an even arrow there. We have the different age groups. So five, six, seven, eight, nine, 10, 11, and 12. And we have a dot to represent, or we use a dot for each student at that age. So there's two five-year-olds, so I'll do two dots."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "We have the different age groups. So five, six, seven, eight, nine, 10, 11, and 12. And we have a dot to represent, or we use a dot for each student at that age. So there's two five-year-olds, so I'll do two dots. One and two. There's one six-year-old, so that's going to be one dot right over here. There's four seven-year-olds, so one, two, three, four dots."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So there's two five-year-olds, so I'll do two dots. One and two. There's one six-year-old, so that's going to be one dot right over here. There's four seven-year-olds, so one, two, three, four dots. There's no eight-year-olds. There's four nine-year-olds, so one, two, three, and four. There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There's four seven-year-olds, so one, two, three, four dots. There's no eight-year-olds. There's four nine-year-olds, so one, two, three, and four. There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old. There's no 11-year-olds, so I'm not going to put any dots there. And then there's two 12-year-olds. So one 12-year-old and another 12-year-old."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old. There's no 11-year-olds, so I'm not going to put any dots there. And then there's two 12-year-olds. So one 12-year-old and another 12-year-old. So there you go. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So one 12-year-old and another 12-year-old. So there you go. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways. And once you have it represented in any of these ways, we can start to ask questions about it. So we could say, what is the most frequent age? Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "These are all showing the same data, just in different ways. And once you have it represented in any of these ways, we can start to ask questions about it. So we could say, what is the most frequent age? Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks. So there's actually seven and nine are tied for the most frequent age. You would have also seen it here, where seven and nine are tied at four. And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks. So there's actually seven and nine are tied for the most frequent age. You would have also seen it here, where seven and nine are tied at four. And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines. That's the largest number. So if you're looking for what's the most frequent age, when you just visually inspect here, it probably pops out at you the fastest. But there's other questions we can ask ourselves."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines. That's the largest number. So if you're looking for what's the most frequent age, when you just visually inspect here, it probably pops out at you the fastest. But there's other questions we can ask ourselves. We can ask ourselves, what is the range of ages in the classroom? And this is, once again, where maybe the dot plot is the most, it jumps out at you the most, because the range is just the maximum age in your, or the maximum data point minus the minimum data point. So what's the maximum age here?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But there's other questions we can ask ourselves. We can ask ourselves, what is the range of ages in the classroom? And this is, once again, where maybe the dot plot is the most, it jumps out at you the most, because the range is just the maximum age in your, or the maximum data point minus the minimum data point. So what's the maximum age here? Well, the maximum age here, we see it from the dot plot, is 12. And the minimum age here, you see, is five. So there's a range of seven."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So what's the maximum age here? Well, the maximum age here, we see it from the dot plot, is 12. And the minimum age here, you see, is five. So there's a range of seven. The difference between the maximum and the minimum is seven. But you could have also done that over here. You could say, hey, the maximum age here is 12."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "So there's a range of seven. The difference between the maximum and the minimum is seven. But you could have also done that over here. You could say, hey, the maximum age here is 12. Minimum age here is five. And so you find the difference between 12 and five, which is seven. Here, you still could have done it."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "You could say, hey, the maximum age here is 12. Minimum age here is five. And so you find the difference between 12 and five, which is seven. Here, you still could have done it. You could say, OK, what's the lowest? Let's look at five. Are there any fours here?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Here, you still could have done it. You could say, OK, what's the lowest? Let's look at five. Are there any fours here? Nope, there's no fours. So five's the minimum age. And what's the largest?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Are there any fours here? Nope, there's no fours. So five's the minimum age. And what's the largest? Is it seven? No, is it nine? Nine, not even 10?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And what's the largest? Is it seven? No, is it nine? Nine, not even 10? Oh, 12. 12. Are there any 13s?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Nine, not even 10? Oh, 12. 12. Are there any 13s? No, 12 is the maximum. So you say 12 minus five is seven to get the range. But then we could ask ourselves other questions."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "Are there any 13s? No, 12 is the maximum. So you say 12 minus five is seven to get the range. But then we could ask ourselves other questions. We could say, how many older than nine is a question we could ask ourselves. And then if we were to look at the dot plot, we say, OK, this is nine. And we care about how many are older than nine."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "But then we could ask ourselves other questions. We could say, how many older than nine is a question we could ask ourselves. And then if we were to look at the dot plot, we say, OK, this is nine. And we care about how many are older than nine. So that would be this one, two, and three. Or you could look over here. How many are older than nine?"}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "And we care about how many are older than nine. So that would be this one, two, and three. Or you could look over here. How many are older than nine? Well, it's the one person who's 10, and then the two who are 12. So there are three. And over here, if you said how many are older than nine, well, then you would just have to go through the list and say, OK, no, no, no, no, no, no, no, no."}, {"video_title": "Frequency tables and dot plots   Data and statistics   6th grade   Khan Academy.mp3", "Sentence": "How many are older than nine? Well, it's the one person who's 10, and then the two who are 12. So there are three. And over here, if you said how many are older than nine, well, then you would just have to go through the list and say, OK, no, no, no, no, no, no, no, no. OK, here, one, two, three. And then not that person right over there. So hopefully, this is just an appreciation for yet another two ways of looking at data, frequency tables and dot plots."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll circle the statistical questions in yellow. And I encourage you to pause this video and try to figure this out yourself first. Look at each of these questions and think about whether you think you need statistics to answer this question or you don't need statistics, whether these are statistical questions or not. So I'm assuming you've given a pass at it, unless we can go through this together. So this first question is, how old are you? So we're talking about how old is a particular person. There is an answer here."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'm assuming you've given a pass at it, unless we can go through this together. So this first question is, how old are you? So we're talking about how old is a particular person. There is an answer here. We don't need any tools of statistics to answer this. So this is not a statistical question. How old are the people who have watched this video in 2013?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "There is an answer here. We don't need any tools of statistics to answer this. So this is not a statistical question. How old are the people who have watched this video in 2013? Now, this is interesting. We're assuming that multiple people will have watched this video in 2013. And they're not all going to be the same age."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "How old are the people who have watched this video in 2013? Now, this is interesting. We're assuming that multiple people will have watched this video in 2013. And they're not all going to be the same age. There's going to be some variability in their age. So one person might be 10 years old. Another person might be 20."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And they're not all going to be the same age. There's going to be some variability in their age. So one person might be 10 years old. Another person might be 20. Another person might be 15. So what answer do you give here? Would you give all of the ages?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Another person might be 20. Another person might be 15. So what answer do you give here? Would you give all of the ages? But we want to get a sense of, in general, how old are the people? So this is where statistics might be valuable. We might want to find some type of central tendency, an average, a median age for this."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Would you give all of the ages? But we want to get a sense of, in general, how old are the people? So this is where statistics might be valuable. We might want to find some type of central tendency, an average, a median age for this. So this is absolutely a statistical question. And you might already be seeing kind of a pattern here. The first question, we were asking about a particular person."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "We might want to find some type of central tendency, an average, a median age for this. So this is absolutely a statistical question. And you might already be seeing kind of a pattern here. The first question, we were asking about a particular person. There was only one answer here. There's no variability in the answer. The second one, we're asking about a trait of a bunch of people."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The first question, we were asking about a particular person. There was only one answer here. There's no variability in the answer. The second one, we're asking about a trait of a bunch of people. And there's variability in that trait. They're not all the same age. And so we'll need statistics to come up with some features of the data set to be able to make some conclusions."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The second one, we're asking about a trait of a bunch of people. And there's variability in that trait. They're not all the same age. And so we'll need statistics to come up with some features of the data set to be able to make some conclusions. We might say, on average, the people who have watched this video in 2013 are 18 years old, or 22 years old, or the median is 24 years old, whatever it might be. Do dogs run faster than cats? So once again, there are many dogs and many cats."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so we'll need statistics to come up with some features of the data set to be able to make some conclusions. We might say, on average, the people who have watched this video in 2013 are 18 years old, or 22 years old, or the median is 24 years old, whatever it might be. Do dogs run faster than cats? So once again, there are many dogs and many cats. And they all run at different speeds. Some dogs run faster than some cats, and some cats run faster than some dogs. So we would need some statistics to get a sense of, in general, or on average, how fast do dogs run?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once again, there are many dogs and many cats. And they all run at different speeds. Some dogs run faster than some cats, and some cats run faster than some dogs. So we would need some statistics to get a sense of, in general, or on average, how fast do dogs run? And then maybe, on average, how fast do cats run? And then we could compare those averages, or we could compare the medians in some way. So this is definitely a statistical question."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we would need some statistics to get a sense of, in general, or on average, how fast do dogs run? And then maybe, on average, how fast do cats run? And then we could compare those averages, or we could compare the medians in some way. So this is definitely a statistical question. Once again, we're talking about, in general, a whole population of dogs, the whole species of dogs, versus cats. And there's variation in how fast dogs run and how fast cats run. If we were talking about a particular dog and a particular cat, well, then there would just be an answer."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is definitely a statistical question. Once again, we're talking about, in general, a whole population of dogs, the whole species of dogs, versus cats. And there's variation in how fast dogs run and how fast cats run. If we were talking about a particular dog and a particular cat, well, then there would just be an answer. Does dog A run faster than cat B? Well, sure. That's not going to be a statistical question."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we were talking about a particular dog and a particular cat, well, then there would just be an answer. Does dog A run faster than cat B? Well, sure. That's not going to be a statistical question. You don't have to use the tools of statistics. And this next question actually fits that pattern. Do wolves weigh?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's not going to be a statistical question. You don't have to use the tools of statistics. And this next question actually fits that pattern. Do wolves weigh? Actually, no. This fits the pattern of the previous one. Do wolves weigh more than dogs?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Do wolves weigh? Actually, no. This fits the pattern of the previous one. Do wolves weigh more than dogs? So once again, there are some very light dogs and some very heavy wolves. So those wolves definitely weigh more than those dogs. But there are some very, very, very heavy dogs."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Do wolves weigh more than dogs? So once again, there are some very light dogs and some very heavy wolves. So those wolves definitely weigh more than those dogs. But there are some very, very, very heavy dogs. And so what you would want to do here, because we have variability in each of these, is you might want to come with some central tendency. On average, what's the median wolf weight? What's the average, the mean wolf weight?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But there are some very, very, very heavy dogs. And so what you would want to do here, because we have variability in each of these, is you might want to come with some central tendency. On average, what's the median wolf weight? What's the average, the mean wolf weight? Compare that to the mean dog's weight. So once again, since we're speaking in general about wolves, not a particular wolf, and in general about dogs, and there's variation in the data, and we're trying to glean some numbers from that to compare, this is definitely a statistical question. Definitely a statistical question."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's the average, the mean wolf weight? Compare that to the mean dog's weight. So once again, since we're speaking in general about wolves, not a particular wolf, and in general about dogs, and there's variation in the data, and we're trying to glean some numbers from that to compare, this is definitely a statistical question. Definitely a statistical question. Does your dog weigh more than that wolf? And we're assuming that we're pointing at a particular wolf. So now this is the particular."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Definitely a statistical question. Does your dog weigh more than that wolf? And we're assuming that we're pointing at a particular wolf. So now this is the particular. We're comparing a dog to a particular dog to a particular wolf. We can put each of them on a weighing machine and come up with an absolute answer. There's no variability in this dog's weight, at least at the moment that we weigh it."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So now this is the particular. We're comparing a dog to a particular dog to a particular wolf. We can put each of them on a weighing machine and come up with an absolute answer. There's no variability in this dog's weight, at least at the moment that we weigh it. No variability in this wolf's weight at the moment that we weigh it. So this is not a statistical question. So I'll put an x next to the ones that are not statistical questions."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's no variability in this dog's weight, at least at the moment that we weigh it. No variability in this wolf's weight at the moment that we weigh it. So this is not a statistical question. So I'll put an x next to the ones that are not statistical questions. Does it rain more in Seattle than Singapore? So once again, there's variation here. And we would also probably want to notice it rained more in Seattle than Singapore in a given year, over a decade, or whatever."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I'll put an x next to the ones that are not statistical questions. Does it rain more in Seattle than Singapore? So once again, there's variation here. And we would also probably want to notice it rained more in Seattle than Singapore in a given year, over a decade, or whatever. But regardless of those questions, however we ask it, in some years it might rain more in Seattle. In other years it might rain more in Singapore. Or if we just pick Seattle, it rains a different amount from year to year."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we would also probably want to notice it rained more in Seattle than Singapore in a given year, over a decade, or whatever. But regardless of those questions, however we ask it, in some years it might rain more in Seattle. In other years it might rain more in Singapore. Or if we just pick Seattle, it rains a different amount from year to year. In Singapore, it rains a different amount from year to year. So how do we compare? Well, that's where the statistics could be valuable."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or if we just pick Seattle, it rains a different amount from year to year. In Singapore, it rains a different amount from year to year. So how do we compare? Well, that's where the statistics could be valuable. There's variability in the data. So we can look at the data set for Seattle and come up with some type of an average, some type of a central tendency, and compare that to the average, the mean, the mode, whatever you want to. The mode probably wouldn't be that useful here."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that's where the statistics could be valuable. There's variability in the data. So we can look at the data set for Seattle and come up with some type of an average, some type of a central tendency, and compare that to the average, the mean, the mode, whatever you want to. The mode probably wouldn't be that useful here. To Singapore. So this is definitely a statistical question. Definitely statistical."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The mode probably wouldn't be that useful here. To Singapore. So this is definitely a statistical question. Definitely statistical. What was the difference in rainfall between Singapore and Seattle in 2013? Well, these two numbers aren't known. They can be measured."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Definitely statistical. What was the difference in rainfall between Singapore and Seattle in 2013? Well, these two numbers aren't known. They can be measured. Both the rainfall in Singapore can be measured. The rainfall in Seattle can be measured. And assuming that this has already happened and we can measure it, then we can just find the difference."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "They can be measured. Both the rainfall in Singapore can be measured. The rainfall in Seattle can be measured. And assuming that this has already happened and we can measure it, then we can just find the difference. So you don't need statistics here. You just have to have both of these measurements and subtract the difference. So not a statistical question."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And assuming that this has already happened and we can measure it, then we can just find the difference. So you don't need statistics here. You just have to have both of these measurements and subtract the difference. So not a statistical question. In general, will I use less gas driving at 55 miles an hour than 70 miles per hour? So this feels statistical because it probably depends on the circumstance. It might depend on the car."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So not a statistical question. In general, will I use less gas driving at 55 miles an hour than 70 miles per hour? So this feels statistical because it probably depends on the circumstance. It might depend on the car. Or even for a given car, when you drive at 55 miles per hour, there's some variation in your gas mileage. It might be how recent an oil change happened, what the wind conditions are like, what the road conditions are like, I mean, exactly how you're driving the car. Are you turning?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "It might depend on the car. Or even for a given car, when you drive at 55 miles per hour, there's some variation in your gas mileage. It might be how recent an oil change happened, what the wind conditions are like, what the road conditions are like, I mean, exactly how you're driving the car. Are you turning? Are you going in a straight line? And same thing for 70 miles an hour. So when we're seeing in general, there's variation in what the gas mileage is at 55 miles an hour and at 70 miles an hour."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Are you turning? Are you going in a straight line? And same thing for 70 miles an hour. So when we're seeing in general, there's variation in what the gas mileage is at 55 miles an hour and at 70 miles an hour. So what you'd probably want to do is say, well, what's my average mileage when I drive at 55 miles an hour? And compare that to the average mileage when I drive at 70. So because we have this variability in each of those cases, this is definitely a statistical question."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when we're seeing in general, there's variation in what the gas mileage is at 55 miles an hour and at 70 miles an hour. So what you'd probably want to do is say, well, what's my average mileage when I drive at 55 miles an hour? And compare that to the average mileage when I drive at 70. So because we have this variability in each of those cases, this is definitely a statistical question. Do English professors get paid less than math professors? So once again, all English professors don't get paid the same amount, and all math professors don't get paid the same amount. Some English professors might do quite well."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So because we have this variability in each of those cases, this is definitely a statistical question. Do English professors get paid less than math professors? So once again, all English professors don't get paid the same amount, and all math professors don't get paid the same amount. Some English professors might do quite well. Some might make very little. Same thing for math professors. So we'd probably want to find some type of an average to represent the central tendency for each of these."}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Some English professors might do quite well. Some might make very little. Same thing for math professors. So we'd probably want to find some type of an average to represent the central tendency for each of these. So once again, this is a statistical question. This is a statistical question. Does the most highly paid English professor at Harvard get paid more than the most highly paid math professor at MIT?"}, {"video_title": "Statistical and non statistical questions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we'd probably want to find some type of an average to represent the central tendency for each of these. So once again, this is a statistical question. This is a statistical question. Does the most highly paid English professor at Harvard get paid more than the most highly paid math professor at MIT? Well, now we're talking about two particular individuals. You could go look at their tax forms, see how much each of them get paid. And especially if we assume that this is in a particular year, let's say, and let's just make it that way, say in 2013, just so that we can remove some variability that they might make from year to year, make it a little bit more concrete."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "Jamie's dad gave her a die for her birthday. She wanted to make sure it was fair, so she took her die to school and rolled it 500 times and kept track of how many times the die rolled each number. Afterwards, she calculated the expected value of the sum of 20 rolls to be 67.4. The expected value of the sum of 20 rolls to be 67.4. On her way home from school, it was raining, and two values were washed away from her data table. Find the two missing absolute frequencies from Jamie's data table. So you see here, she rolled her die 500 times, and she wrote down how many times she got a two."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "The expected value of the sum of 20 rolls to be 67.4. On her way home from school, it was raining, and two values were washed away from her data table. Find the two missing absolute frequencies from Jamie's data table. So you see here, she rolled her die 500 times, and she wrote down how many times she got a two. She got a two 110 times, a 395 times, a 470 times, a 575 times, and then she had written down what she got, how many times she got a one and a six, but then it got washed away, so we need to figure out how many times she got a one and a six. Given the information on this table right over here, and given the information that the expected value of the sum of 20 rolls is 67.4. So I encourage you to pause this video and think about it on your own before I give a go at it."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you see here, she rolled her die 500 times, and she wrote down how many times she got a two. She got a two 110 times, a 395 times, a 470 times, a 575 times, and then she had written down what she got, how many times she got a one and a six, but then it got washed away, so we need to figure out how many times she got a one and a six. Given the information on this table right over here, and given the information that the expected value of the sum of 20 rolls is 67.4. So I encourage you to pause this video and think about it on your own before I give a go at it. So first, let's think about what this expected value, the sum of 20 rolls, being 67.4 tells us. That means that the expected value of one roll, the expected value of the sum of 20 rolls is just 20 times the expected value of one roll. So the expected value of a roll, let me do it here, expected value of a roll is going to be equal to 67.4 divided by 20."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I encourage you to pause this video and think about it on your own before I give a go at it. So first, let's think about what this expected value, the sum of 20 rolls, being 67.4 tells us. That means that the expected value of one roll, the expected value of the sum of 20 rolls is just 20 times the expected value of one roll. So the expected value of a roll, let me do it here, expected value of a roll is going to be equal to 67.4 divided by 20. We can get our calculator out. Let's see, so we have 67.4 divided by 20 is 3.37. So this is equal to 3.37."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the expected value of a roll, let me do it here, expected value of a roll is going to be equal to 67.4 divided by 20. We can get our calculator out. Let's see, so we have 67.4 divided by 20 is 3.37. So this is equal to 3.37. So how does that help us? Well, we know how to calculate an expected value given this frequency table right over here. If we say that this number right over here, let's say that's capital A, and let's say that this number here is capital B."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to 3.37. So how does that help us? Well, we know how to calculate an expected value given this frequency table right over here. If we say that this number right over here, let's say that's capital A, and let's say that this number here is capital B. If we were to try to calculate the expected value of a roll, what we really want to do is take the weighted frequency of each of these values, the weighted sum. So for example, if we got a one A out of 500 times, it would be A out of 500 times one, plus, I'll do these in different colors, plus 110 out of 500 times two, plus 110 out of 500 times two. Notice, this is the frequency, which was they got two, times two, we're taking a weighted sum of these values."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we say that this number right over here, let's say that's capital A, and let's say that this number here is capital B. If we were to try to calculate the expected value of a roll, what we really want to do is take the weighted frequency of each of these values, the weighted sum. So for example, if we got a one A out of 500 times, it would be A out of 500 times one, plus, I'll do these in different colors, plus 110 out of 500 times two, plus 110 out of 500 times two. Notice, this is the frequency, which was they got two, times two, we're taking a weighted sum of these values. And then plus 95 out of 500 times three, plus 95 out of 500 times three, plus, I think you see where this is going, 70 over 500 times four, plus 70 over 500 times four, almost there, plus, let's see, I haven't used this brown color, plus 75 over 500 times, I'll do it here, plus 75 over 500 times five, finally, plus B over 500 times six, this is going to give us our expected value of a roll, which is going to be equal to 3.37. So all of this is equal to 3.37. So one thing that we can do, since we have all these 500s in this denominator right over here, let's multiply both sides of this equation times 500."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "Notice, this is the frequency, which was they got two, times two, we're taking a weighted sum of these values. And then plus 95 out of 500 times three, plus 95 out of 500 times three, plus, I think you see where this is going, 70 over 500 times four, plus 70 over 500 times four, almost there, plus, let's see, I haven't used this brown color, plus 75 over 500 times, I'll do it here, plus 75 over 500 times five, finally, plus B over 500 times six, this is going to give us our expected value of a roll, which is going to be equal to 3.37. So all of this is equal to 3.37. So one thing that we can do, since we have all these 500s in this denominator right over here, let's multiply both sides of this equation times 500. If we do that, the left-hand side becomes, well, 500 times A over 500 is just going to be A, plus 110, plus, oh, 110 times two, so it's going to be 220, plus 95 times three, that's going to be 15 less than 300, so it's going to be plus 285, plus 285, and then 70 times four is 280, plus 280. 75 times five is going to be 350 plus 25, 375, so plus 375, plus 6B, make sure I'm not skipping any steps here, plus 6B is going to be equal to, is going to be equal to this times 500, and that is going to be equal to 3.37 times 500 is equal to 1685, 1,685. So all I did to go from this step right over here, which I set up saying, hey, this is the expected value of one roll, which we already know to be 3.37, is I just multiplied both sides of this equation by 500."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So one thing that we can do, since we have all these 500s in this denominator right over here, let's multiply both sides of this equation times 500. If we do that, the left-hand side becomes, well, 500 times A over 500 is just going to be A, plus 110, plus, oh, 110 times two, so it's going to be 220, plus 95 times three, that's going to be 15 less than 300, so it's going to be plus 285, plus 285, and then 70 times four is 280, plus 280. 75 times five is going to be 350 plus 25, 375, so plus 375, plus 6B, make sure I'm not skipping any steps here, plus 6B is going to be equal to, is going to be equal to this times 500, and that is going to be equal to 3.37 times 500 is equal to 1685, 1,685. So all I did to go from this step right over here, which I set up saying, hey, this is the expected value of one roll, which we already know to be 3.37, is I just multiplied both sides of this equation by 500. I just did this times 500, and I did this times, I did this times 500, and this 500 obviously cancels with all of these, and then 500 times 3.37 is 1685, and so I got this right over here. Now, I got one, two, three, four, five, six, yep, I did enough, I have the right number of terms. I just want to make sure I'm not making a careless mistake, and so if we want to simplify this, we can subtract 220, 285, 280, and 375 from both sides, and so we would be, if we did that, we would get A, if we subtract that from the left-hand side, we're just going to get A plus 6B, and on the right-hand side, we are going to get, let's get our calculator out, 1,685 minus 220, 220 minus 285, minus 285, minus 280, minus 280, minus 375, minus 375 gets us to 525."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So all I did to go from this step right over here, which I set up saying, hey, this is the expected value of one roll, which we already know to be 3.37, is I just multiplied both sides of this equation by 500. I just did this times 500, and I did this times, I did this times 500, and this 500 obviously cancels with all of these, and then 500 times 3.37 is 1685, and so I got this right over here. Now, I got one, two, three, four, five, six, yep, I did enough, I have the right number of terms. I just want to make sure I'm not making a careless mistake, and so if we want to simplify this, we can subtract 220, 285, 280, and 375 from both sides, and so we would be, if we did that, we would get A, if we subtract that from the left-hand side, we're just going to get A plus 6B, and on the right-hand side, we are going to get, let's get our calculator out, 1,685 minus 220, 220 minus 285, minus 285, minus 280, minus 280, minus 375, minus 375 gets us to 525. So we get A plus 6B is equal to 525, and you say, okay, you did all that work, but we still have one equation with two unknowns. How do we figure out what A and B, how do we figure out what A and B actually are? Well, we know something else."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "I just want to make sure I'm not making a careless mistake, and so if we want to simplify this, we can subtract 220, 285, 280, and 375 from both sides, and so we would be, if we did that, we would get A, if we subtract that from the left-hand side, we're just going to get A plus 6B, and on the right-hand side, we are going to get, let's get our calculator out, 1,685 minus 220, 220 minus 285, minus 285, minus 280, minus 280, minus 375, minus 375 gets us to 525. So we get A plus 6B is equal to 525, and you say, okay, you did all that work, but we still have one equation with two unknowns. How do we figure out what A and B, how do we figure out what A and B actually are? Well, we know something else. We know, and this was actually much easier to figure out, we know that the sum of this whole table right over here, A plus 110 plus 95 plus 70 plus 75 plus B is equal to 500, or if we, let me write that down. So we know that A plus 110 plus 95 plus 70 plus 75 plus B needs to be equal to 500, needs to be equal to 500, or we could subtract 110 plus 95 plus 70 plus 75 from both sides and get, if you subtract it from the left-hand side, you're just left with A plus B, A plus B, and on the right-hand side, if we start with 500, so 500 minus 110, minus 95, minus 70, minus 75 gets us to 150. So A plus B must be equal to 150, is equal to 150."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we know something else. We know, and this was actually much easier to figure out, we know that the sum of this whole table right over here, A plus 110 plus 95 plus 70 plus 75 plus B is equal to 500, or if we, let me write that down. So we know that A plus 110 plus 95 plus 70 plus 75 plus B needs to be equal to 500, needs to be equal to 500, or we could subtract 110 plus 95 plus 70 plus 75 from both sides and get, if you subtract it from the left-hand side, you're just left with A plus B, A plus B, and on the right-hand side, if we start with 500, so 500 minus 110, minus 95, minus 70, minus 75 gets us to 150. So A plus B must be equal to 150, is equal to 150. And now we have a system of two equations and two unknowns, and so we know how to solve those. We could do it by substitution, or we could subtract the second equation from the first. So let's do that."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So A plus B must be equal to 150, is equal to 150. And now we have a system of two equations and two unknowns, and so we know how to solve those. We could do it by substitution, or we could subtract the second equation from the first. So let's do that. Let's subtract the left-hand side of this equation from that. So, or essentially, we could add these two, so we can multiply this one times a negative one, and then add these two equations. The A's are going to cancel out, and we are going to be left with six B minus B is five B, is equal to 375, is equal to 375."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's do that. Let's subtract the left-hand side of this equation from that. So, or essentially, we could add these two, so we can multiply this one times a negative one, and then add these two equations. The A's are going to cancel out, and we are going to be left with six B minus B is five B, is equal to 375, is equal to 375. Did I do that right? If I add 125 to this, I get to 500, and then another 25, I get to 525. So five B is equal to 375, or if we divide both sides by five, we get B is equal to, B is equal to 75."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "The A's are going to cancel out, and we are going to be left with six B minus B is five B, is equal to 375, is equal to 375. Did I do that right? If I add 125 to this, I get to 500, and then another 25, I get to 525. So five B is equal to 375, or if we divide both sides by five, we get B is equal to, B is equal to 75. B is equal to 75. So this right over here is equal to 75. If B is equal to 75, what is A?"}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "So five B is equal to 375, or if we divide both sides by five, we get B is equal to, B is equal to 75. B is equal to 75. So this right over here is equal to 75. If B is equal to 75, what is A? Well, we know that A plus B is equal to 500. We figured that out a little while ago before we multiplied both sides of this times a negative one. We knew that A plus B, when B is now 75, so we could say A plus 75, is equal to 150."}, {"video_title": "Getting data from expected value   Probability and Statistics   Khan Academy.mp3", "Sentence": "If B is equal to 75, what is A? Well, we know that A plus B is equal to 500. We figured that out a little while ago before we multiplied both sides of this times a negative one. We knew that A plus B, when B is now 75, so we could say A plus 75, is equal to 150. And that's just, from this, we figured out that A plus B is equal to 150 before we multiplied both sides times a negative. Subtract 75 from both sides, you get A is also equal to 75. And we are done."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which of the following plots suits the above description? So let's see, this looks like a negative linear correlation or association. As the years go by, you have a smaller percent of smokers. This one does too. As years go by, you have the number of smokers go down and down. This one down here also looks like that. Although it's not as smooth."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This one does too. As years go by, you have the number of smokers go down and down. This one down here also looks like that. Although it's not as smooth. If you were to fit a line here, it looks like you have a few outliers. Well, this is a positive correlation. So we can definitely rule out graph four."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Although it's not as smooth. If you were to fit a line here, it looks like you have a few outliers. Well, this is a positive correlation. So we can definitely rule out graph four. Now the other thing that they told us is that there are no outliers. Suggests a negative linear association with no outliers. If you were to try to fit a line to graph three, you could fit a line pretty reasonably that would go someplace like that."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can definitely rule out graph four. Now the other thing that they told us is that there are no outliers. Suggests a negative linear association with no outliers. If you were to try to fit a line to graph three, you could fit a line pretty reasonably that would go someplace like that. But it would have this outlier right over here. It looks like it's 12 or 13 years. It looks about 13 years after 1967."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you were to try to fit a line to graph three, you could fit a line pretty reasonably that would go someplace like that. But it would have this outlier right over here. It looks like it's 12 or 13 years. It looks about 13 years after 1967. So that would be 1980. Looks like an outlier there. But they said it doesn't have any outliers."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It looks about 13 years after 1967. So that would be 1980. Looks like an outlier there. But they said it doesn't have any outliers. So we would rule out graph number three. And so we have to pick between graph one and graph two. And so the other hint they give us or piece of data they give us is that the percent drops by 0.5% each year."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "But they said it doesn't have any outliers. So we would rule out graph number three. And so we have to pick between graph one and graph two. And so the other hint they give us or piece of data they give us is that the percent drops by 0.5% each year. So here, what's happening? As 10 years go by, let's see. As we go from, let's start with, let's see, in 1967, where it looks like we're at about 55%."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so the other hint they give us or piece of data they give us is that the percent drops by 0.5% each year. So here, what's happening? As 10 years go by, let's see. As we go from, let's start with, let's see, in 1967, where it looks like we're at about 55%. And then 10 years go by, we are roughly at around 45%, a little under 45%. So we dropped 10% in 10 years. That seems to be how much this is dropping, about roughly 10% in 10 years."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "As we go from, let's start with, let's see, in 1967, where it looks like we're at about 55%. And then 10 years go by, we are roughly at around 45%, a little under 45%. So we dropped 10% in 10 years. That seems to be how much this is dropping, about roughly 10% in 10 years. Another 10 years go by. We go from 45% or a little more than 10% in 10 years. And so that would mean that we're dropping, on average, more than 1 percentage point per year."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "That seems to be how much this is dropping, about roughly 10% in 10 years. Another 10 years go by. We go from 45% or a little more than 10% in 10 years. And so that would mean that we're dropping, on average, more than 1 percentage point per year. That seems more than what's going on here. Now let's look over here. Over here, we're starting, it looks like, at around 42%."}, {"video_title": "People smoking less over time scatter plot   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so that would mean that we're dropping, on average, more than 1 percentage point per year. That seems more than what's going on here. Now let's look over here. Over here, we're starting, it looks like, at around 42%. And then after 10 years, it looks like we're at 37%. So it looks like we've dropped about 5% in 10 years, which is consistent with this. If you drop 5% in 10 years, that means you drop half a percent per year."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "But because it's so applicable to so many things, it's often a misused law or sometimes a slightly misunderstood. So just to be a little bit formal in our mathematics, let me just define it for you first. And then we'll talk a little bit about the intuition. So let's say I have a random variable X. And we know its expected value or its population mean. The law of large numbers just says that if we take a sample of n observations of our random variable, and if we were to average all of those observations, and let me define another variable, let's call that X sub n with a line on top of it, this is the mean of n observations of our random variable. So it's literally, this is my first observation."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say I have a random variable X. And we know its expected value or its population mean. The law of large numbers just says that if we take a sample of n observations of our random variable, and if we were to average all of those observations, and let me define another variable, let's call that X sub n with a line on top of it, this is the mean of n observations of our random variable. So it's literally, this is my first observation. So you could kind of say I run the experiment once and I get this observation and I run it again, I get that observation. And I keep running it n times. And then I divide by my number of observations."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's literally, this is my first observation. So you could kind of say I run the experiment once and I get this observation and I run it again, I get that observation. And I keep running it n times. And then I divide by my number of observations. So this is my sample mean. This is the mean of all of the observations I've made. The law of large numbers just tells us that my sample mean will approach my expected value of the random variable."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then I divide by my number of observations. So this is my sample mean. This is the mean of all of the observations I've made. The law of large numbers just tells us that my sample mean will approach my expected value of the random variable. Or I could also write it as my sample mean will approach my population mean for n approaching infinity. And I'll be a little informal with what is approach or what is convergence mean, but I think you have the general intuitive sense that if I take a large enough sample here, that I'm going to end up getting the expected value of the population as a whole. And I think to a lot of us that's kind of intuitive, that if I do enough trials that over large samples, that the trials would kind of give me the numbers that I would expect given the expected value and the probability of all that."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "The law of large numbers just tells us that my sample mean will approach my expected value of the random variable. Or I could also write it as my sample mean will approach my population mean for n approaching infinity. And I'll be a little informal with what is approach or what is convergence mean, but I think you have the general intuitive sense that if I take a large enough sample here, that I'm going to end up getting the expected value of the population as a whole. And I think to a lot of us that's kind of intuitive, that if I do enough trials that over large samples, that the trials would kind of give me the numbers that I would expect given the expected value and the probability of all that. But I think it's often a little bit misunderstood in terms of why that happens. And before I go into that, let me give you a particular example. So the law of large numbers will just tell us that, let's say I have a random variable x is equal to the number of heads after 100 tosses of a fair coin."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I think to a lot of us that's kind of intuitive, that if I do enough trials that over large samples, that the trials would kind of give me the numbers that I would expect given the expected value and the probability of all that. But I think it's often a little bit misunderstood in terms of why that happens. And before I go into that, let me give you a particular example. So the law of large numbers will just tell us that, let's say I have a random variable x is equal to the number of heads after 100 tosses of a fair coin. Tosses or flips of a fair coin. The law of large numbers just tells us, well, first of all, we know what the expected value of this random variable is. It's the number of tosses, the number of trials, times the probability of success of any trial."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the law of large numbers will just tell us that, let's say I have a random variable x is equal to the number of heads after 100 tosses of a fair coin. Tosses or flips of a fair coin. The law of large numbers just tells us, well, first of all, we know what the expected value of this random variable is. It's the number of tosses, the number of trials, times the probability of success of any trial. So that's equal to 50. So the law of large numbers just says if I were to take a sample, or if I were to average the sample of a bunch of these trials, so I get, I don't know, my first time I run this trial I flip 100 coins or I have 100 coins in a shoebox and I shake the shoebox and I count the number of heads and I get 55. So that would be x1."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's the number of tosses, the number of trials, times the probability of success of any trial. So that's equal to 50. So the law of large numbers just says if I were to take a sample, or if I were to average the sample of a bunch of these trials, so I get, I don't know, my first time I run this trial I flip 100 coins or I have 100 coins in a shoebox and I shake the shoebox and I count the number of heads and I get 55. So that would be x1. Then I shake the box again and I get 65. Then I shake the box again and I get 45. And I do this n times and then I divide it by the number of times I did it."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that would be x1. Then I shake the box again and I get 65. Then I shake the box again and I get 45. And I do this n times and then I divide it by the number of times I did it. The law of large numbers just tells us that this average, the average of all of my observations, is going to converge to 50 as n approaches infinity. Or for n approaching infinity. And I want to talk a little bit about why this happens or intuitively why this is."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I do this n times and then I divide it by the number of times I did it. The law of large numbers just tells us that this average, the average of all of my observations, is going to converge to 50 as n approaches infinity. Or for n approaching infinity. And I want to talk a little bit about why this happens or intuitively why this is. A lot of people kind of feel that like, oh, this means that if after 100 trials, that if I'm above the average, that somehow the laws of probability are going to give me more heads or fewer heads to kind of make up the difference. And that's not quite what's going to happen. And that's often called the gambler's fallacy."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I want to talk a little bit about why this happens or intuitively why this is. A lot of people kind of feel that like, oh, this means that if after 100 trials, that if I'm above the average, that somehow the laws of probability are going to give me more heads or fewer heads to kind of make up the difference. And that's not quite what's going to happen. And that's often called the gambler's fallacy. And let me differentiate. And I'll use this example. So let's say, let me make a graph."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's often called the gambler's fallacy. And let me differentiate. And I'll use this example. So let's say, let me make a graph. And I'll switch colors. So let's say that this is, so let me make, so on the, this is n, my x-axis is n. This is the number of trials I take. And my y-axis, let me make that the sample mean."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say, let me make a graph. And I'll switch colors. So let's say that this is, so let me make, so on the, this is n, my x-axis is n. This is the number of trials I take. And my y-axis, let me make that the sample mean. And we know what the expected value is, right? We know the expected value of this random variable is. It's 50."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And my y-axis, let me make that the sample mean. And we know what the expected value is, right? We know the expected value of this random variable is. It's 50. Let me draw that here. This is 50. So just going to the example I did."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's 50. Let me draw that here. This is 50. So just going to the example I did. So when n is equal to, let me just plot it here. So my first trial, I got 55. And so that was my average, right?"}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just going to the example I did. So when n is equal to, let me just plot it here. So my first trial, I got 55. And so that was my average, right? I only had one data point. Then after two trials, let's see, then I have 65. And so my average is going to be 65 plus 55 divided by 2."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so that was my average, right? I only had one data point. Then after two trials, let's see, then I have 65. And so my average is going to be 65 plus 55 divided by 2. Which is 60. So then my average went up a little bit. Then I had a 45, which will bring my average down a little bit."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so my average is going to be 65 plus 55 divided by 2. Which is 60. So then my average went up a little bit. Then I had a 45, which will bring my average down a little bit. I won't plot a 45 here. Now I have to average all of these out. What's 45 plus 65?"}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then I had a 45, which will bring my average down a little bit. I won't plot a 45 here. Now I have to average all of these out. What's 45 plus 65? Well, let me actually just get the numbers, just so you get the point. So it's 55 plus 65 is 120. Plus 45 is 165."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's 45 plus 65? Well, let me actually just get the numbers, just so you get the point. So it's 55 plus 65 is 120. Plus 45 is 165. Divided by 3 is 53. Right? No, no, no."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus 45 is 165. Divided by 3 is 53. Right? No, no, no. 15. 55. So the average goes down back down to 55."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "No, no, no. 15. 55. So the average goes down back down to 55. And we could keep doing these trials, right? So you might say that the law of large numbers tells us, OK, after we've done three trials and our average is there. So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the average goes down back down to 55. And we could keep doing these trials, right? So you might say that the law of large numbers tells us, OK, after we've done three trials and our average is there. So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future. That somehow the next couple of trials are going to have to be down here in order to bring our average down. And that's not necessarily the case. Going forward, the probabilities are always the same."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future. That somehow the next couple of trials are going to have to be down here in order to bring our average down. And that's not necessarily the case. Going forward, the probabilities are always the same. The probabilities are always 50% that I'm going to get heads. It's not like if I had a bunch of heads to start off with, or more than I would have expected to start off with, that all of a sudden things would be made up and I'd get more tails. And that would be the gambler's fallacy."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "Going forward, the probabilities are always the same. The probabilities are always 50% that I'm going to get heads. It's not like if I had a bunch of heads to start off with, or more than I would have expected to start off with, that all of a sudden things would be made up and I'd get more tails. And that would be the gambler's fallacy. That if you have a long streak of heads, or you have a disproportionate number of heads, that at some point you're going to have a higher likelihood of having a disproportionate number of tails. And that's not quite true. What the law of large numbers tells us is that it doesn't care, let's say after some finite number of trials, your average actually, it's a low probability of this happening, but let's say your average is actually up here."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that would be the gambler's fallacy. That if you have a long streak of heads, or you have a disproportionate number of heads, that at some point you're going to have a higher likelihood of having a disproportionate number of tails. And that's not quite true. What the law of large numbers tells us is that it doesn't care, let's say after some finite number of trials, your average actually, it's a low probability of this happening, but let's say your average is actually up here. It's actually at 70, right? Like, wow, we really diverged a good bit from the expected value. But what the law of large numbers says, well, I don't care how many trials this is, we have an infinite number of trials left, right?"}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "What the law of large numbers tells us is that it doesn't care, let's say after some finite number of trials, your average actually, it's a low probability of this happening, but let's say your average is actually up here. It's actually at 70, right? Like, wow, we really diverged a good bit from the expected value. But what the law of large numbers says, well, I don't care how many trials this is, we have an infinite number of trials left, right? And the expected value for that infinite number of trials, or especially in this type of situation, is going to be this. So when you average a finite number that averages out to some high number, and then an infinite number that's going to converge to this, you're going to, over time, converge back to the expected value. And that was a very informal way of describing it."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what the law of large numbers says, well, I don't care how many trials this is, we have an infinite number of trials left, right? And the expected value for that infinite number of trials, or especially in this type of situation, is going to be this. So when you average a finite number that averages out to some high number, and then an infinite number that's going to converge to this, you're going to, over time, converge back to the expected value. And that was a very informal way of describing it. But that's what the law of large numbers tells you. And it's an important thing. It's not telling you that if you get a bunch of heads, that somehow the probability of getting tails is going to increase to kind of make up for the heads."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that was a very informal way of describing it. But that's what the law of large numbers tells you. And it's an important thing. It's not telling you that if you get a bunch of heads, that somehow the probability of getting tails is going to increase to kind of make up for the heads. What it's telling you is that no matter what happened over a finite number of trials, no matter what the average is over a finite number of trials, you have an infinite number of trials left. And if you do enough of them, it's going to converge back to your expected value. And this is an important thing to think about."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's not telling you that if you get a bunch of heads, that somehow the probability of getting tails is going to increase to kind of make up for the heads. What it's telling you is that no matter what happened over a finite number of trials, no matter what the average is over a finite number of trials, you have an infinite number of trials left. And if you do enough of them, it's going to converge back to your expected value. And this is an important thing to think about. But this isn't used in practice every day with the lottery and with casinos. Because they know that if you do large enough samples, and we could even calculate if you do large enough samples, what's the probability that things deviate significantly? But casinos and the lottery every day operate on this principle that if you take enough people, sure, in the short term or with a few samples, a couple people might beat the house."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is an important thing to think about. But this isn't used in practice every day with the lottery and with casinos. Because they know that if you do large enough samples, and we could even calculate if you do large enough samples, what's the probability that things deviate significantly? But casinos and the lottery every day operate on this principle that if you take enough people, sure, in the short term or with a few samples, a couple people might beat the house. But over the long term, the house is always going to win because of the parameters of the games that they're making you play. Anyway, this is an important thing in probability. And I think it's fairly intuitive."}, {"video_title": "Law of large numbers   Probability and Statistics   Khan Academy.mp3", "Sentence": "But casinos and the lottery every day operate on this principle that if you take enough people, sure, in the short term or with a few samples, a couple people might beat the house. But over the long term, the house is always going to win because of the parameters of the games that they're making you play. Anyway, this is an important thing in probability. And I think it's fairly intuitive. Although sometimes when you see it formally explained like this with the random variables, then that's a little bit confusing. All it's saying is that as you take more and more samples, the average of that sample is going to approximate the true average, or I should be a little bit more particular, the mean of your sample is going to converge to the true mean of the population, or to the expected value of the random variable. Anyway, see you in the next video."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's not quite what random variables are. Random variables are really ways to map outcomes of random processes to numbers. So if you have a random process, like you're flipping a coin, or you are rolling dice, or you are measuring the rain that might fall tomorrow. So random process. You're really just mapping outcomes of that to numbers. You're quantifying the outcomes. So what's an example of a random variable?"}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So random process. You're really just mapping outcomes of that to numbers. You're quantifying the outcomes. So what's an example of a random variable? Well, let's define one right over here. So I'm going to define random variable capital X. And they tend to be denoted by capital letters."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's an example of a random variable? Well, let's define one right over here. So I'm going to define random variable capital X. And they tend to be denoted by capital letters. So random variable capital X, I will define it as, is going to be equal to 1 if my fair die rolls heads. And it's going to be equal to 0 if tails. I could have defined this any way I wanted to."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And they tend to be denoted by capital letters. So random variable capital X, I will define it as, is going to be equal to 1 if my fair die rolls heads. And it's going to be equal to 0 if tails. I could have defined this any way I wanted to. This is actually a fairly typical way of defining a random variable, especially for a coin flip. But I could have defined this as 100. And I could have defined this as 703."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could have defined this any way I wanted to. This is actually a fairly typical way of defining a random variable, especially for a coin flip. But I could have defined this as 100. And I could have defined this as 703. And this would still be a legitimate random variable. It might not be as pure a way of thinking about it as defining 1 as heads and 0 as tails. But that would have been a random variable."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I could have defined this as 703. And this would still be a legitimate random variable. It might not be as pure a way of thinking about it as defining 1 as heads and 0 as tails. But that would have been a random variable. Notice, we have taken this random process, flipping a coin, and we've mapped the outcomes of that random process. And we've quantified them, 1 if heads, 0 if tails. So we can define another random variable, capital Y, as equal to, let's say, the sum of rolls of, let's say, 7 dice."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "But that would have been a random variable. Notice, we have taken this random process, flipping a coin, and we've mapped the outcomes of that random process. And we've quantified them, 1 if heads, 0 if tails. So we can define another random variable, capital Y, as equal to, let's say, the sum of rolls of, let's say, 7 dice. And when we talk about the sum, we're talking about the sum of the 7. Let me write this. The sum of the upward face after rolling 7 dice."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can define another random variable, capital Y, as equal to, let's say, the sum of rolls of, let's say, 7 dice. And when we talk about the sum, we're talking about the sum of the 7. Let me write this. The sum of the upward face after rolling 7 dice. Once again, we are quantifying an outcome for a random process. We are the random processes rolling these 7 dice and seeing what sides show up on top. And then we are taking those, and we are taking the sum, and we are defining a random variable in that way."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "The sum of the upward face after rolling 7 dice. Once again, we are quantifying an outcome for a random process. We are the random processes rolling these 7 dice and seeing what sides show up on top. And then we are taking those, and we are taking the sum, and we are defining a random variable in that way. So the natural question you might ask is, why are we doing this? What's so useful about defining random variables like this? It will become more apparent as we get a little bit deeper in probability."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we are taking those, and we are taking the sum, and we are defining a random variable in that way. So the natural question you might ask is, why are we doing this? What's so useful about defining random variables like this? It will become more apparent as we get a little bit deeper in probability. But the simple way of thinking about it is, as soon as you quantify outcomes, you can start to do a little bit more math on the outcomes. And you can start to use a little bit more mathematical notation on the outcome. So for example, if you cared about the probability that the sum of the upward faces after rolling 7 dice, if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of, and you would have to write all of what I just wrote here, is less than or equal to 30."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "It will become more apparent as we get a little bit deeper in probability. But the simple way of thinking about it is, as soon as you quantify outcomes, you can start to do a little bit more math on the outcomes. And you can start to use a little bit more mathematical notation on the outcome. So for example, if you cared about the probability that the sum of the upward faces after rolling 7 dice, if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of, and you would have to write all of what I just wrote here, is less than or equal to 30. You would have had to write that big thing. And if you wanted to write, and then you would try to figure it out somehow if you had some information. But now we can just write the probability that capital Y is less than or equal to 30."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, if you cared about the probability that the sum of the upward faces after rolling 7 dice, if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of, and you would have to write all of what I just wrote here, is less than or equal to 30. You would have had to write that big thing. And if you wanted to write, and then you would try to figure it out somehow if you had some information. But now we can just write the probability that capital Y is less than or equal to 30. It's a little bit cleaner notation. And if someone else cares about the probability that the sum of the upward face after rolling 7 dice, if they say, hey, what's the probability that that's even, instead of having to write all of that over, they can say, well, what's the probability that Y is even? Now, the one thing that I do want to emphasize is how these are different than traditional variables, traditional variables that you see in your algebra class, like x plus 5 is equal to 6, usually denoted by lowercase variables."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "But now we can just write the probability that capital Y is less than or equal to 30. It's a little bit cleaner notation. And if someone else cares about the probability that the sum of the upward face after rolling 7 dice, if they say, hey, what's the probability that that's even, instead of having to write all of that over, they can say, well, what's the probability that Y is even? Now, the one thing that I do want to emphasize is how these are different than traditional variables, traditional variables that you see in your algebra class, like x plus 5 is equal to 6, usually denoted by lowercase variables. Y is equal to x plus 7. These variables, you can essentially assign values. You either can solve for them."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, the one thing that I do want to emphasize is how these are different than traditional variables, traditional variables that you see in your algebra class, like x plus 5 is equal to 6, usually denoted by lowercase variables. Y is equal to x plus 7. These variables, you can essentially assign values. You either can solve for them. So in this case, x is an unknown. You can subtract 5 from both sides and solve for x. Say that x is going to be equal to 1."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "You either can solve for them. So in this case, x is an unknown. You can subtract 5 from both sides and solve for x. Say that x is going to be equal to 1. In this case, you could say, well, x is going to vary. We can assign a value to x and see how y varies as a function of x. You can either assign variable, you can assign values to them, or you can solve for them."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "Say that x is going to be equal to 1. In this case, you could say, well, x is going to vary. We can assign a value to x and see how y varies as a function of x. You can either assign variable, you can assign values to them, or you can solve for them. You could say, hey, x is going to be 1 in this case. That's not going to be the case with a random variable. A random variable can take on many, many, many, many, many different values with different probabilities."}, {"video_title": "Random variables   Probability and Statistics   Khan Academy.mp3", "Sentence": "You can either assign variable, you can assign values to them, or you can solve for them. You could say, hey, x is going to be 1 in this case. That's not going to be the case with a random variable. A random variable can take on many, many, many, many, many different values with different probabilities. And it makes much more sense to talk about the probability of a random variable equaling a value, or the probability that is less than or greater than something, or the probability that it has some property. And you see that in either of these cases. In the next video, we'll continue this discussion."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And all this error means is that you've rejected, this is the error of rejecting, let me do this in a different color, rejecting the null hypothesis even though it is true. So for example, in actually all of the hypothesis testing examples we've seen, we start assuming that the null hypothesis is true. We always assume that the null hypothesis is true. And given that the null hypothesis is true, we say OK, if the null hypothesis is true, then the mean is usually going to be equal to some value. So we create some distribution assuming that the null hypothesis is true. It normally has some mean value right over there. And then we have some statistic, and we are seeing if the null hypothesis is true, what is the probability of getting that statistic, or getting a result that extreme or more extreme than that statistic?"}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And given that the null hypothesis is true, we say OK, if the null hypothesis is true, then the mean is usually going to be equal to some value. So we create some distribution assuming that the null hypothesis is true. It normally has some mean value right over there. And then we have some statistic, and we are seeing if the null hypothesis is true, what is the probability of getting that statistic, or getting a result that extreme or more extreme than that statistic? So let's say that the statistic gives us some value over here, and we say, gee, you know what? There's only, I don't know, there might be a 1% chance, there's only a 1% probability of getting a result that extreme or greater. And then if that's, I guess, low enough of a threshold for us, we will reject the null hypothesis."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have some statistic, and we are seeing if the null hypothesis is true, what is the probability of getting that statistic, or getting a result that extreme or more extreme than that statistic? So let's say that the statistic gives us some value over here, and we say, gee, you know what? There's only, I don't know, there might be a 1% chance, there's only a 1% probability of getting a result that extreme or greater. And then if that's, I guess, low enough of a threshold for us, we will reject the null hypothesis. So in this case, we will. So let's say that, actually, let's think of it this way. Let's say that 1% is our threshold."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then if that's, I guess, low enough of a threshold for us, we will reject the null hypothesis. So in this case, we will. So let's say that, actually, let's think of it this way. Let's say that 1% is our threshold. We say, look, we're going to assume that the null hypothesis is true. There's some threshold that if we get a value any more extreme than that value, there's less than a 1% chance of that happening. So let's say we're looking at sample means, and we get a sample mean that is way out here."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that 1% is our threshold. We say, look, we're going to assume that the null hypothesis is true. There's some threshold that if we get a value any more extreme than that value, there's less than a 1% chance of that happening. So let's say we're looking at sample means, and we get a sample mean that is way out here. We say, well, there's less than a 1% chance of that happening, given that the null hypothesis is true. So we are going to reject the null hypothesis. So we will reject the null hypothesis."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say we're looking at sample means, and we get a sample mean that is way out here. We say, well, there's less than a 1% chance of that happening, given that the null hypothesis is true. So we are going to reject the null hypothesis. So we will reject the null hypothesis. Now what does that mean, though? Let's say that this area, the probability of getting a result like that or that much more extreme is just this area right here. So let's say that's half a percent."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we will reject the null hypothesis. Now what does that mean, though? Let's say that this area, the probability of getting a result like that or that much more extreme is just this area right here. So let's say that's half a percent. Maybe I can write it this way. Let's say it's half a percent. And because it's so unlikely to get a statistic like that, assuming that the null hypothesis is true, we decide to reject the null hypothesis."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that's half a percent. Maybe I can write it this way. Let's say it's half a percent. And because it's so unlikely to get a statistic like that, assuming that the null hypothesis is true, we decide to reject the null hypothesis. Or another way to view it is, there's a 0.5% chance that we have made a type I error in rejecting the null hypothesis. Because if the null hypothesis is true, there's a 0.5% chance that this could still happen. So in rejecting it, we would make a mistake."}, {"video_title": "Type 1 errors   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And because it's so unlikely to get a statistic like that, assuming that the null hypothesis is true, we decide to reject the null hypothesis. Or another way to view it is, there's a 0.5% chance that we have made a type I error in rejecting the null hypothesis. Because if the null hypothesis is true, there's a 0.5% chance that this could still happen. So in rejecting it, we would make a mistake. There's a 0.5% chance we've made a type I error. I just want to clear that up. Hopefully that clarified it for you."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "I have this article right here from WebMD. And the point of this isn't to poke holes at WebMD. I think they have some great articles and they have some great information on their site. But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese?"}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying?"}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows?"}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore?"}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast."}, {"video_title": "Correlation and causality   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast. And this also leads to obesity. But once again, if this is the actual reality, doing the advice that that article's saying might actually be a bad thing. If you eat an unhealthy dinner and then force yourself to eat a breakfast when you're not hungry, that might make the obesity even worse."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "We are told that a zookeeper took a random sample of 30 days and observed how much food an elephant ate on each of those days. The sample mean was 350 kilograms and the sample standard deviation was 25 kilograms. The resulting 90% confidence interval for the mean amount of food was from 341 kilograms to 359 kilograms. Which of the following statements is a correct interpretation of the 90% confidence level? So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Which of the following statements is a correct interpretation of the 90% confidence level? So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. So let's think about it this way. Let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. So let's think about it this way. Let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is. So let's say sample mean and this is their first sample. It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is. So let's say sample mean and this is their first sample. It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample and then if I did another sample, let's say this is the mean of that next sample so that's sample mean two and I have an associated confidence interval and that interval, not only the start and end points will change but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample and then if I did another sample, let's say this is the mean of that next sample so that's sample mean two and I have an associated confidence interval and that interval, not only the start and end points will change but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So now with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So now with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval. So it causes a little bit of unease. See choice, so I'm just gonna put a question mark here."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval. So it causes a little bit of unease. See choice, so I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals, yep, that's what it does, every time you sample, you produce an interval, that capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "See choice, so I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals, yep, that's what it does, every time you sample, you produce an interval, that capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about. If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out."}, {"video_title": "Interpreting confidence level example   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Yeah, that's exactly what we're talking about. If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out. In repeated sampling, this method produces a sample mean between 341 kilograms and 359 kilograms in about 90% of samples. No, the confidence interval does not put a constrain on that 90% of the time you will have a sample mean between these values. It is not trying to do that."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "He randomly selects 20 students at his school and records their caffeine intake in milligrams and the amount of time studying in a given week. Here is a computer output from a least squares regression analysis on his sample. Assume that all conditions for inference have been met. What is a 95% confidence interval for the slope of the least squares regression line? So if you feel inspired, pause the video and see if you can have a go at it. Otherwise, we'll do this together. Okay, so let's first remind ourselves what's even going on."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "What is a 95% confidence interval for the slope of the least squares regression line? So if you feel inspired, pause the video and see if you can have a go at it. Otherwise, we'll do this together. Okay, so let's first remind ourselves what's even going on. So let's visualize the regression. So our horizontal axis, or our x-axis, that would be our caffeine intake in milligrams. And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Okay, so let's first remind ourselves what's even going on. So let's visualize the regression. So our horizontal axis, or our x-axis, that would be our caffeine intake in milligrams. And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. And Moussa here, he randomly selects 20 students. And so for each of those students, he sees how much caffeine they consumed and how much time they spent studying and plots them here. And so there'll be 20 data points."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. And Moussa here, he randomly selects 20 students. And so for each of those students, he sees how much caffeine they consumed and how much time they spent studying and plots them here. And so there'll be 20 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. He inputs these data points into a computer in order to fit a least squares regression line. And let's say the least squares regression line looks something like this."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And so there'll be 20 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. He inputs these data points into a computer in order to fit a least squares regression line. And let's say the least squares regression line looks something like this. And a least squares regression line comes from trying to minimize the square distance between the line and all of these points. And then this is giving us information on that least squares regression line. And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And let's say the least squares regression line looks something like this. And a least squares regression line comes from trying to minimize the square distance between the line and all of these points. And then this is giving us information on that least squares regression line. And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column. The constant coefficient tells us essentially what is the y-intercept here, so 2.544. And then the coefficient on the caffeine, this is one way of thinking about, well, for every incremental increase in caffeine, how much does the time studying increase? Or you might recognize this as the slope of the least squares regression line."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column. The constant coefficient tells us essentially what is the y-intercept here, so 2.544. And then the coefficient on the caffeine, this is one way of thinking about, well, for every incremental increase in caffeine, how much does the time studying increase? Or you might recognize this as the slope of the least squares regression line. So this is the slope, and this would be equal to 0.164. Now this information right over here, it tells us how well our least squares regression line fits the data. R squared you might already be familiar with."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Or you might recognize this as the slope of the least squares regression line. So this is the slope, and this would be equal to 0.164. Now this information right over here, it tells us how well our least squares regression line fits the data. R squared you might already be familiar with. It says how much of the variance in the y variable is explainable by the x variable. If it was one or 100%, that means all of it could be explained, and it's a very good fit. If it was zero, that means none of it can be explained."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "R squared you might already be familiar with. It says how much of the variance in the y variable is explainable by the x variable. If it was one or 100%, that means all of it could be explained, and it's a very good fit. If it was zero, that means none of it can be explained. It would be a very bad fit. Capital S, this is the standard deviation of the residuals, and it's another measure of how much these data points vary from this regression line. Now this column right over here is going to prove to be useful for answering the question at hand."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "If it was zero, that means none of it can be explained. It would be a very bad fit. Capital S, this is the standard deviation of the residuals, and it's another measure of how much these data points vary from this regression line. Now this column right over here is going to prove to be useful for answering the question at hand. This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line. You could view this as the estimate of the standard deviation of the sampling distribution of the slope of the regression line. Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Now this column right over here is going to prove to be useful for answering the question at hand. This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line. You could view this as the estimate of the standard deviation of the sampling distribution of the slope of the regression line. Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population. This would sometimes also be called the standard error of the slope of the least squares regression line. Now these last two columns you don't have to worry about in the context of this video."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population. This would sometimes also be called the standard error of the slope of the least squares regression line. Now these last two columns you don't have to worry about in the context of this video. This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low. It's about a 1% chance that you would have gotten these results if there truly was not a relationship between caffeine intake and time studying. But with all of that out of the way, let's actually answer the question."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "Now these last two columns you don't have to worry about in the context of this video. This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low. It's about a 1% chance that you would have gotten these results if there truly was not a relationship between caffeine intake and time studying. But with all of that out of the way, let's actually answer the question. Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate. We don't actually know the standard deviation of the sampling distribution. So the last thing we have to do is figure out what is this critical t value?"}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "But with all of that out of the way, let's actually answer the question. Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate. We don't actually know the standard deviation of the sampling distribution. So the last thing we have to do is figure out what is this critical t value? You can figure it out using either a calculator or using a table. I'll do it using a table, and to do that, we need to know what the degrees of freedom. Well, when you're doing this with a regression slope like we're doing right now, your degrees of freedom are going to be the number of data points you have minus two so our degrees of freedom are going to be 20 minus two which is equal to 18."}, {"video_title": "Confidence interval for the slope of a regression line   AP Statistics   Khan Academy.mp3", "Sentence": "So the last thing we have to do is figure out what is this critical t value? You can figure it out using either a calculator or using a table. I'll do it using a table, and to do that, we need to know what the degrees of freedom. Well, when you're doing this with a regression slope like we're doing right now, your degrees of freedom are going to be the number of data points you have minus two so our degrees of freedom are going to be 20 minus two which is equal to 18. I'm not gonna go into a bunch of depth right now. It actually is beyond the scope of this video for sure as to why you subtract two here, but just so that we can look it up on a table, this is our degrees of freedom. So we care about a 95% confidence level."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "The amount of fuel he uses follows a normal distribution for each part of his commute, but the amount of fuel he uses on the way home varies more. The amounts of fuel he uses for each part of the commute are also independent of each other. Here are summary statistics for the amount of fuel Shinji uses for each part of his commute. So when he goes to work, he uses a mean of 10 liters of fuel with a standard deviation of 1.5 liters. And on the way home, he also has a mean of 10 liters, but there is more variation, there is more spread. He has a standard deviation of two liters. Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So when he goes to work, he uses a mean of 10 liters of fuel with a standard deviation of 1.5 liters. And on the way home, he also has a mean of 10 liters, but there is more variation, there is more spread. He has a standard deviation of two liters. Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. What is the probability that Shinji runs out of fuel? All right, this is really interesting. We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. What is the probability that Shinji runs out of fuel? All right, this is really interesting. We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. They say that right over here, follows a normal distribution. But here we're talking about the total amount of fuel he has to go to work and to go home. So what we wanna do is come up with a total distribution, home and back, I guess you could say."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. They say that right over here, follows a normal distribution. But here we're talking about the total amount of fuel he has to go to work and to go home. So what we wanna do is come up with a total distribution, home and back, I guess you could say. We could call this work plus home, home and back. If you have two random variables that can be described by normal distributions, and you were to define a new random variable as their sum, the distribution of that new random variable will still be a normal distribution, and its mean will be the sum of the means of those other random variables. So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So what we wanna do is come up with a total distribution, home and back, I guess you could say. We could call this work plus home, home and back. If you have two random variables that can be described by normal distributions, and you were to define a new random variable as their sum, the distribution of that new random variable will still be a normal distribution, and its mean will be the sum of the means of those other random variables. So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. He will use a mean of 20 liters in the round trip. Now for the standard deviation from home plus work, you can't just add the standard deviations, going and coming back. But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. He will use a mean of 20 liters in the round trip. Now for the standard deviation from home plus work, you can't just add the standard deviations, going and coming back. But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. And only because they are independent can we add the variances. So what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home. So what's the variance of going to work?"}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. And only because they are independent can we add the variances. So what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home. So what's the variance of going to work? Well, 1.5 squared is, so this will be 1.5 squared. And what's the variance coming home? Well, this is going to be two squared, two squared."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So what's the variance of going to work? Well, 1.5 squared is, so this will be 1.5 squared. And what's the variance coming home? Well, this is going to be two squared, two squared. Well, this is 2.25 plus four, which is equal to 6.25. So the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "Well, this is going to be two squared, two squared. Well, this is 2.25 plus four, which is equal to 6.25. So the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. So we have this normal distribution that might look something like this. We know its mean is 20 liters. So this is 20 liters."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. So we have this normal distribution that might look something like this. We know its mean is 20 liters. So this is 20 liters. And we wanna know what is the probability that Shinji runs out of fuel? Well, to run out of fuel, he would need to require more than 25 liters of fuel. So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So this is 20 liters. And we wanna know what is the probability that Shinji runs out of fuel? Well, to run out of fuel, he would need to require more than 25 liters of fuel. So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters. He actually has 25 liters in his tank. So how do we figure out that area right over there?"}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters. He actually has 25 liters in his tank. So how do we figure out that area right over there? Well, we could use a z-table. We could say how many standard deviations above the mean is 25 liters? Well, it is five liters above the mean."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So how do we figure out that area right over there? Well, we could use a z-table. We could say how many standard deviations above the mean is 25 liters? Well, it is five liters above the mean. So let me write this down. So the z here, the z is equal to 25 minus the mean, minus 20, divided by the standard deviation for, I guess you could say, this combined normal distribution. This is two standard deviations above the mean, or a z-score of plus two."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "Well, it is five liters above the mean. So let me write this down. So the z here, the z is equal to 25 minus the mean, minus 20, divided by the standard deviation for, I guess you could say, this combined normal distribution. This is two standard deviations above the mean, or a z-score of plus two. So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean. And then if we subtract that from one, we will get the area that we care about. So let's get our z-table out."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "This is two standard deviations above the mean, or a z-score of plus two. So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean. And then if we subtract that from one, we will get the area that we care about. So let's get our z-table out. We care about a z-score of exactly two. So 2.00 is right over here,.9772. So that tells us that this area right over here is 0.9772."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So let's get our z-table out. We care about a z-score of exactly two. So 2.00 is right over here,.9772. So that tells us that this area right over here is 0.9772. And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables   Khan Academy.mp3", "Sentence": "So that tells us that this area right over here is 0.9772. And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228. Did I do that right? I think I did that right. Yes, 0.0228 is the probability that Shinji runs out of fuel."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So you can have a sample study, and we've already talked about this in several videos, but we'll go over it again in this one. You can have an observational study. Observational study. Or you can have an experiment. Experiment. So let's go through each of these, and always pause this video and see if you can think about what these words likely mean, or you might already know. Well, sample study we have looked at, this is really where you're trying to estimate the value of a parameter for a population."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Or you can have an experiment. Experiment. So let's go through each of these, and always pause this video and see if you can think about what these words likely mean, or you might already know. Well, sample study we have looked at, this is really where you're trying to estimate the value of a parameter for a population. So what's an example of that? So let's say we take the population of people in a city, and so that could be hundreds of thousands of people, and the parameter that you care about is how much time, on average, do they spend on a computer. So the parameter would be for the entire population."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Well, sample study we have looked at, this is really where you're trying to estimate the value of a parameter for a population. So what's an example of that? So let's say we take the population of people in a city, and so that could be hundreds of thousands of people, and the parameter that you care about is how much time, on average, do they spend on a computer. So the parameter would be for the entire population. If it was possible, you would go talk to every, maybe there's a million people in the city, you would talk to all million of those people, and ask them how much time they spend on a computer, and you would get the average, and then that would be the parameter. So population parameter, would be average time on a computer per day. Average daily time on a computer."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So the parameter would be for the entire population. If it was possible, you would go talk to every, maybe there's a million people in the city, you would talk to all million of those people, and ask them how much time they spend on a computer, and you would get the average, and then that would be the parameter. So population parameter, would be average time on a computer per day. Average daily time on a computer. Now you determine that it's impractical to go talk to everyone, so you're not going to be able to figure out the exact population parameter, average daily time on a computer, so instead you do a sample study. You randomly sample, and there's a lot of thought in thinking about whether your sample is truly random, so you randomly sample, but there's also different techniques of randomly sampling. So you randomly sample people from your population, and then you take the average daily time on a computer for your sample, and that is going to be an estimate for the population parameter."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Average daily time on a computer. Now you determine that it's impractical to go talk to everyone, so you're not going to be able to figure out the exact population parameter, average daily time on a computer, so instead you do a sample study. You randomly sample, and there's a lot of thought in thinking about whether your sample is truly random, so you randomly sample, but there's also different techniques of randomly sampling. So you randomly sample people from your population, and then you take the average daily time on a computer for your sample, and that is going to be an estimate for the population parameter. So that's your classic sample study. Now in an observational study, you're not trying to estimate a parameter, you're trying to understand how two parameters in a population might move together or not. So let's say that you have a population now."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So you randomly sample people from your population, and then you take the average daily time on a computer for your sample, and that is going to be an estimate for the population parameter. So that's your classic sample study. Now in an observational study, you're not trying to estimate a parameter, you're trying to understand how two parameters in a population might move together or not. So let's say that you have a population now. Let's say you have a population of 1,000 people. And you're curious about whether average daily time on a computer, how it relates to people's blood pressure. So average computer time, actually let me write it this way, instead of average computer time, it should just be computer time."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say that you have a population now. Let's say you have a population of 1,000 people. And you're curious about whether average daily time on a computer, how it relates to people's blood pressure. So average computer time, actually let me write it this way, instead of average computer time, it should just be computer time. Computer time versus blood pressure. So what you do is you apply a survey to all 1,000 people, and you ask them how much time you spend on a computer, and what is your blood pressure? Or maybe you measure it in some way."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So average computer time, actually let me write it this way, instead of average computer time, it should just be computer time. Computer time versus blood pressure. So what you do is you apply a survey to all 1,000 people, and you ask them how much time you spend on a computer, and what is your blood pressure? Or maybe you measure it in some way. And then you plot it all, you look at the data, and you see if those two variables move together. So what does that mean? Well, let me draw."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Or maybe you measure it in some way. And then you plot it all, you look at the data, and you see if those two variables move together. So what does that mean? Well, let me draw. So if this axis is, let's say this is computer time, computer time, and this axis is blood pressure. Blood pressure. So let's say that there's one person who doesn't spend a lot of time on a computer, and they have a relatively low blood pressure."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Well, let me draw. So if this axis is, let's say this is computer time, computer time, and this axis is blood pressure. Blood pressure. So let's say that there's one person who doesn't spend a lot of time on a computer, and they have a relatively low blood pressure. There's another person who spends a lot of time, has high blood pressure. There could be someone who doesn't spend much time on a computer, but has a reasonably high blood pressure. But you keep doing this, and you get all these data points for those 1,000 people."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say that there's one person who doesn't spend a lot of time on a computer, and they have a relatively low blood pressure. There's another person who spends a lot of time, has high blood pressure. There could be someone who doesn't spend much time on a computer, but has a reasonably high blood pressure. But you keep doing this, and you get all these data points for those 1,000 people. And I'm not going to sit here and draw 1,000 points. But you see something like this. And so you see, hey look, it looks like, there's definitely some outliers, but it looks like these two variables move together."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "But you keep doing this, and you get all these data points for those 1,000 people. And I'm not going to sit here and draw 1,000 points. But you see something like this. And so you see, hey look, it looks like, there's definitely some outliers, but it looks like these two variables move together. It looks like in general, the more computer time, the higher the blood pressure, or the higher the blood pressure, the more computer time. And so you can make a conclusion here about these two variables correlating, that they're positively correlated. There is a positive, a reasonable conclusion, if you did the study appropriately, would be that more computer time correlates with higher blood pressure, or that higher blood pressure correlates with more computer time."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so you see, hey look, it looks like, there's definitely some outliers, but it looks like these two variables move together. It looks like in general, the more computer time, the higher the blood pressure, or the higher the blood pressure, the more computer time. And so you can make a conclusion here about these two variables correlating, that they're positively correlated. There is a positive, a reasonable conclusion, if you did the study appropriately, would be that more computer time correlates with higher blood pressure, or that higher blood pressure correlates with more computer time. Now when you do these observational studies, or when you interpret these observational studies, when you read about someone else's, it's very important not to say, oh well, this shows me that computer time causes blood pressure. Because this is not showing causality. And you also can't say, maybe you might say somehow blood pressure causes more people to spend time in front of a computer."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "There is a positive, a reasonable conclusion, if you did the study appropriately, would be that more computer time correlates with higher blood pressure, or that higher blood pressure correlates with more computer time. Now when you do these observational studies, or when you interpret these observational studies, when you read about someone else's, it's very important not to say, oh well, this shows me that computer time causes blood pressure. Because this is not showing causality. And you also can't say, maybe you might say somehow blood pressure causes more people to spend time in front of a computer. That seems even a little bit sillier, but they're actually the same. Because all you're saying is that there's a correlation, these two variables move together. You can't make a conclusion about causality, that computer time causes blood pressure, or that blood pressure causes some time, high blood pressure causes more computer time."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And you also can't say, maybe you might say somehow blood pressure causes more people to spend time in front of a computer. That seems even a little bit sillier, but they're actually the same. Because all you're saying is that there's a correlation, these two variables move together. You can't make a conclusion about causality, that computer time causes blood pressure, or that blood pressure causes some time, high blood pressure causes more computer time. Why can't you make that? Well, there could be what's called a confounding variable. Sometimes called a lurking variable."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "You can't make a conclusion about causality, that computer time causes blood pressure, or that blood pressure causes some time, high blood pressure causes more computer time. Why can't you make that? Well, there could be what's called a confounding variable. Sometimes called a lurking variable. Where, let's say that, so this is computer time. Computer time. And this is blood pressure, I'll just write it like that."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Sometimes called a lurking variable. Where, let's say that, so this is computer time. Computer time. And this is blood pressure, I'll just write it like that. Blood, looks like building. So blood, blood pressure. And it looks like these two things move together."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And this is blood pressure, I'll just write it like that. Blood, looks like building. So blood, blood pressure. And it looks like these two things move together. We saw that right over here in our data. But there could be a root variable that drives both of these. A confounding variable."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And it looks like these two things move together. We saw that right over here in our data. But there could be a root variable that drives both of these. A confounding variable. And that could just be the amount of physical activity someone has. So there could just be a lack of physical activity driving both. Lack of activity."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "A confounding variable. And that could just be the amount of physical activity someone has. So there could just be a lack of physical activity driving both. Lack of activity. People who are less active spend more time in front of a computer, and people who are less active have higher, have higher blood pressure. And if you were to control for this, if you were to take a bunch of people who had a similar lack of activity, or had a similar level of activity, you might see that computer time does not correlate with blood pressure. That these are just both driven by the same thing."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Lack of activity. People who are less active spend more time in front of a computer, and people who are less active have higher, have higher blood pressure. And if you were to control for this, if you were to take a bunch of people who had a similar lack of activity, or had a similar level of activity, you might see that computer time does not correlate with blood pressure. That these are just both driven by the same thing. And what you're really seeing here is like, okay, people with high lack of activity who aren't active, well it drives both of these variables. So once again, when you do this observational study, and if you do it well, you can draw correlations, and that might give you decent hypotheses for causality, but this does not show causality. Because you could have these confounding variables."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "That these are just both driven by the same thing. And what you're really seeing here is like, okay, people with high lack of activity who aren't active, well it drives both of these variables. So once again, when you do this observational study, and if you do it well, you can draw correlations, and that might give you decent hypotheses for causality, but this does not show causality. Because you could have these confounding variables. Now experiments, and experiments are the basis of the scientific method. Experiments are all about trying to establish causality. And so what you would do is, if you wanted to do an experiment, you would take, and you probably wouldn't be able to do it with a thousand people, experiments in some way are the hardest to do of all of these."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Because you could have these confounding variables. Now experiments, and experiments are the basis of the scientific method. Experiments are all about trying to establish causality. And so what you would do is, if you wanted to do an experiment, you would take, and you probably wouldn't be able to do it with a thousand people, experiments in some way are the hardest to do of all of these. Maybe you take a hundred people. A hundred people. And to avoid having this confounding variable introduce error into your experiment, you randomly assign these hundred people into two groups."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And so what you would do is, if you wanted to do an experiment, you would take, and you probably wouldn't be able to do it with a thousand people, experiments in some way are the hardest to do of all of these. Maybe you take a hundred people. A hundred people. And to avoid having this confounding variable introduce error into your experiment, you randomly assign these hundred people into two groups. So random assign. So random. It's very important that they're randomly assigned."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And to avoid having this confounding variable introduce error into your experiment, you randomly assign these hundred people into two groups. So random assign. So random. It's very important that they're randomly assigned. And that's nice, you might not know all of the confounding variables there, but it makes it likely that each group will have a same amount of people with lack of activity, or that the lack of activity, or the activity levels on average in each of the groups, when they're randomly assigned, it gives you a better chance that one group doesn't have a significantly different activity level than the other. And then what you do is, you have a control group, and you have a treatment group. Once again, you've randomly assigned them."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "It's very important that they're randomly assigned. And that's nice, you might not know all of the confounding variables there, but it makes it likely that each group will have a same amount of people with lack of activity, or that the lack of activity, or the activity levels on average in each of the groups, when they're randomly assigned, it gives you a better chance that one group doesn't have a significantly different activity level than the other. And then what you do is, you have a control group, and you have a treatment group. Once again, you've randomly assigned them. So control, and then treatment. And what you might say is, okay, for some amount of time, all of you in the control group can only spend a max of 30 minutes in front of a computer. And on the, or maybe if you really wanted to do it, you say you have to spend exactly 30 minutes on a computer, and that's maybe a little unrealistic."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, you've randomly assigned them. So control, and then treatment. And what you might say is, okay, for some amount of time, all of you in the control group can only spend a max of 30 minutes in front of a computer. And on the, or maybe if you really wanted to do it, you say you have to spend exactly 30 minutes on a computer, and that's maybe a little unrealistic. And then the treatment group, you have to say, you have to spend exactly two hours in front of a computer. And I'm making up these numbers at random. And it would be nice to see, okay, what was everyone's blood pressure before the experiment?"}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And on the, or maybe if you really wanted to do it, you say you have to spend exactly 30 minutes on a computer, and that's maybe a little unrealistic. And then the treatment group, you have to say, you have to spend exactly two hours in front of a computer. And I'm making up these numbers at random. And it would be nice to see, okay, what was everyone's blood pressure before the experiment? And you can say, okay, well, the averages are similar going into the experiment. And then you go some amount of time, and you measure blood pressure. And if you see that, wow, this group definitely has a higher blood pressure."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And it would be nice to see, okay, what was everyone's blood pressure before the experiment? And you can say, okay, well, the averages are similar going into the experiment. And then you go some amount of time, and you measure blood pressure. And if you see that, wow, this group definitely has a higher blood pressure. This group has a higher blood pressure. So the blood pressure, the blood pressure is higher here. And once again, some of this might have just happened randomly."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And if you see that, wow, this group definitely has a higher blood pressure. This group has a higher blood pressure. So the blood pressure, the blood pressure is higher here. And once again, some of this might have just happened randomly. It might have been, you know, the people you happened to put in there, et cetera, et cetera. But depending, if this was a large enough experiment, and you conducted it well, this says, hey, look, there's, I'm feeling like there's a causality here. That by making these people spend more time in front of a computer, that that actually raised their blood pressure."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, some of this might have just happened randomly. It might have been, you know, the people you happened to put in there, et cetera, et cetera. But depending, if this was a large enough experiment, and you conducted it well, this says, hey, look, there's, I'm feeling like there's a causality here. That by making these people spend more time in front of a computer, that that actually raised their blood pressure. So once again, sample study, you're trying to estimate a population parameter. Observation study, you are seeing if there is a correlation between two things. And you have to be careful not to say, hey, one is causing the other, because you could have confounding variables."}, {"video_title": "Types of statistical studies   Study design   AP Statistics   Khan Academy.mp3", "Sentence": "That by making these people spend more time in front of a computer, that that actually raised their blood pressure. So once again, sample study, you're trying to estimate a population parameter. Observation study, you are seeing if there is a correlation between two things. And you have to be careful not to say, hey, one is causing the other, because you could have confounding variables. Experiment, you're trying to establish or show causality. And you do that by taking your group, randomly assigning to a controller treatment that should evenly, or hopefully evenly distribute, not always, there's some chance it doesn't, but distribute the confounding variables. And then you, on each group, you change how much of one of these variables they get, and you see if it drives the other variable."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The table below displays the probability distribution of X, the number of shots that Anoush makes in a set of two attempts, along with some summary statistics. So here's the random variable X. It's a discrete random variable. It only takes on a finite number of values. Sometimes people say it takes on a countable number of values, but we see he can either make zero free throws, one, or two of the two. And the probability that he makes zero is here, one is here, two is here. And then they also give us the mean of X and the standard deviation of X."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It only takes on a finite number of values. Sometimes people say it takes on a countable number of values, but we see he can either make zero free throws, one, or two of the two. And the probability that he makes zero is here, one is here, two is here. And then they also give us the mean of X and the standard deviation of X. Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N? All right, so let's define a new random variable N, which is equal to his net gain, net gain. We can define N in terms of X."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And then they also give us the mean of X and the standard deviation of X. Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N? All right, so let's define a new random variable N, which is equal to his net gain, net gain. We can define N in terms of X. What is his net gain going to be? Well, let's see, N, it's going to be equal to 10 times however many shots he makes. So it's gonna be 10 times X."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We can define N in terms of X. What is his net gain going to be? Well, let's see, N, it's going to be equal to 10 times however many shots he makes. So it's gonna be 10 times X. And then no matter what, he has to pay $15 to play, minus 15. In fact, we could set up a little table here for the probability distribution of N. So let me make it right over here. So I'll make it look just like this one."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it's gonna be 10 times X. And then no matter what, he has to pay $15 to play, minus 15. In fact, we could set up a little table here for the probability distribution of N. So let me make it right over here. So I'll make it look just like this one. N is equal to net gain. And here we'll have the probability of N. And there's three outcomes here. The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So I'll make it look just like this one. N is equal to net gain. And here we'll have the probability of N. And there's three outcomes here. The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. And it would have the same probability, 0.16. When he makes one shot, the net gain is gonna be 10 times one minus 15, which is negative five. But it's gonna have the same probability."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. And it would have the same probability, 0.16. When he makes one shot, the net gain is gonna be 10 times one minus 15, which is negative five. But it's gonna have the same probability. He has a 48% chance of making one shot, and so it's a 48% chance of losing $5. And then last but not least, when X is two, his net gain is going to be positive five, plus five. And so this is a 0.36 chance."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But it's gonna have the same probability. He has a 48% chance of making one shot, and so it's a 48% chance of losing $5. And then last but not least, when X is two, his net gain is going to be positive five, plus five. And so this is a 0.36 chance. So what they want us to figure out are what are the mean and standard deviation of his net gain? So let's first figure out the mean of N. Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is a 0.36 chance. So what they want us to figure out are what are the mean and standard deviation of his net gain? So let's first figure out the mean of N. Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15. This is 1.2. So it is 12 minus 15, which is equal to negative three."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15. This is 1.2. So it is 12 minus 15, which is equal to negative three. Now the standard deviation of N is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it is 12 minus 15, which is equal to negative three. Now the standard deviation of N is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. So this is going to be equal to 10 times the standard deviation of X. Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. So this is going to be equal to 10 times the standard deviation of X. Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, well, your spread should grow by the amount that you're scaling it. But by shifting it, it doesn't affect how much you disperse from the mean. So standard deviation is only affected by the scaling, but not by the shifting here."}, {"video_title": "Example Transforming a discrete random variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, well, your spread should grow by the amount that you're scaling it. But by shifting it, it doesn't affect how much you disperse from the mean. So standard deviation is only affected by the scaling, but not by the shifting here. So this is going to be 10 times 0.69, which is going to, this was an approximation, so I'll say this is approximately equal to 6.9. So this is our new distribution for our net gain. This is the mean of our net gain, and this is roughly the standard deviation of our net gain."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It's the number of successes from n trials, so it's a finite number of trials, where the probability of success is equal to p, so the probability is constant across the trials, for each of these independent trials. So the probability of success in one trial is not dependent on what happened in the other trials. And we also talked in that previous video where we talked about the expected value of this binomial variable, is we said, hey, it could be viewed, this binomial variable can be viewed as the sum of n of what you could really consider to be a Bernoulli variable here. So this variable, this random variable y, the probability that it's equal to one, you could view that as a success, is equal to p. The probability that it's a failure, that y is equal to zero, is one minus p. So you could view y, the outcome of y is really the, or whether y is one or zero is really whether we had a success or not in each of these trials. So if you add up n y's, then you are going to get x. And we use that information to figure out what the expected value of x is going to be. Because the expected value of y is pretty straightforward to directly compute."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this variable, this random variable y, the probability that it's equal to one, you could view that as a success, is equal to p. The probability that it's a failure, that y is equal to zero, is one minus p. So you could view y, the outcome of y is really the, or whether y is one or zero is really whether we had a success or not in each of these trials. So if you add up n y's, then you are going to get x. And we use that information to figure out what the expected value of x is going to be. Because the expected value of y is pretty straightforward to directly compute. Expected value of y is just probability-weighted outcomes. So it's p times one plus one minus p, one minus p, times zero, times zero. This whole term's gonna be zero."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Because the expected value of y is pretty straightforward to directly compute. Expected value of y is just probability-weighted outcomes. So it's p times one plus one minus p, one minus p, times zero, times zero. This whole term's gonna be zero. And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review. We could say that the expected value of x is just going to be equal to, we know from our expected value properties, that's going to be equal to the sum of the expected values of these n y's. Or you could say it is n times the expected value, times the expected value of y."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This whole term's gonna be zero. And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review. We could say that the expected value of x is just going to be equal to, we know from our expected value properties, that's going to be equal to the sum of the expected values of these n y's. Or you could say it is n times the expected value, times the expected value of y. The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to. Because we could see, we know from our variance properties, you can't do this with standard deviation, but you could do it with variance. And then once you figure out the variance, you just take the square root for the standard deviation."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Or you could say it is n times the expected value, times the expected value of y. The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to. Because we could see, we know from our variance properties, you can't do this with standard deviation, but you could do it with variance. And then once you figure out the variance, you just take the square root for the standard deviation. The variance of x is similarly going to be the sum of the variances of these n y's. So it's going to be similarly n times the variance, n times the variance of y. So this all boils down to what is the variance of y going to be equal to?"}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And then once you figure out the variance, you just take the square root for the standard deviation. The variance of x is similarly going to be the sum of the variances of these n y's. So it's going to be similarly n times the variance, n times the variance of y. So this all boils down to what is the variance of y going to be equal to? So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here. All right, so we want to figure out the variance of y. So variance of y is going to be equal to what?"}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this all boils down to what is the variance of y going to be equal to? So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here. All right, so we want to figure out the variance of y. So variance of y is going to be equal to what? Well here it's going to be the probability squared distances from the expected value. So we have a probability of p, where what is going to be our squared distance from the expected value? Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So variance of y is going to be equal to what? Well here it's going to be the probability squared distances from the expected value. So we have a probability of p, where what is going to be our squared distance from the expected value? Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight. And then we have, actually let me scroll over a little bit, well I'll just do it right over here. Plus we have a probability of one minus p, one minus p for the other possible outcome."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight. And then we have, actually let me scroll over a little bit, well I'll just do it right over here. Plus we have a probability of one minus p, one minus p for the other possible outcome. So in that outcome, we are at zero. And the difference between zero and our expected value, well that's just going to be zero minus p. And once again, we are going to square that distance. And so this is the expression, or square that quantity."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Plus we have a probability of one minus p, one minus p for the other possible outcome. So in that outcome, we are at zero. And the difference between zero and our expected value, well that's just going to be zero minus p. And once again, we are going to square that distance. And so this is the expression, or square that quantity. And so this is the expression for the variance of y, and we can simplify it a little bit. So this is all going to be equal to, so let me just, p times one minus p squared. And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with?"}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is the expression, or square that quantity. And so this is the expression for the variance of y, and we can simplify it a little bit. So this is all going to be equal to, so let me just, p times one minus p squared. And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out. This is just, this whole thing is just a one. So you're left with p times one minus p, which is indeed the variance for a binomial variable."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out. This is just, this whole thing is just a one. So you're left with p times one minus p, which is indeed the variance for a binomial variable. We actually proved that in other videos. I guess it doesn't hurt to see it again. But there you have it."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So you're left with p times one minus p, which is indeed the variance for a binomial variable. We actually proved that in other videos. I guess it doesn't hurt to see it again. But there you have it. We know what the variance of y is. It is p times one minus p. And the variance of x is just n times the variance of y. So there we go."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But there you have it. We know what the variance of y is. It is p times one minus p. And the variance of x is just n times the variance of y. So there we go. We deserve a little bit of a drum roll. The variance of x is equal to n times p times one minus p. So if we were to take the concrete example of the last video, where if I were to take 10 free throws, so each trial is a shot, is a free throw. So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So there we go. We deserve a little bit of a drum roll. The variance of x is equal to n times p times one minus p. So if we were to take the concrete example of the last video, where if I were to take 10 free throws, so each trial is a shot, is a free throw. So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. And so that would be what? This right over here, so this would be equal to 10 times 0.3 times 0.7 times 0.21. So my variance in this situation is going to be equal to 2.1."}, {"video_title": "Variance of a binomial variable   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. And so that would be what? This right over here, so this would be equal to 10 times 0.3 times 0.7 times 0.21. So my variance in this situation is going to be equal to 2.1. It is equal to 2.1. And if I wanted to figure out the standard deviation of this right over here, I would just take the square root of this. So if you want the standard deviation, just take the square root of this expression right over here."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "The rest are fair coins. So if three are unfair, the rest are eight coins. And when I say that, or when this problem says that they are fair coins, it means that they have a 50-50 chance of coming up either heads or tails. You randomly choose one coin from the bag and flip it two times. What is the percent probability of getting two heads? So this is an interesting question, but if we break it down, essentially with a decision tree, it'll help break it down a little bit better. So let's say that, so we have a bag."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "You randomly choose one coin from the bag and flip it two times. What is the percent probability of getting two heads? So this is an interesting question, but if we break it down, essentially with a decision tree, it'll help break it down a little bit better. So let's say that, so we have a bag. Three of them are unfair. So we can even visualize the bag. You don't have to do this all the time."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that, so we have a bag. Three of them are unfair. So we can even visualize the bag. You don't have to do this all the time. So we have, out of the fair coins in white, one, two, three, four, four, five fair coins. And we have three unfair coins. One, two, three, and this whole thing is my bag right over here."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "You don't have to do this all the time. So we have, out of the fair coins in white, one, two, three, four, four, five fair coins. And we have three unfair coins. One, two, three, and this whole thing is my bag right over here. That is my bag of coins. If I were to, when I take my hand in, if I were to take any of these white coins, there's a 50% chance that it gets heads on any flip. The odds of getting two heads in a row would be 50% times 50% for these white coins."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "One, two, three, and this whole thing is my bag right over here. That is my bag of coins. If I were to, when I take my hand in, if I were to take any of these white coins, there's a 50% chance that it gets heads on any flip. The odds of getting two heads in a row would be 50% times 50% for these white coins. But I don't know I'm going to get a white coin. If I get one of these orange coins, I have a 60% chance of coming up heads. And the odds of coming up 60%, if I have picked one of these orange coins, the probability of getting heads twice is going to be 60% times 60%."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "The odds of getting two heads in a row would be 50% times 50% for these white coins. But I don't know I'm going to get a white coin. If I get one of these orange coins, I have a 60% chance of coming up heads. And the odds of coming up 60%, if I have picked one of these orange coins, the probability of getting heads twice is going to be 60% times 60%. So how do I factor in this idea that I don't know if I've picked a white fair coin or an orange unfair coin? We'll assume that the coins actually aren't white and orange. They all look like regular coins."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the odds of coming up 60%, if I have picked one of these orange coins, the probability of getting heads twice is going to be 60% times 60%. So how do I factor in this idea that I don't know if I've picked a white fair coin or an orange unfair coin? We'll assume that the coins actually aren't white and orange. They all look like regular coins. So what I'll do is I'll draw a little bit of a decision tree here. I guess maybe I could call it a probability tree. So there's some probability that I pick a fair coin."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "They all look like regular coins. So what I'll do is I'll draw a little bit of a decision tree here. I guess maybe I could call it a probability tree. So there's some probability that I pick a fair coin. And there's some probability that I pick an unfair coin. And so what is the probability that I pick a fair coin? Well, 1, 2, 3, 4, 5 out of the total 8 coins are fair."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's some probability that I pick a fair coin. And there's some probability that I pick an unfair coin. And so what is the probability that I pick a fair coin? Well, 1, 2, 3, 4, 5 out of the total 8 coins are fair. So there is a 5 8's probability. I'll write it here on the branch, actually. So there's a 5 8's chance that I pick a fair coin."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, 1, 2, 3, 4, 5 out of the total 8 coins are fair. So there is a 5 8's probability. I'll write it here on the branch, actually. So there's a 5 8's chance that I pick a fair coin. And then there is a 3, 1, 2, 3 out of 8 chance that I pick an unfair coin. So if I were to just tell you what's the probability of picking a fair coin, you'd say, oh, 5 8's. What's the probability of an unfair coin?"}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a 5 8's chance that I pick a fair coin. And then there is a 3, 1, 2, 3 out of 8 chance that I pick an unfair coin. So if I were to just tell you what's the probability of picking a fair coin, you'd say, oh, 5 8's. What's the probability of an unfair coin? 3 8's. And you could convert that to a decimal or a percentage or whatever you'd like. Now, given that I have picked a fair coin, what is the probability that I will get heads twice?"}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's the probability of an unfair coin? 3 8's. And you could convert that to a decimal or a percentage or whatever you'd like. Now, given that I have picked a fair coin, what is the probability that I will get heads twice? So let me write it this way. And this is just notation right here. So the probability of, I'll call it heads, heads, so I get two heads in a row, given that I have a fair coin."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, given that I have picked a fair coin, what is the probability that I will get heads twice? So let me write it this way. And this is just notation right here. So the probability of, I'll call it heads, heads, so I get two heads in a row, given that I have a fair coin. It looks like very fancy notation, but it's just like, look, if you knew for a fact that that coin you had is absolutely fair, that it has a 50% chance of coming up heads, what is the probability of getting two heads in a row? Well, then we can just say, well, that's just going to be 50%. So 50% times 50%, which is equal to 25%."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of, I'll call it heads, heads, so I get two heads in a row, given that I have a fair coin. It looks like very fancy notation, but it's just like, look, if you knew for a fact that that coin you had is absolutely fair, that it has a 50% chance of coming up heads, what is the probability of getting two heads in a row? Well, then we can just say, well, that's just going to be 50%. So 50% times 50%, which is equal to 25%. Now, what is the probability that you, so if you want to know what is the probability that you picked a fair coin and you got two heads in a row? So given that you have a fair coin, it's a 25% chance that you have two heads in a row. But the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5 8ths times the 25%."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 50% times 50%, which is equal to 25%. Now, what is the probability that you, so if you want to know what is the probability that you picked a fair coin and you got two heads in a row? So given that you have a fair coin, it's a 25% chance that you have two heads in a row. But the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5 8ths times the 25%. So the probability, so this whole branch, I should maybe draw it this way, the probability of this whole series of events happening. So starting with you picking the fair coin and then getting two heads in a row will be, I'll write it this way, it will be 5 8ths, 5 over 8, times this right over here, times the 0.25. I want to make it very clear."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5 8ths times the 25%. So the probability, so this whole branch, I should maybe draw it this way, the probability of this whole series of events happening. So starting with you picking the fair coin and then getting two heads in a row will be, I'll write it this way, it will be 5 8ths, 5 over 8, times this right over here, times the 0.25. I want to make it very clear. The 0.25 is the probability of getting two heads in a row given that you knew that you got a fair coin. But the probability of this whole series of events happening, you would have to multiply this times the probability that you actually got a fair coin. So another way of thinking about it is this is the probability that you got a fair coin and that you have two heads in a row."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "I want to make it very clear. The 0.25 is the probability of getting two heads in a row given that you knew that you got a fair coin. But the probability of this whole series of events happening, you would have to multiply this times the probability that you actually got a fair coin. So another way of thinking about it is this is the probability that you got a fair coin and that you have two heads in a row. Now let's do the same thing for the unfair coin. So the probability, I'll do that in the same green color, the probability that I get heads heads given that my coin is unfair. So if you were to somehow know that your coin is unfair, what is the probability of getting two heads in a row?"}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So another way of thinking about it is this is the probability that you got a fair coin and that you have two heads in a row. Now let's do the same thing for the unfair coin. So the probability, I'll do that in the same green color, the probability that I get heads heads given that my coin is unfair. So if you were to somehow know that your coin is unfair, what is the probability of getting two heads in a row? Well, in this unfair coin, it has a 60% chance of coming up heads. So it will be equal to 0.6 times 0.6, which is 0.36. So if you have an unfair coin, if you know for a fact that you have an unfair coin, if that is a given, you have a 36% chance of getting two heads in a row."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you were to somehow know that your coin is unfair, what is the probability of getting two heads in a row? Well, in this unfair coin, it has a 60% chance of coming up heads. So it will be equal to 0.6 times 0.6, which is 0.36. So if you have an unfair coin, if you know for a fact that you have an unfair coin, if that is a given, you have a 36% chance of getting two heads in a row. Now, if you want to know the probability of this whole series of events, the probability that you picked an unfair coin and you get two heads in a row, so the probability of unfair and two heads in a row, given that you had that unfair coin, you would multiply this 3 8ths times the 0.36. So this will be equal to the 3 8ths times 0.36. And so let's get a calculator out and calculate these."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you have an unfair coin, if you know for a fact that you have an unfair coin, if that is a given, you have a 36% chance of getting two heads in a row. Now, if you want to know the probability of this whole series of events, the probability that you picked an unfair coin and you get two heads in a row, so the probability of unfair and two heads in a row, given that you had that unfair coin, you would multiply this 3 8ths times the 0.36. So this will be equal to the 3 8ths times 0.36. And so let's get a calculator out and calculate these. So if I take 5 divided by 8 times 0.25, I get 15 point, or I'll just write it as a decimal, 0.15625. So this is equal to 0.15625. And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so let's get a calculator out and calculate these. So if I take 5 divided by 8 times 0.25, I get 15 point, or I'll just write it as a decimal, 0.15625. So this is equal to 0.15625. And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135. So this is 0.135. So if someone were to ask you, what's the probability of picking the fair coin and then getting two heads in a row with that fair coin, you would get this number. If someone were to say, what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin, you would get this number."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135. So this is 0.135. So if someone were to ask you, what's the probability of picking the fair coin and then getting two heads in a row with that fair coin, you would get this number. If someone were to say, what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin, you would get this number. Now if someone were to say, either way, what's the probability of getting two heads in a row? Because that's what they're asking us here. What is the probability of getting two heads?"}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "If someone were to say, what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin, you would get this number. Now if someone were to say, either way, what's the probability of getting two heads in a row? Because that's what they're asking us here. What is the probability of getting two heads? So we could get it through this method, by chance picking the fair coin, or through this method, by chance picking the unfair coin. So since we can do it either way, we can sum up the probabilities. Either of these events meet our constraints."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of getting two heads? So we could get it through this method, by chance picking the fair coin, or through this method, by chance picking the unfair coin. So since we can do it either way, we can sum up the probabilities. Either of these events meet our constraints. So we can just add these two things up. So let's do that. So we can add 0.135 plus 0.15625 gives us 0.29125."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "Either of these events meet our constraints. So we can just add these two things up. So let's do that. So we can add 0.135 plus 0.15625 gives us 0.29125. So 0.29125, that's when we add 0.15625 plus 0.135 will equal this. And if we want to write it as a percentage, you essentially just multiply this times 100 and add the percentage sign there. So this is equal to 29.125%."}, {"video_title": "Dependent probability example   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can add 0.135 plus 0.15625 gives us 0.29125. So 0.29125, that's when we add 0.15625 plus 0.135 will equal this. And if we want to write it as a percentage, you essentially just multiply this times 100 and add the percentage sign there. So this is equal to 29.125%. Or if we were to round to the nearest hundredths, then this would be the exact number, or we could say it's approximately 29.13%, depending on how much we need to round it. So we have a little less than a third chance of this happening. And the reason why, remember, if it was a fair coin, there would only be, if everything in the bag was a fair coin, there'd be a 25% chance of this happening."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to test the hypothesis that more than 30% of US households have internet access, with a significance level of 5%. We collect a sample of 150 households and find that 57 have access. So to do our hypothesis test, let's just establish our null hypothesis and our alternative hypothesis. So our null hypothesis is that the hypothesis is not correct. Our null hypothesis is that the proportion of US households that have internet access is less than or equal to 30%. And our alternative hypothesis is what our hypothesis actually is, is that the proportion is greater than 30%. We see it over here."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our null hypothesis is that the hypothesis is not correct. Our null hypothesis is that the proportion of US households that have internet access is less than or equal to 30%. And our alternative hypothesis is what our hypothesis actually is, is that the proportion is greater than 30%. We see it over here. We want to test the hypothesis that more than 30% of US households have internet access. That's that right here. This is what we're testing."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "We see it over here. We want to test the hypothesis that more than 30% of US households have internet access. That's that right here. This is what we're testing. We're testing the alternative hypothesis. And the way we're going to do it is we're going to assume a p-value based on the null hypothesis. We're going to assume a proportion based on the null hypothesis for the population."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is what we're testing. We're testing the alternative hypothesis. And the way we're going to do it is we're going to assume a p-value based on the null hypothesis. We're going to assume a proportion based on the null hypothesis for the population. And given that assumption, what is the probability that 57 out of 150 of our samples actually have internet access? And if that probability is less than 5%, if it's less than our significance level, then we're going to reject the null hypothesis in favor of the alternative one. So let's think about this a little bit."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to assume a proportion based on the null hypothesis for the population. And given that assumption, what is the probability that 57 out of 150 of our samples actually have internet access? And if that probability is less than 5%, if it's less than our significance level, then we're going to reject the null hypothesis in favor of the alternative one. So let's think about this a little bit. So we're going to start off assuming the null hypothesis is true. And in that assumption, we're going to have to pick a population proportion or a population mean. We know that for Bernoulli distributions, they're the same thing."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about this a little bit. So we're going to start off assuming the null hypothesis is true. And in that assumption, we're going to have to pick a population proportion or a population mean. We know that for Bernoulli distributions, they're the same thing. And what I'm going to do is I'm going to pick a proportion so high so that it maximizes the probability of getting this over here. And we actually don't even know what that number is. And actually, so that we can think about it a little more intelligent, let's just find out what our sample portion even is."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that for Bernoulli distributions, they're the same thing. And what I'm going to do is I'm going to pick a proportion so high so that it maximizes the probability of getting this over here. And we actually don't even know what that number is. And actually, so that we can think about it a little more intelligent, let's just find out what our sample portion even is. We had 57 people out of 150 having internet access. So 57 households out of 150. So our sample proportion is 0.38."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And actually, so that we can think about it a little more intelligent, let's just find out what our sample portion even is. We had 57 people out of 150 having internet access. So 57 households out of 150. So our sample proportion is 0.38. So let me write that over here. Our sample proportion is equal to 0.38. So when we assume our null hypothesis to be true, we're going to assume a population proportion that maximizes the probability that we get this over here."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our sample proportion is 0.38. So let me write that over here. Our sample proportion is equal to 0.38. So when we assume our null hypothesis to be true, we're going to assume a population proportion that maximizes the probability that we get this over here. So the highest population proportion that's within our null hypothesis that will maximize the probability of getting this is actually if we're right at 30%. So if we say our population proportion, we're going to assume this is true. This is our null hypothesis."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So when we assume our null hypothesis to be true, we're going to assume a population proportion that maximizes the probability that we get this over here. So the highest population proportion that's within our null hypothesis that will maximize the probability of getting this is actually if we're right at 30%. So if we say our population proportion, we're going to assume this is true. This is our null hypothesis. We're going to assume that it is 0.3 or 30%. And I want you to understand that. If we said 29% would have been in our null hypothesis."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is our null hypothesis. We're going to assume that it is 0.3 or 30%. And I want you to understand that. If we said 29% would have been in our null hypothesis. 28%, that would have been in our null hypothesis. But for 29% or 28%, the probability of getting this would have been even lower. So it wouldn't have been as strong of a test."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we said 29% would have been in our null hypothesis. 28%, that would have been in our null hypothesis. But for 29% or 28%, the probability of getting this would have been even lower. So it wouldn't have been as strong of a test. If we take the maximum proportion that still satisfies our null hypothesis, we're maximizing the probability that we get this. So if that number is still low, if it's still less than 5%, we can feel pretty good about the alternative hypothesis. So just to refresh ourselves, we're going to assume a population proportion of 0.3."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it wouldn't have been as strong of a test. If we take the maximum proportion that still satisfies our null hypothesis, we're maximizing the probability that we get this. So if that number is still low, if it's still less than 5%, we can feel pretty good about the alternative hypothesis. So just to refresh ourselves, we're going to assume a population proportion of 0.3. And if we just think about the distribution, sometimes it's helpful to draw these things. So I will draw it. So this is what the population distribution looks like."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So just to refresh ourselves, we're going to assume a population proportion of 0.3. And if we just think about the distribution, sometimes it's helpful to draw these things. So I will draw it. So this is what the population distribution looks like. Based on our assumption, based on this assumption right over here. Our population distribution has point, or maybe I should write 30% have internet access. And I'll just signify that with a 1."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is what the population distribution looks like. Based on our assumption, based on this assumption right over here. Our population distribution has point, or maybe I should write 30% have internet access. And I'll just signify that with a 1. And then the rest don't have internet access. 70% do not have internet access. This is just a Bernoulli distribution."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll just signify that with a 1. And then the rest don't have internet access. 70% do not have internet access. This is just a Bernoulli distribution. We know that the mean over here is going to be the same thing as the proportion that has internet access. So the mean over here is going to be 0.3, same thing as 30%. This is the population mean."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is just a Bernoulli distribution. We know that the mean over here is going to be the same thing as the proportion that has internet access. So the mean over here is going to be 0.3, same thing as 30%. This is the population mean. And maybe I should write it this way. The mean assuming our null hypothesis is 0.3. And then the population standard deviation, let me write this over here in yellow."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the population mean. And maybe I should write it this way. The mean assuming our null hypothesis is 0.3. And then the population standard deviation, let me write this over here in yellow. The population standard deviation, assuming our null hypothesis, and we've seen this when we first learned about Bernoulli distributions. It is going to be the square root of the percentage of the population that has internet access, so 0.3, times the proportion of the population that does not have internet access, times 0.7 right over here. So this would be the square root of 0.21."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the population standard deviation, let me write this over here in yellow. The population standard deviation, assuming our null hypothesis, and we've seen this when we first learned about Bernoulli distributions. It is going to be the square root of the percentage of the population that has internet access, so 0.3, times the proportion of the population that does not have internet access, times 0.7 right over here. So this would be the square root of 0.21. And we could deal with this later using our calculator. Now, with that out of the way, we want to figure out the probability of getting a sample proportion that has of 0.38. So let's look at the distribution of sample proportions."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this would be the square root of 0.21. And we could deal with this later using our calculator. Now, with that out of the way, we want to figure out the probability of getting a sample proportion that has of 0.38. So let's look at the distribution of sample proportions. So you could literally look at every combination of getting 150 households from this, and you would actually get a binomial distribution. And we've also seen this before. You would actually get a binomial distribution where you'd get a bunch of bars like that."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's look at the distribution of sample proportions. So you could literally look at every combination of getting 150 households from this, and you would actually get a binomial distribution. And we've also seen this before. You would actually get a binomial distribution where you'd get a bunch of bars like that. But if your n is suitably large, and in particular, and this is kind of the test for it, the test if n times p, and in this case, we're saying p is 30%, if n times p is greater than 5, and n times 1 minus p is greater than 5, you can assume that the distribution of the sample proportion is going to be normal. So if you looked at all of the different ways you could sample 150 households from this population, you'd get all of these bars. But since our n is pretty big, it's 150, and 150 times 0.3 is obviously greater than 5, 150 times 0.7 is also greater than 5."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You would actually get a binomial distribution where you'd get a bunch of bars like that. But if your n is suitably large, and in particular, and this is kind of the test for it, the test if n times p, and in this case, we're saying p is 30%, if n times p is greater than 5, and n times 1 minus p is greater than 5, you can assume that the distribution of the sample proportion is going to be normal. So if you looked at all of the different ways you could sample 150 households from this population, you'd get all of these bars. But since our n is pretty big, it's 150, and 150 times 0.3 is obviously greater than 5, 150 times 0.7 is also greater than 5. You can approximate that with a normal distribution. So let me do that. So you can approximate it with a normal distribution."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "But since our n is pretty big, it's 150, and 150 times 0.3 is obviously greater than 5, 150 times 0.7 is also greater than 5. You can approximate that with a normal distribution. So let me do that. So you can approximate it with a normal distribution. So this is a normal distribution right over there. Now, the mean of the distribution of the proportion data that we're assuming is a normal distribution is going to be, and remember, we're working under the context that the null hypothesis is true. So this mean is going to be this mean right here."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you can approximate it with a normal distribution. So this is a normal distribution right over there. Now, the mean of the distribution of the proportion data that we're assuming is a normal distribution is going to be, and remember, we're working under the context that the null hypothesis is true. So this mean is going to be this mean right here. So the mean of our sample proportions is going to be the same thing as our population mean. So this is going to be 0.3, same value as that. And the standard deviation, this comes straight from the central limit theorem."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this mean is going to be this mean right here. So the mean of our sample proportions is going to be the same thing as our population mean. So this is going to be 0.3, same value as that. And the standard deviation, this comes straight from the central limit theorem. So the standard deviation of our sample proportions is going to be the square root. It's going to be our population standard deviation. The standard deviation we're assuming with our null hypothesis divided by the square root of the number of samples we have."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the standard deviation, this comes straight from the central limit theorem. So the standard deviation of our sample proportions is going to be the square root. It's going to be our population standard deviation. The standard deviation we're assuming with our null hypothesis divided by the square root of the number of samples we have. And in this case, we have 150 samples. It's going to be 150 samples, and we can calculate this. This value on top, we just figured out, is the square root of 0.21."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "The standard deviation we're assuming with our null hypothesis divided by the square root of the number of samples we have. And in this case, we have 150 samples. It's going to be 150 samples, and we can calculate this. This value on top, we just figured out, is the square root of 0.21. So this is the square root of 0.21 over the square root of 150, and I can get the calculator out to calculate this, so I'll just do it the way I wrote it. Square root of 0.21, and I'm going to divide that, so whatever answer is, I'm going to divide that by the square root of 150. So it's 0.037."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "This value on top, we just figured out, is the square root of 0.21. So this is the square root of 0.21 over the square root of 150, and I can get the calculator out to calculate this, so I'll just do it the way I wrote it. Square root of 0.21, and I'm going to divide that, so whatever answer is, I'm going to divide that by the square root of 150. So it's 0.037. So we figured out the standard deviation here of the distribution of our sample proportions is going to be, let me write this down, I'll scroll over to the right a little bit. It is 0.037. I think I'm falling off the screen a little bit."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 0.037. So we figured out the standard deviation here of the distribution of our sample proportions is going to be, let me write this down, I'll scroll over to the right a little bit. It is 0.037. I think I'm falling off the screen a little bit. So we'll just say 0.037. Now, to figure out the probability of having a sample proportion of 0.38, we just have to figure out how many standard deviations that is away from our mean, or essentially calculate a z statistic for our sample, because a z statistic or a z score is really just how many standard deviations you are away from the mean, and then figure out whether the probability of getting that z statistic is more or less than 5%. So let's figure out how many standard deviations we are away from the mean."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "I think I'm falling off the screen a little bit. So we'll just say 0.037. Now, to figure out the probability of having a sample proportion of 0.38, we just have to figure out how many standard deviations that is away from our mean, or essentially calculate a z statistic for our sample, because a z statistic or a z score is really just how many standard deviations you are away from the mean, and then figure out whether the probability of getting that z statistic is more or less than 5%. So let's figure out how many standard deviations we are away from the mean. So just to remind ourselves, this sample proportion we got, we can view as just a sample from this distribution of all of the possible sample proportions. So how many standard deviations away from the mean is this? So if we take our sample proportion, subtract from that the mean of the distribution of sample proportions, and divide it by the standard deviation of the distribution of the sample proportions, we get 0.38 minus 0.3, all of that over this value, which we just figured out, was 0.037."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's figure out how many standard deviations we are away from the mean. So just to remind ourselves, this sample proportion we got, we can view as just a sample from this distribution of all of the possible sample proportions. So how many standard deviations away from the mean is this? So if we take our sample proportion, subtract from that the mean of the distribution of sample proportions, and divide it by the standard deviation of the distribution of the sample proportions, we get 0.38 minus 0.3, all of that over this value, which we just figured out, was 0.037. So what does that give us? The numerator over here is just 0.08. The denominator is 0.037."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if we take our sample proportion, subtract from that the mean of the distribution of sample proportions, and divide it by the standard deviation of the distribution of the sample proportions, we get 0.38 minus 0.3, all of that over this value, which we just figured out, was 0.037. So what does that give us? The numerator over here is just 0.08. The denominator is 0.037. So let's figure this out. So our numerator is 0.08 divided by this last number right here, which is the 0.037. So second answer, and we get 2.14 standard deviations."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "The denominator is 0.037. So let's figure this out. So our numerator is 0.08 divided by this last number right here, which is the 0.037. So second answer, and we get 2.14 standard deviations. So this is equal to 2.14 standard deviations. Or we could say that our z statistic, we could call this our z score or our z statistic, the number of standard deviations we are away from our mean is 2.14. And to be exact, we're 2.14 standard deviations above the mean."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So second answer, and we get 2.14 standard deviations. So this is equal to 2.14 standard deviations. Or we could say that our z statistic, we could call this our z score or our z statistic, the number of standard deviations we are away from our mean is 2.14. And to be exact, we're 2.14 standard deviations above the mean. We're going to care about a one-tailed distribution. Now, is the probability of getting this more or less than 5%? If it's less than 5%, we're going to reject the null hypothesis in favor of our alternative."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to be exact, we're 2.14 standard deviations above the mean. We're going to care about a one-tailed distribution. Now, is the probability of getting this more or less than 5%? If it's less than 5%, we're going to reject the null hypothesis in favor of our alternative. So how do we think about that? Well, let's think about just a normalized normal distribution. Or maybe you could call it a z distribution if you want."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's less than 5%, we're going to reject the null hypothesis in favor of our alternative. So how do we think about that? Well, let's think about just a normalized normal distribution. Or maybe you could call it a z distribution if you want. If you look at a normal distribution, a completely normalized normal distribution, its mean is at 0. And essentially, each of these values are essentially z scores, because if you're a value of 1, it literally means you're one standard deviation away from this mean over here. So we need to find a critical z value right over here."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or maybe you could call it a z distribution if you want. If you look at a normal distribution, a completely normalized normal distribution, its mean is at 0. And essentially, each of these values are essentially z scores, because if you're a value of 1, it literally means you're one standard deviation away from this mean over here. So we need to find a critical z value right over here. Let me call that a critical z. We could even say a critical z score or critical z value, so that the probability of getting a z value higher than that is 5%. So that this whole area right here is 5%."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we need to find a critical z value right over here. Let me call that a critical z. We could even say a critical z score or critical z value, so that the probability of getting a z value higher than that is 5%. So that this whole area right here is 5%. And that's because that's what our significance level is. Anything that has a lower than 5% chance of occurring, for us, will be validation to reject our null hypothesis. Or another way of thinking about it, if that area is 5%, this whole area right over here is 95%."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that this whole area right here is 5%. And that's because that's what our significance level is. Anything that has a lower than 5% chance of occurring, for us, will be validation to reject our null hypothesis. Or another way of thinking about it, if that area is 5%, this whole area right over here is 95%. And once again, this is a one-tailed test, because we only care about values greater than this. Z values greater than that will make us reject the null hypothesis. And to figure out this critical z value, we can literally just go to a z table."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way of thinking about it, if that area is 5%, this whole area right over here is 95%. And once again, this is a one-tailed test, because we only care about values greater than this. Z values greater than that will make us reject the null hypothesis. And to figure out this critical z value, we can literally just go to a z table. And we say, OK, the probability of getting a z value less than that is 95%. And that's exactly the number that this gives us, the cumulative probability of getting a value less than that. And so if we just scan this, we're looking for 95%."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And to figure out this critical z value, we can literally just go to a z table. And we say, OK, the probability of getting a z value less than that is 95%. And that's exactly the number that this gives us, the cumulative probability of getting a value less than that. And so if we just scan this, we're looking for 95%. We have 0.9495, we have 0.9505. So I'll go with this, just to make sure we're a little bit closer. So this z value, and the z value here is 1.6, and the next digit is 5."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so if we just scan this, we're looking for 95%. We have 0.9495, we have 0.9505. So I'll go with this, just to make sure we're a little bit closer. So this z value, and the z value here is 1.6, and the next digit is 5. 1.65. So this critical z value is equal to 1.65. So the probability of getting a z value less than 1.65, or even in a completely normalized normal distribution, the probability of getting a value less than 1.65, or in any normal distribution, the probability of being less than 1.65 standard deviations away from the mean is going to be 95%."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this z value, and the z value here is 1.6, and the next digit is 5. 1.65. So this critical z value is equal to 1.65. So the probability of getting a z value less than 1.65, or even in a completely normalized normal distribution, the probability of getting a value less than 1.65, or in any normal distribution, the probability of being less than 1.65 standard deviations away from the mean is going to be 95%. So that's our critical z value. Now, the z value, or the z statistic for our actual sample, is 2.14. Our actual z value we got is 2.14."}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of getting a z value less than 1.65, or even in a completely normalized normal distribution, the probability of getting a value less than 1.65, or in any normal distribution, the probability of being less than 1.65 standard deviations away from the mean is going to be 95%. So that's our critical z value. Now, the z value, or the z statistic for our actual sample, is 2.14. Our actual z value we got is 2.14. It's sitting all the way out here someplace. So the probability of getting that was definitely less than 5%. And actually, we could even say, what's the probability of getting that or something, or a more extreme result?"}, {"video_title": "Large sample proportion hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our actual z value we got is 2.14. It's sitting all the way out here someplace. So the probability of getting that was definitely less than 5%. And actually, we could even say, what's the probability of getting that or something, or a more extreme result? And if you figured out this area, and you could actually figure it out by looking at a z table, you could figure out the p value of this result. But anyway, the whole exercise here is just to figure out if we're going to reject the null hypothesis with a significance level of 5%. We can."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "And so what we could do is we could set up some buckets of time studied and some buckets of percent correct, and then we could survey the students and or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students."}, {"video_title": "Marginal and conditional distributions   Analyzing categorical data   AP Statistics   Khan Academy.mp3", "Sentence": "So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%. And then to get the full conditional distribution, we would keep doing that."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "We are told a restaurant owner installed a new automated drink machine. The machine is designed to dispense 530 milliliters of liquid on the medium-sized setting. The owner suspects that the machine may be dispensing too much in medium drinks. They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something interesting going on here. There is a difference."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something interesting going on here. There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be equal to 530 milliliters because that's what the drink machine is supposed to do. And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be equal to 530 milliliters because that's what the drink machine is supposed to do. And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. We see it up here."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. And so this one is looking like this right over here. Only these two are even dealing with the mean."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. And so this one is looking like this right over here. Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what the owner is afraid of, and so that's not the kind of the news that we're trying to find some evidence for. So I would definitely pick choice C. Let's do another example."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what the owner is afraid of, and so that's not the kind of the news that we're trying to find some evidence for. So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less than eight hours of sleep on average. To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less than eight hours of sleep on average. To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis?"}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care about the population of students at the school, and so we would say, well, the null hypothesis is that the parameter for the students at that school, the mean amount of sleep that they're getting, is indeed greater than or equal to eight hours. And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care about the population of students at the school, and so we would say, well, the null hypothesis is that the parameter for the students at that school, the mean amount of sleep that they're getting, is indeed greater than or equal to eight hours. And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at their school are getting less than eight hours of sleep on average. And so they suspect that the population parameter, the population mean for their school is actually less than eight hours. And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at their school are getting less than eight hours of sleep on average. And so they suspect that the population parameter, the population mean for their school is actually less than eight hours. And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. But the other thing that sometimes folks get stuck up on, but the other thing that sometimes confuses folks is, well, we are measuring, is that we are calculating a statistic from a sample."}, {"video_title": "Examples of null and alternative hypotheses   AP Statistics   Khan Academy.mp3", "Sentence": "And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. But the other thing that sometimes folks get stuck up on, but the other thing that sometimes confuses folks is, well, we are measuring, is that we are calculating a statistic from a sample. Here we're calculating the sample mean, but the sample statistics are not what should be involved in your hypotheses. Your hypotheses are claims about your population that you care about. Here, the population is the students at the high school."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "So they all agree to put in their salaries into a computer, and so these are their salaries, they're measured in thousands, so one makes 35,000, 50,000, 50,000, 50,000, 56,000, two make 60,000, one make 75,000, and one makes 250,000, so she's doing very well for herself. And the computer spits out a bunch of parameters based on this data here. So it spits out two typical measures of central tendency. The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure?"}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread?"}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range?"}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency."}, {"video_title": "Mean and standard deviation versus median and IQR   AP Statistics   Khan Academy.mp3", "Sentence": "So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency. And that's why you'll see when people talk about salaries, they'll often talk about median because you could have some skewed salaries, especially on the upside. When you talk about things like home prices, you'll see median often measured more typically than mean because home prices in a neighborhood or in a city, a lot of the houses might be in the $200,000, $300,000 range but maybe there's one ginormous mansion that is $100 million. And if you calculated mean, that would skew and give a false impression of the average or the central tendency of prices in that city."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "What is the smallest sample size required to obtain the desired margin of error? So let's just remind ourselves what the confidence interval will look like and what part of it is the margin of error and then we can think about what is her sample size that she would need. So she wants to estimate the true population proportion that favor a tax increase. She doesn't know what this is so she is going to take a sample size of size n and in fact this question is all about what n does she need in order to have the desired margin of error. Well whatever sample she takes there she's going to calculate a sample proportion and then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value and this critical value is based on the confidence level. We'll talk about that in a second. What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "She doesn't know what this is so she is going to take a sample size of size n and in fact this question is all about what n does she need in order to have the desired margin of error. Well whatever sample she takes there she's going to calculate a sample proportion and then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value and this critical value is based on the confidence level. We'll talk about that in a second. What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. It would be the standard error of her sample proportion which is the sample proportion times one minus the sample proportion, all of that over her sample size. Now she wants the margin of error to be no more than 2%. So the margin of error is this part right over here."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. It would be the standard error of her sample proportion which is the sample proportion times one minus the sample proportion, all of that over her sample size. Now she wants the margin of error to be no more than 2%. So the margin of error is this part right over here. So this part right over there she wants to be no more than 2%. Has to be less than or equal to 2%. That green color is kind of too shocking."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So the margin of error is this part right over here. So this part right over there she wants to be no more than 2%. Has to be less than or equal to 2%. That green color is kind of too shocking. It's unpleasant. All right, less than or equal to 2% right over here. So how do we figure that out?"}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "That green color is kind of too shocking. It's unpleasant. All right, less than or equal to 2% right over here. So how do we figure that out? Well the first thing, let's just make sure we incorporate the 95% confidence level. So we could look at a z-table. Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So how do we figure that out? Well the first thing, let's just make sure we incorporate the 95% confidence level. So we could look at a z-table. Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. So this would be 2.5% that is unshaded at the top right over there, and then this would be 2.5% right over here. And we could look up in a z-table, and if you were to look up in a z-table, you would not look up 95%. You would look up the percentage that would leave 2.5% unshaded at the top."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. So this would be 2.5% that is unshaded at the top right over there, and then this would be 2.5% right over here. And we could look up in a z-table, and if you were to look up in a z-table, you would not look up 95%. You would look up the percentage that would leave 2.5% unshaded at the top. So you would actually look up 97.5%. But it's good to know in general that at a 95% confidence level, you're looking at a critical value of 1.96. And that's just something good to know."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "You would look up the percentage that would leave 2.5% unshaded at the top. So you would actually look up 97.5%. But it's good to know in general that at a 95% confidence level, you're looking at a critical value of 1.96. And that's just something good to know. We could of course look it up on a z-table. So this is 1.96. And so this is going to be 1.96 right over here."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And that's just something good to know. We could of course look it up on a z-table. So this is 1.96. And so this is going to be 1.96 right over here. But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take? Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is going to be 1.96 right over here. But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take? Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. We want this thing to be less than or equal to 2%. This is our margin of error. And so what we could do is we could pick a sample proportion."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. We want this thing to be less than or equal to 2%. This is our margin of error. And so what we could do is we could pick a sample proportion. We don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe. So the p hat, the maximum p hat."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And so what we could do is we could pick a sample proportion. We don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe. So the p hat, the maximum p hat. And so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0.5."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So the p hat, the maximum p hat. And so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0.5. And I wanna emphasize, we don't know. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "It's actually going to be p hat is 0.5. And I wanna emphasize, we don't know. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion. But we wanna figure out what n to take. And so to be safe, she says, okay, well, what sample proportion would maximize my margin of error? And so let me just assume that, and then let me calculate n. So let me set up an inequality here."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "She didn't even take the random sample and calculate the sample proportion. But we wanna figure out what n to take. And so to be safe, she says, okay, well, what sample proportion would maximize my margin of error? And so let me just assume that, and then let me calculate n. So let me set up an inequality here. We want 1.96, that's our critical value, times the square root of, we're just going to assume 0.5 for our sample proportion, although, of course, we don't know what it is yet until we actually take the sample. So that's our sample proportion. That's one minus our sample proportion."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And so let me just assume that, and then let me calculate n. So let me set up an inequality here. We want 1.96, that's our critical value, times the square root of, we're just going to assume 0.5 for our sample proportion, although, of course, we don't know what it is yet until we actually take the sample. So that's our sample proportion. That's one minus our sample proportion. All of that over n needs to be less than or equal to 2%. We don't want our margin of error to be any larger than 2%. And let me just write this as a decimal, 0.02."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "That's one minus our sample proportion. All of that over n needs to be less than or equal to 2%. We don't want our margin of error to be any larger than 2%. And let me just write this as a decimal, 0.02. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And let me just write this as a decimal, 0.02. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. And so this would be equal to, on the left-hand side, we'd have the square root of all of this. But that's the same thing as the square root of 0.5 times 0.5, so that'd just be 0.5 over the square root of n. Needs to be less than or equal to, actually, let me write it this way. This is the same thing as two over 100."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. And so this would be equal to, on the left-hand side, we'd have the square root of all of this. But that's the same thing as the square root of 0.5 times 0.5, so that'd just be 0.5 over the square root of n. Needs to be less than or equal to, actually, let me write it this way. This is the same thing as two over 100. So two over 100 times one over 1.96 needs to be less than or equal to two over 196. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "This is the same thing as two over 100. So two over 100 times one over 1.96 needs to be less than or equal to two over 196. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. All right, so let's see. We could take the reciprocal of both sides. We could say the square root of n over 0.5 and 1.96 over two."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. All right, so let's see. We could take the reciprocal of both sides. We could say the square root of n over 0.5 and 1.96 over two. And let's see, what's 196 divided by two? That is going to be 98. So this would be 98."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "We could say the square root of n over 0.5 and 1.96 over two. And let's see, what's 196 divided by two? That is going to be 98. So this would be 98. And so if we take the reciprocal of both sides, then you're gonna swap the inequality, so it's gonna be greater than or equal to. Let's see, I can multiply both sides of this by 0.5. So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So this would be 98. And so if we take the reciprocal of both sides, then you're gonna swap the inequality, so it's gonna be greater than or equal to. Let's see, I can multiply both sides of this by 0.5. So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. Let's see, 0.5. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared?"}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. Let's see, 0.5. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared? Well, you know 50 squared is 2,500, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be d. But if you wanna multiply it out, we can. 49 times 49, nine times nine is 81. Nine times four is 36, plus eight is 44."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And what's 49 squared? Well, you know 50 squared is 2,500, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be d. But if you wanna multiply it out, we can. 49 times 49, nine times nine is 81. Nine times four is 36, plus eight is 44. Four times nine, 36. Four times four is 16 plus three, we have 19. And then you add all of that together, and you indeed do get, so that's 10."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "Nine times four is 36, plus eight is 44. Four times nine, 36. Four times four is 16 plus three, we have 19. And then you add all of that together, and you indeed do get, so that's 10. And so this is a 14. You do indeed get 2,401. So that's the minimum sample size that Della should take if she genuinely wanted her margin of error to be no more than 2%."}, {"video_title": "Determining sample size based on confidence and margin of error   AP Statistics   Khan Academy.mp3", "Sentence": "And then you add all of that together, and you indeed do get, so that's 10. And so this is a 14. You do indeed get 2,401. So that's the minimum sample size that Della should take if she genuinely wanted her margin of error to be no more than 2%. Now, it might turn out that her margin of error, when she actually takes the sample of size 2,401, if her sample proportion is less than 0.5, or greater than 0.5, well, then she's going to be in a situation where her margin of error might be less than this, but she just wanted it to be no more than that. Another important thing to appreciate is, it just, the math all worked out very nicely just now, where I got our n to be actually a whole number, but if I got 2,401.5, then you would have to round up to the nearest whole number because you can't have a, your sample size is always going to be a whole number value. So I will leave you there."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "If the significance level was lowered to 1 hundredth, which of the following would be true? So pause this video and see if you can answer it on your own. Okay, now let's do this together. And let's see, they're talking about how the probability of a type two error or the power, and or the power would change. So before I even look at the choices, let's think about this. We have talked about in previous videos that if we increase our level of significance, that will increase our power. And power is the probability of not making a type two error."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And let's see, they're talking about how the probability of a type two error or the power, and or the power would change. So before I even look at the choices, let's think about this. We have talked about in previous videos that if we increase our level of significance, that will increase our power. And power is the probability of not making a type two error. So that would decrease the probability of making a type two error. But in this question, we're going the other way. We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And power is the probability of not making a type two error. So that would decrease the probability of making a type two error. But in this question, we're going the other way. We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. It actually would increase, it actually would increase the probability of making a type two, a type two error. And so which of these choices are consistent with that? Well, choice A says that both the type two error and the power would decrease."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. It actually would increase, it actually would increase the probability of making a type two, a type two error. And so which of these choices are consistent with that? Well, choice A says that both the type two error and the power would decrease. Well, those don't, these two things don't move together. If one increases, the other decreases. So we rule that one out."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Well, choice A says that both the type two error and the power would decrease. Well, those don't, these two things don't move together. If one increases, the other decreases. So we rule that one out. Choice B also has these two things moving together, which can't be true. If one increases, the other decreases. Choice C, the probability of a type two error would increase."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So we rule that one out. Choice B also has these two things moving together, which can't be true. If one increases, the other decreases. Choice C, the probability of a type two error would increase. That's consistent with what we have here. And the power of the test would decrease. Yep, that's consistent with what we have here."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Choice C, the probability of a type two error would increase. That's consistent with what we have here. And the power of the test would decrease. Yep, that's consistent with what we have here. So that looks good. And choice D is the opposite of that. The probability of a type two error would decrease."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Yep, that's consistent with what we have here. So that looks good. And choice D is the opposite of that. The probability of a type two error would decrease. So this is, they're talking about this scenario over here, and that would have happened if they increased our significance level, not decreased it. So we could rule that one out as well. Let's do another example."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "The probability of a type two error would decrease. So this is, they're talking about this scenario over here, and that would have happened if they increased our significance level, not decreased it. So we could rule that one out as well. Let's do another example. Asha owns a car wash and is trying to decide whether or not to purchase a vending machine so that customers can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her customers would buy coffee. She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "Let's do another example. Asha owns a car wash and is trying to decide whether or not to purchase a vending machine so that customers can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her customers would buy coffee. She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. And she'll then do a significance test using alpha equals 0.05 to see if the sample proportion who say yes is significantly greater than 30%. Which situation below would result in the highest power for her test? So again, pause this video and try to answer it."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. And she'll then do a significance test using alpha equals 0.05 to see if the sample proportion who say yes is significantly greater than 30%. Which situation below would result in the highest power for her test? So again, pause this video and try to answer it. Well, before I even look at the choices, we could think about what her hypotheses would be. Her null hypothesis is, you could kind of view it as a status quo, no news here. And that would be that the true population proportion of people who want to buy coffee is 30%."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So again, pause this video and try to answer it. Well, before I even look at the choices, we could think about what her hypotheses would be. Her null hypothesis is, you could kind of view it as a status quo, no news here. And that would be that the true population proportion of people who want to buy coffee is 30%. And that her alternative hypothesis is that no, the true population proportion, the true population parameter there, is greater than, is greater than 30%. And so if we're talking about what would result in the highest power for her test. So a high power, a high power means the lowest probability of making a type two error."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And that would be that the true population proportion of people who want to buy coffee is 30%. And that her alternative hypothesis is that no, the true population proportion, the true population parameter there, is greater than, is greater than 30%. And so if we're talking about what would result in the highest power for her test. So a high power, a high power means the lowest probability of making a type two error. And in other videos, we've talked about it. It looks like she's dealing with the sample size and what is the true proportion of customers that would buy coffee. And the sample size is under her control."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "So a high power, a high power means the lowest probability of making a type two error. And in other videos, we've talked about it. It looks like she's dealing with the sample size and what is the true proportion of customers that would buy coffee. And the sample size is under her control. The true proportion isn't. Don't wanna make it seem like somehow you can change the true proportion in order to get a higher power. You can change the sample size."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "And the sample size is under her control. The true proportion isn't. Don't wanna make it seem like somehow you can change the true proportion in order to get a higher power. You can change the sample size. But the general principle is, the higher the sample size, the higher the power. So you want a highest possible sample size. And you're going to have a higher power if the true proportion is further from your hypothesis, your null hypothesis proportion."}, {"video_title": "Examples thinking about power in significance tests   AP Statistics   Khan Academy.mp3", "Sentence": "You can change the sample size. But the general principle is, the higher the sample size, the higher the power. So you want a highest possible sample size. And you're going to have a higher power if the true proportion is further from your hypothesis, your null hypothesis proportion. And so we want the highest possible N, and that looks like an N of 200, which is there and there. And we want a true proportion of customers that would actually buy coffee as far away as possible from our null hypothesis, which once again would not be under Asha's control. But you can clearly see that 50% is further from 30 than 32 is, so this one, choice D, is the one that looks good."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "A group of four friends likes to bowl together, and each friend keeps track of his all-time highest score in a single game. Their high scores are all between 180 and 220, except for Adam, whose high score is 250. Adam then bowls a great game and has a new high score of 290. How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they're each going to have, they each keep track of their all-time high score."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they're each going to have, they each keep track of their all-time high score. So we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends. This is the second lowest, second to highest, and this is the highest scoring of the friends."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "We have four friends, and they're each going to have, they each keep track of their all-time high score. So we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends. This is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super awesome game, the scores look something like this."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "This is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "So before Adam bowls this super awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember, the median is the middle number."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember, the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're going to take the average of whatever this question mark is in 220. That's going to be the median."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "Well, remember, the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're going to take the average of whatever this question mark is in 220. That's going to be the median. Now, over here, after Adam has scored a new high score, how would we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers, whatever this friend's high score was, and it hasn't changed. And so we're going to have the same median."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "That's going to be the median. Now, over here, after Adam has scored a new high score, how would we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers, whatever this friend's high score was, and it hasn't changed. And so we're going to have the same median. It's going to be 220 plus question mark divided by two. It's going to be halfway between question mark and 220. So our median won't change."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "And so we're going to have the same median. It's going to be 220 plus question mark divided by two. It's going to be halfway between question mark and 220. So our median won't change. So median no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "So our median won't change. So median no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four. And then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list, you have a higher number, 290."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "Well, the mean, you take the sum of all these numbers and then you divide by four. And then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list, you have a higher number, 290. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four because their sum is going to be larger. And so the mean is going to go up."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "Well, the first three numbers are the same, but in the second list, you have a higher number, 290. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four because their sum is going to be larger. And so the mean is going to go up. The mean will increase. So median no change and mean increase. All right, so this says both increase."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "And so the mean is going to go up. The mean will increase. So median no change and mean increase. All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change. The mean will increase, yep, and the median will stay the same. Yep, that's exactly what we're talking about. And if you want to make it a little bit more tangible, you could replace question mark with some number."}, {"video_title": "Impact on median and mean when increasing highest value   6th grade   Khan Academy.mp3", "Sentence": "No, median doesn't change. The mean will increase, yep, and the median will stay the same. Yep, that's exactly what we're talking about. And if you want to make it a little bit more tangible, you could replace question mark with some number. You could replace it, maybe this question mark is 200. And if you try it out with 200 just to make things tangible, you're going to see that that is indeed going to be the case. The median would be halfway between these two numbers, and I just arbitrarily picked 200."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "An ecologist surveys the age of about 100 trees in a local forest. He uses a box and whisker plot to map his data, shown below. What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21."}, {"video_title": "Box and whisker plot   Descriptive statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages. So if you view median as your central tendency measurement, it's only at 21 years. And you can even see it."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So ideally, you would like to know the mean height of men in the United States. Let me write this down. Mean height, height of men, men in the United States. So how would you do that? When I talk about the mean, when I talk about the mean, I'm talking about the arithmetic mean. If I were to talk about some other types of means, and there are other types of means, like the geometric mean, I would specify. But when people just say mean, they're usually talking about the arithmetic mean."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how would you do that? When I talk about the mean, when I talk about the mean, I'm talking about the arithmetic mean. If I were to talk about some other types of means, and there are other types of means, like the geometric mean, I would specify. But when people just say mean, they're usually talking about the arithmetic mean. So how would you go about finding the mean height of men in the United States? Well, the obvious one is, is well, you go and ask every, or measure every man in the United States, take their height, add them all together, and then divide by the number of men there are in the United States. But the question you need to ask yourself is whether that is practical."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "But when people just say mean, they're usually talking about the arithmetic mean. So how would you go about finding the mean height of men in the United States? Well, the obvious one is, is well, you go and ask every, or measure every man in the United States, take their height, add them all together, and then divide by the number of men there are in the United States. But the question you need to ask yourself is whether that is practical. Because you have on the order, let's see, there's about 300 million people in the United States, roughly half of them will be men, or at least they'll be male. And so you will have 150 million, roughly 150 million men in the United States. So if you wanted the true mean height of all of the men in the United States, you would have to somehow survey, or not even, you would have to be able to go and measure all 150 million men."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the question you need to ask yourself is whether that is practical. Because you have on the order, let's see, there's about 300 million people in the United States, roughly half of them will be men, or at least they'll be male. And so you will have 150 million, roughly 150 million men in the United States. So if you wanted the true mean height of all of the men in the United States, you would have to somehow survey, or not even, you would have to be able to go and measure all 150 million men. And even if you did try to do that, by the time you're done, many of them might have passed away, the new men will have been born, and so your data will go stale immediately. So it is seemingly impossible, or almost impossible to get the exact height of every man in the United States in a snapshot of time. And so instead, what you do is say, well look, okay, I can't get every man, but maybe I can take a sample, I could take a sample of the men in the United States, and I'm going to make an effort, I'm gonna make an effort that it's a random sample."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you wanted the true mean height of all of the men in the United States, you would have to somehow survey, or not even, you would have to be able to go and measure all 150 million men. And even if you did try to do that, by the time you're done, many of them might have passed away, the new men will have been born, and so your data will go stale immediately. So it is seemingly impossible, or almost impossible to get the exact height of every man in the United States in a snapshot of time. And so instead, what you do is say, well look, okay, I can't get every man, but maybe I can take a sample, I could take a sample of the men in the United States, and I'm going to make an effort, I'm gonna make an effort that it's a random sample. I don't wanna just go sample 100 people who happen to play basketball, or play basketball for their college. I don't wanna go sample 100 people who are volleyball players. I wanna randomly sample, just maybe the first person who comes out of a mall in a random town, or in several towns, or something like that, something that should not be based in any way, or skewed in any way, by height."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so instead, what you do is say, well look, okay, I can't get every man, but maybe I can take a sample, I could take a sample of the men in the United States, and I'm going to make an effort, I'm gonna make an effort that it's a random sample. I don't wanna just go sample 100 people who happen to play basketball, or play basketball for their college. I don't wanna go sample 100 people who are volleyball players. I wanna randomly sample, just maybe the first person who comes out of a mall in a random town, or in several towns, or something like that, something that should not be based in any way, or skewed in any way, by height. So you take a sample, and from that sample, you can calculate a mean of at least the sample, and you'll hope that that is indicative of, especially if this was a reasonably random sample, you'll hope that that was indicative of the mean of the entire population. And what you're going to see in much of statistics, in much of statistics, it is all about, it is all about using information, using things that we can calculate about a sample, to infer things about a population, because we can't directly measure the entire population. So for example, let's say, and I wouldn't, if you're actually trying to do this, I would recommend doing at least 100 data points, or 1,000, and later on we'll talk about how you can think about whether you've measured enough, or how confident you can be."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "I wanna randomly sample, just maybe the first person who comes out of a mall in a random town, or in several towns, or something like that, something that should not be based in any way, or skewed in any way, by height. So you take a sample, and from that sample, you can calculate a mean of at least the sample, and you'll hope that that is indicative of, especially if this was a reasonably random sample, you'll hope that that was indicative of the mean of the entire population. And what you're going to see in much of statistics, in much of statistics, it is all about, it is all about using information, using things that we can calculate about a sample, to infer things about a population, because we can't directly measure the entire population. So for example, let's say, and I wouldn't, if you're actually trying to do this, I would recommend doing at least 100 data points, or 1,000, and later on we'll talk about how you can think about whether you've measured enough, or how confident you can be. But let's just say you're a little bit lazy, and you just sample five men. And so you get their five heights. Let's say one is 6.2 feet."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, let's say, and I wouldn't, if you're actually trying to do this, I would recommend doing at least 100 data points, or 1,000, and later on we'll talk about how you can think about whether you've measured enough, or how confident you can be. But let's just say you're a little bit lazy, and you just sample five men. And so you get their five heights. Let's say one is 6.2 feet. Let's say one is 5.5 feet. 5.5 feet would be five foot six inches. One would be, let's say one ends up being 5.75, seven five feet."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say one is 6.2 feet. Let's say one is 5.5 feet. 5.5 feet would be five foot six inches. One would be, let's say one ends up being 5.75, seven five feet. Another one is 6.3 feet. Another is 5.9 feet. Now, if these are the ones that you happen to sample, what would you get for the mean of this sample?"}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "One would be, let's say one ends up being 5.75, seven five feet. Another one is 6.3 feet. Another is 5.9 feet. Now, if these are the ones that you happen to sample, what would you get for the mean of this sample? Well, let's get our calculator out, and we get 6.2 plus 5.5 plus 5.75 plus 6.3 plus 5.9. The sum is 29.65, and then we wanna divide by the number of data points we have. So we have five data points."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, if these are the ones that you happen to sample, what would you get for the mean of this sample? Well, let's get our calculator out, and we get 6.2 plus 5.5 plus 5.75 plus 6.3 plus 5.9. The sum is 29.65, and then we wanna divide by the number of data points we have. So we have five data points. So let's divide 29.65 divided by five, and we get 5.93 feet. So here our sample mean, and I'm going to denote it with an X with a bar over it, is, and I already forgot the number, 5.93 feet. 5.93 feet."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have five data points. So let's divide 29.65 divided by five, and we get 5.93 feet. So here our sample mean, and I'm going to denote it with an X with a bar over it, is, and I already forgot the number, 5.93 feet. 5.93 feet. This is our sample mean, or if we wanna make it clear, sample arithmetic mean. And when we're taking this calculation based on a sample, and then somehow we're trying to estimate it for the entire population, we call this right over here, we call it a statistic. We call it a statistic."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "5.93 feet. This is our sample mean, or if we wanna make it clear, sample arithmetic mean. And when we're taking this calculation based on a sample, and then somehow we're trying to estimate it for the entire population, we call this right over here, we call it a statistic. We call it a statistic. Now, you might be saying, well, what notation do we use if somehow we are able to measure it for the population? Well, let's say we can't even measure it for the population, but we at least wanna denote what the population mean is. Well, if you wanna do that, the population mean is usually denoted by the Greek letter mu."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "We call it a statistic. Now, you might be saying, well, what notation do we use if somehow we are able to measure it for the population? Well, let's say we can't even measure it for the population, but we at least wanna denote what the population mean is. Well, if you wanna do that, the population mean is usually denoted by the Greek letter mu. So the population mean is usually denoted by the Greek letter mu. And so in a lot of statistics, it's calculating a sample mean in an attempt to estimate this thing that you might not know, the population mean. And these calculations on the entire population, sometimes you might be able to do it, oftentimes you will not be able to do it, these are called parameters."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, if you wanna do that, the population mean is usually denoted by the Greek letter mu. So the population mean is usually denoted by the Greek letter mu. And so in a lot of statistics, it's calculating a sample mean in an attempt to estimate this thing that you might not know, the population mean. And these calculations on the entire population, sometimes you might be able to do it, oftentimes you will not be able to do it, these are called parameters. This is called parameters. So what you're going to find in much of statistics, it's all about calculating statistics for a sample, finding these sample statistics in order to estimate parameters for an entire population. Now, the last thing I wanna do is introduce you to some of the notation that you might see in a statistics textbook that looks very mathy and very difficult."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "And these calculations on the entire population, sometimes you might be able to do it, oftentimes you will not be able to do it, these are called parameters. This is called parameters. So what you're going to find in much of statistics, it's all about calculating statistics for a sample, finding these sample statistics in order to estimate parameters for an entire population. Now, the last thing I wanna do is introduce you to some of the notation that you might see in a statistics textbook that looks very mathy and very difficult. But hopefully, after the next few minutes, you'll appreciate that it's really just doing exactly what we did here, adding up the numbers and dividing by the number of numbers you had. If you had to do the population mean, it's the exact same thing, it's just many, many more numbers in this context, you would have to add up 150 million numbers and divide by 150 million. So how do mathematicians talk about an operation like that, adding up a bunch of numbers and then dividing by the number of numbers?"}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, the last thing I wanna do is introduce you to some of the notation that you might see in a statistics textbook that looks very mathy and very difficult. But hopefully, after the next few minutes, you'll appreciate that it's really just doing exactly what we did here, adding up the numbers and dividing by the number of numbers you had. If you had to do the population mean, it's the exact same thing, it's just many, many more numbers in this context, you would have to add up 150 million numbers and divide by 150 million. So how do mathematicians talk about an operation like that, adding up a bunch of numbers and then dividing by the number of numbers? Well, let's first think about the sample mean, the sample, well, the sample mean, because that's where we actually did the calculation. So a mathematician might call each of these data points, they'll call it, let's say they'll call this first one right over here, they'll call this x sub one, they'll call this one x sub two, they'll call this one x sub three, they'll call this one, when I say sub, I'm literally saying subscript one, subscript two, subscript three. They could call this x subscript four, they could call this x subscript five, and so if you had n of these, you would just keep going, x subscript six, x subscript seven, all the way to x subscript n. And so to take the sum of all of these, they would denote it as the sum, let me write it right over here, so they will say that the sample mean is equal to the sum, the sum of all of my x sub i's, my x sub i's, so the way you can conceptualize it, these i's will change, so the i's are going to, in this case, the i started at one, the i's are going to start at one until the size of our actual sample, so all the way until n. In this case, n was equal to five."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how do mathematicians talk about an operation like that, adding up a bunch of numbers and then dividing by the number of numbers? Well, let's first think about the sample mean, the sample, well, the sample mean, because that's where we actually did the calculation. So a mathematician might call each of these data points, they'll call it, let's say they'll call this first one right over here, they'll call this x sub one, they'll call this one x sub two, they'll call this one x sub three, they'll call this one, when I say sub, I'm literally saying subscript one, subscript two, subscript three. They could call this x subscript four, they could call this x subscript five, and so if you had n of these, you would just keep going, x subscript six, x subscript seven, all the way to x subscript n. And so to take the sum of all of these, they would denote it as the sum, let me write it right over here, so they will say that the sample mean is equal to the sum, the sum of all of my x sub i's, my x sub i's, so the way you can conceptualize it, these i's will change, so the i's are going to, in this case, the i started at one, the i's are going to start at one until the size of our actual sample, so all the way until n. In this case, n was equal to five. So this is literally saying, this is literally saying, this is equal to x sub one plus x sub two plus x sub three, all the way, all the way to the nth one. Once again, in this case, we only had five. Now, are we done?"}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "They could call this x subscript four, they could call this x subscript five, and so if you had n of these, you would just keep going, x subscript six, x subscript seven, all the way to x subscript n. And so to take the sum of all of these, they would denote it as the sum, let me write it right over here, so they will say that the sample mean is equal to the sum, the sum of all of my x sub i's, my x sub i's, so the way you can conceptualize it, these i's will change, so the i's are going to, in this case, the i started at one, the i's are going to start at one until the size of our actual sample, so all the way until n. In this case, n was equal to five. So this is literally saying, this is literally saying, this is equal to x sub one plus x sub two plus x sub three, all the way, all the way to the nth one. Once again, in this case, we only had five. Now, are we done? Is this what the sample mean is? Well, no, we aren't done. We don't just add up all of the data points."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, are we done? Is this what the sample mean is? Well, no, we aren't done. We don't just add up all of the data points. We then have to divide by the number of data points there are. So we then have to divide, we then have to divide by the number of data points that there actually are. So this might look like very fancy notation, but it's really just saying, add up your data points and divide by the number of data points you have."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "We don't just add up all of the data points. We then have to divide by the number of data points there are. So we then have to divide, we then have to divide by the number of data points that there actually are. So this might look like very fancy notation, but it's really just saying, add up your data points and divide by the number of data points you have. And this capital Greek letter, sigma, literally means sum, sum all of the x i's from x sub one all the way to x sub n, and then divide by the number of data points you have. Now let's think about how we would denote the same thing, but instead of for the sample mean, doing it for the population mean. So the population mean, they will denote it with mu."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this might look like very fancy notation, but it's really just saying, add up your data points and divide by the number of data points you have. And this capital Greek letter, sigma, literally means sum, sum all of the x i's from x sub one all the way to x sub n, and then divide by the number of data points you have. Now let's think about how we would denote the same thing, but instead of for the sample mean, doing it for the population mean. So the population mean, they will denote it with mu. We already talked about that. And here, once again, you're gonna take the sum, but this time it's going to be the sum of all of the elements in your population. So your x sub i's, and you'll still start at i equals one, but it usually gets denoted that, hey, you're taking the whole population."}, {"video_title": "Inferring population mean from sample mean   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the population mean, they will denote it with mu. We already talked about that. And here, once again, you're gonna take the sum, but this time it's going to be the sum of all of the elements in your population. So your x sub i's, and you'll still start at i equals one, but it usually gets denoted that, hey, you're taking the whole population. So they'll often put a capital N right over here to somehow denote that this is a bigger number than maybe this smaller n. But once again, we are not done. We have to divide by the number of data points that we are actually summing. And so this, once again, is the same thing as x sub one plus x sub two plus x sub three all the way to x sub capital N, all of that divided by our capital N. And once again, in this situation, we found this practical, we found this impractical."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "So to figure out this probability, a good place to start is just to think about all of the different possible ways that we can flip three coins. So we could get all tails. Tails, tails, tails. We could get tails, tails, heads. We could get tails, heads, tails. We could get tails, heads, heads. We could get heads, tails, tails."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "We could get tails, tails, heads. We could get tails, heads, tails. We could get tails, heads, heads. We could get heads, tails, tails. We could get heads, tails, heads. We could get heads, heads, tails. And then we could get all heads."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "We could get heads, tails, tails. We could get heads, tails, heads. We could get heads, heads, tails. And then we could get all heads. So there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. Now, how many of the outcomes involve flipping exactly two heads? Let's see, that's all tails."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "And then we could get all heads. So there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. Now, how many of the outcomes involve flipping exactly two heads? Let's see, that's all tails. That's one head, one head. This has two heads right there. Two heads."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "Let's see, that's all tails. That's one head, one head. This has two heads right there. Two heads. That's one head. This is two heads right over there. Then this is two heads right over here."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "Two heads. That's one head. This is two heads right over there. Then this is two heads right over here. And then this is three heads, so that doesn't count. So there are three outcomes where we, with exactly two heads. So, let me spell heads properly."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "Then this is two heads right over here. And then this is three heads, so that doesn't count. So there are three outcomes where we, with exactly two heads. So, let me spell heads properly. Two heads. So the probability of flipping of exactly two heads, and the word exactly is important, because if you didn't say exactly, then maybe three heads, well, you flip two heads, so you have to say exactly two heads. So you don't include the situation where you get three heads."}, {"video_title": "Example Probability through counting outcomes   Precalculus   Khan Academy.mp3", "Sentence": "So, let me spell heads properly. Two heads. So the probability of flipping of exactly two heads, and the word exactly is important, because if you didn't say exactly, then maybe three heads, well, you flip two heads, so you have to say exactly two heads. So you don't include the situation where you get three heads. So the probability of flipping exactly two heads is equal to the three outcomes with two heads divided by the eight possible outcomes, or 3 8ths. So it is equal to 3 8ths. And we are done."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And so I went to the insurance company and I said, I want to get a $1 million policy. And what I'm actually getting a quote on is a term life policy, which is really, I just care about the next 20 years. After those 20 years, hopefully I can pay off my mortgage and there'll be money saved up. And hopefully my kids would kind of at least have maybe gotten to college or I would have saved up enough money for college. So that's why I'm willing to do a term life policy. The other option is to do a whole life policy, where you could pay a certain amount per year for the rest of your life. And at any point you die, you get the $1 million."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And hopefully my kids would kind of at least have maybe gotten to college or I would have saved up enough money for college. So that's why I'm willing to do a term life policy. The other option is to do a whole life policy, where you could pay a certain amount per year for the rest of your life. And at any point you die, you get the $1 million. In a term life, I'm only going to pay $500 per year for the next 20 years. If at any point over those 20 years I die, my family gets $1 million. At the 21st year, I have to get a new policy."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And at any point you die, you get the $1 million. In a term life, I'm only going to pay $500 per year for the next 20 years. If at any point over those 20 years I die, my family gets $1 million. At the 21st year, I have to get a new policy. And since I'm going to be older and I'd have a higher chance of dying at that point, then it's probably going to be more expensive for me to get insurance. But I really am just worried about the next 20 years. But what I want to do in this video is think about, given these numbers that have been quoted to me by the insurance company, what do they think that my odds of dying are over the next 20 years?"}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "At the 21st year, I have to get a new policy. And since I'm going to be older and I'd have a higher chance of dying at that point, then it's probably going to be more expensive for me to get insurance. But I really am just worried about the next 20 years. But what I want to do in this video is think about, given these numbers that have been quoted to me by the insurance company, what do they think that my odds of dying are over the next 20 years? So what I want to think about is the probability of Sal's death in 20 years, based on what the people at the insurance company are telling me. Or at least, what's the maximum probability of my death in order for them to make money? And the way to think about it, or one way to think about it, kind of a back of the envelope way, is to think about what's the total premiums they're getting over the life of this policy, divided by how much they're insuring me for."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "But what I want to do in this video is think about, given these numbers that have been quoted to me by the insurance company, what do they think that my odds of dying are over the next 20 years? So what I want to think about is the probability of Sal's death in 20 years, based on what the people at the insurance company are telling me. Or at least, what's the maximum probability of my death in order for them to make money? And the way to think about it, or one way to think about it, kind of a back of the envelope way, is to think about what's the total premiums they're getting over the life of this policy, divided by how much they're insuring me for. So they're getting $500 times 20 years is equal to $10,000 over the life of this policy. And they're insuring me for $1 million. So they're getting, let's see, those zeros cancel out, this zero cancels out."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And the way to think about it, or one way to think about it, kind of a back of the envelope way, is to think about what's the total premiums they're getting over the life of this policy, divided by how much they're insuring me for. So they're getting $500 times 20 years is equal to $10,000 over the life of this policy. And they're insuring me for $1 million. So they're getting, let's see, those zeros cancel out, this zero cancels out. They're getting, over the life of the policy, $1 in premiums for every $100 in insurance. Or another way to think about it, let's say that there were 100 Sal's, 134-year-olds looking to get 20-year-term life insurance. And they insured all of them."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "So they're getting, let's see, those zeros cancel out, this zero cancels out. They're getting, over the life of the policy, $1 in premiums for every $100 in insurance. Or another way to think about it, let's say that there were 100 Sal's, 134-year-olds looking to get 20-year-term life insurance. And they insured all of them. So if you multiplied this times 100, if you multiplied this by 100, they would get $100 in premiums. $100 in premiums. This is the case where you have 100 Sal's, or 100 people who are pretty similar to me."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And they insured all of them. So if you multiplied this times 100, if you multiplied this by 100, they would get $100 in premiums. $100 in premiums. This is the case where you have 100 Sal's, or 100 people who are pretty similar to me. 100 Sal's. They would get $100 in premium. And the only way that they could make money is if, at most, one of those Sal's, or really just break even, if at most one of those Sal's were to die."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "This is the case where you have 100 Sal's, or 100 people who are pretty similar to me. 100 Sal's. They would get $100 in premium. And the only way that they could make money is if, at most, one of those Sal's, or really just break even, if at most one of those Sal's were to die. So break even if only one Sal dies. I don't like talking about this. It's a little bit morbid."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And the only way that they could make money is if, at most, one of those Sal's, or really just break even, if at most one of those Sal's were to die. So break even if only one Sal dies. I don't like talking about this. It's a little bit morbid. So one way to think about it, they're getting $1 in premium for $100 insurance. Or if they had 100 Sal's, they would get $100 in premium. And the only way they would break even if only one of those Sal's dies."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "It's a little bit morbid. So one way to think about it, they're getting $1 in premium for $100 insurance. Or if they had 100 Sal's, they would get $100 in premium. And the only way they would break even if only one of those Sal's dies. So what they're really saying is that the only way they can break even is if the probability of Sal dying in the next 20 years is less than or equal to 1 in 100. And this is an insurance company. They're trying to make money."}, {"video_title": "Term life insurance and death probability   Finance & Capital Markets   Khan Academy.mp3", "Sentence": "And the only way they would break even if only one of those Sal's dies. So what they're really saying is that the only way they can break even is if the probability of Sal dying in the next 20 years is less than or equal to 1 in 100. And this is an insurance company. They're trying to make money. So they're probably giving these numbers because they think the probability of me dying is a good, maybe it's 1 in 200, or it's 1 in 300, something lower. So that they can insure, one way to think about it, they could insure more Sal's for every $100 in premium they have to pay out. But either way, it's a back of the envelope way of thinking about it."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Lucio wants to test whether playing violent video games makes people more violent. He asks his friends whether they play violent video games and whether they have been in a fight in the last month. He recorded the results in the table shown below. Fill in the table to show the fraction of each group of students who have been in a fight. Then decide whether there's an association between violent video games and getting in a fight amongst Lucio's friends. So let's see what they're doing here. So students who play video games, fractions who have been in a fight, fraction who haven't."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Fill in the table to show the fraction of each group of students who have been in a fight. Then decide whether there's an association between violent video games and getting in a fight amongst Lucio's friends. So let's see what they're doing here. So students who play video games, fractions who have been in a fight, fraction who haven't. Students who don't play violent video games, fraction who have been in a fight, fraction who haven't. So let's answer the first part of this. Students who play violent video games."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So students who play video games, fractions who have been in a fight, fraction who haven't. Students who don't play violent video games, fraction who have been in a fight, fraction who haven't. So let's answer the first part of this. Students who play violent video games. So let's look at those students. So the students who play violent video games, it looks like Ellen plays violent video games. Actually, let me just focus on the data that we care about."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Students who play violent video games. So let's look at those students. So the students who play violent video games, it looks like Ellen plays violent video games. Actually, let me just focus on the data that we care about. So Ellen, so let's look at all the people who play violent video games. So let's see. This column is violent video games, so we have a yes here."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me just focus on the data that we care about. So Ellen, so let's look at all the people who play violent video games. So let's see. This column is violent video games, so we have a yes here. So we have both of these right over here. And then we have down here. And then that's all of them."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "This column is violent video games, so we have a yes here. So we have both of these right over here. And then we have down here. And then that's all of them. There's one, two, three, four, five people who play violent video games. Now what fraction of them have been in a fight? Well, it looks like one out of the five have been in a fight."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then that's all of them. There's one, two, three, four, five people who play violent video games. Now what fraction of them have been in a fight? Well, it looks like one out of the five have been in a fight. The rest of them have not been in a fight. So we could say 1 5th. Let's just write that down."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it looks like one out of the five have been in a fight. The rest of them have not been in a fight. So we could say 1 5th. Let's just write that down. So 1 5th have been in a fight. 1 5th have been in a fight. Fraction who haven't, 4 5ths."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's just write that down. So 1 5th have been in a fight. 1 5th have been in a fight. Fraction who haven't, 4 5ths. So that's all these other nos. They play violent video games, but they haven't been in a fight, one, two, three, four, 4 5ths. So students who don't play violent video games."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "Fraction who haven't, 4 5ths. So that's all these other nos. They play violent video games, but they haven't been in a fight, one, two, three, four, 4 5ths. So students who don't play violent video games. Well, that's everyone else. And let's see how many data points that is. That is one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So students who don't play violent video games. Well, that's everyone else. And let's see how many data points that is. That is one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. So there's a total of 15 students. And how many of them have been in a fight? So let's see."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. So there's a total of 15 students. And how many of them have been in a fight? So let's see. We have one, and then let's see, one, and two, and three. So three out of the 15 have been in a fight. So three out of 15 is the same thing as one out of five."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's see. We have one, and then let's see, one, and two, and three. So three out of the 15 have been in a fight. So three out of 15 is the same thing as one out of five. Those are equivalent fractions. And then the fraction who haven't, well, that's just going to be everyone else. That's going to be 12 out of 15 or four out of five."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "So three out of 15 is the same thing as one out of five. Those are equivalent fractions. And then the fraction who haven't, well, that's just going to be everyone else. That's going to be 12 out of 15 or four out of five. So based on Lucio's data, and this wasn't a huge sample size, obviously, he only found five kids who were playing violent video games, and one of them had gotten into a fight. So this isn't a super rigorous study. But at least based on his data, if we're trying to decide whether there's an association between violent games and getting into a fight amongst Lucio's friends, it doesn't seem like there is."}, {"video_title": "Video games and violence   Statistical studies   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's going to be 12 out of 15 or four out of five. So based on Lucio's data, and this wasn't a huge sample size, obviously, he only found five kids who were playing violent video games, and one of them had gotten into a fight. So this isn't a super rigorous study. But at least based on his data, if we're trying to decide whether there's an association between violent games and getting into a fight amongst Lucio's friends, it doesn't seem like there is. It seems like relatively, whether or not they play violent video games or not, one fifth of them have been in a fight in the last month. So it really doesn't seem any difference. If this number was, I don't know, four fifths or five fifths or all of them, then I would say, hey, even with Lucio's fairly small sample, I would say, hey, maybe there is some type of a strong association between playing violent video games and fighting."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And as we will see as we build up our understanding of them, not only are they interesting in their own right, but there's a lot of very powerful probability and statistics that we can do based on our understanding of binomial variables. So to make things concrete as quickly as possible, I'll start with a very tangible example of a binomial variable, and then we'll think a little bit more abstractly about what makes it binomial. So let's say that I have a coin. So this is my coin here. Doesn't even have to be a fair coin. Let me just draw this really fast. So that's my coin."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this is my coin here. Doesn't even have to be a fair coin. Let me just draw this really fast. So that's my coin. And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4. And what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after 10 flips of my coin. Now what makes this a binomial variable?"}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So that's my coin. And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4. And what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after 10 flips of my coin. Now what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials. So it's made up, made up of independent, independent trials. Now what do I mean by independent trials?"}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials. So it's made up, made up of independent, independent trials. Now what do I mean by independent trials? Well, a trial is each flip of my coin. So a flip is equal to a trial in the language of this statement that I just made. And what do I mean by each flip or each trial being independent?"}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now what do I mean by independent trials? Well, a trial is each flip of my coin. So a flip is equal to a trial in the language of this statement that I just made. And what do I mean by each flip or each trial being independent? Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip. So in this case, we are made up of independent trials. So another condition is each trial can be clearly classified as either a success or failure."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And what do I mean by each flip or each trial being independent? Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip. So in this case, we are made up of independent trials. So another condition is each trial can be clearly classified as either a success or failure. Or another way of thinking about it, each trial clearly has one of two discrete outcomes. So each trial, and in the example I'm giving, the flip is a trial, can be classified, can be classified as either success or failure. So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So another condition is each trial can be clearly classified as either a success or failure. Or another way of thinking about it, each trial clearly has one of two discrete outcomes. So each trial, and in the example I'm giving, the flip is a trial, can be classified, can be classified as either success or failure. So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. And so you're either going to have success or failure. You're either gonna have heads or tails on each of these trials. Now another condition for being a binomial variable is that you have a fixed number of trials."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. And so you're either going to have success or failure. You're either gonna have heads or tails on each of these trials. Now another condition for being a binomial variable is that you have a fixed number of trials. Fixed number of trials. So in this case, we're saying that we have 10 trials, 10 flips of our coin. And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Now another condition for being a binomial variable is that you have a fixed number of trials. Fixed number of trials. So in this case, we're saying that we have 10 trials, 10 flips of our coin. And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. And we've already talked about it. On each trial on each flip, the probability of heads is going to stay at 0.6. If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. And we've already talked about it. On each trial on each flip, the probability of heads is going to stay at 0.6. If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. And so you might say, okay, that's reasonable. I get why this is a binomial variable. Can you give me an example of something that is not a binomial variable?"}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. And so you might say, okay, that's reasonable. I get why this is a binomial variable. Can you give me an example of something that is not a binomial variable? Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement. So you might immediately say, well, this feels like it could be binomial. We have each trial can be classified as either a success or failure."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Can you give me an example of something that is not a binomial variable? Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement. So you might immediately say, well, this feels like it could be binomial. We have each trial can be classified as either a success or failure. For each trials, when I take a card out, if I get a king, that looks like that would be a success. If I don't get a king, that would be a failure. So it seems to meet that right over there."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "We have each trial can be classified as either a success or failure. For each trials, when I take a card out, if I get a king, that looks like that would be a success. If I don't get a king, that would be a failure. So it seems to meet that right over there. It has a fixed number of trials. I'm taking two cards out of the deck, so it seems to meet that. But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant?"}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it seems to meet that right over there. It has a fixed number of trials. I'm taking two cards out of the deck, so it seems to meet that. But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it. So the probability of a king on the first trial would be four out of 52. But what about the probability of getting a king on the second, on the second trial?"}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it. So the probability of a king on the first trial would be four out of 52. But what about the probability of getting a king on the second, on the second trial? What would this be equal to? Well, it depends on what happened on the first trial. If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "But what about the probability of getting a king on the second, on the second trial? What would this be equal to? Well, it depends on what happened on the first trial. If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. Well, now, there would be three kings left in a deck of 51 cards. But if you did not get a king on the first trial, now you have four kings in a deck of 51 cards, because remember, we're doing it without replacement. You're just taking that first card, whatever you did, and you're taking it aside."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. Well, now, there would be three kings left in a deck of 51 cards. But if you did not get a king on the first trial, now you have four kings in a deck of 51 cards, because remember, we're doing it without replacement. You're just taking that first card, whatever you did, and you're taking it aside. So what's interesting here is this is not made up of independent trials. It does not meet this condition. The probability on your second trial is dependent on what happens on your first trial."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You're just taking that first card, whatever you did, and you're taking it aside. So what's interesting here is this is not made up of independent trials. It does not meet this condition. The probability on your second trial is dependent on what happens on your first trial. And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant. And so that's why this right over here is not a binomial variable. Now, if y, if we got rid of, without replacement, and if we said we did replace every card after we picked it, then things would be different."}, {"video_title": "Binomial variables   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The probability on your second trial is dependent on what happens on your first trial. And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant. And so that's why this right over here is not a binomial variable. Now, if y, if we got rid of, without replacement, and if we said we did replace every card after we picked it, then things would be different. Then we actually would be looking at a binomial variable. So instead of without replacement, if I just said with replacement, well then, your probability of a king on each trial is going to be four out of 52. You have a finite number of trials."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "So we have some data here that we can plot on a scatter plot that looks something like that. And so the next question, given that we've been talking a lot about lines of regression or regression lines, is can we fit a regression line to this? Well, if we try to, we might get something that looks like this, or maybe something that looks like this. I'm just eyeballing it. Obviously, we could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line. But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "I'm just eyeballing it. Obviously, we could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line. But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this data set. And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is this exact same data set."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this data set. And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is this exact same data set. You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "So this is this exact same data set. You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23. I did that for all of these data points. I did it on a spreadsheet."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "So 10 to the what power is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23. I did that for all of these data points. I did it on a spreadsheet. And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear. Now, be clear."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "I did it on a spreadsheet. And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear. Now, be clear. The true relationship between x and y is not linear. It looks like some type of an exponential relationship. But the value of transforming the data, and there's different ways you can do it."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "Now, be clear. The true relationship between x and y is not linear. It looks like some type of an exponential relationship. But the value of transforming the data, and there's different ways you can do it. In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression. Because this data set, you could actually fit a linear regression line to this quite well. You could imagine a line that looks something like this."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "But the value of transforming the data, and there's different ways you can do it. In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression. Because this data set, you could actually fit a linear regression line to this quite well. You could imagine a line that looks something like this. It would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals. And so this might be the direction you wanna go at."}, {"video_title": "Transforming nonlinear data   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "You could imagine a line that looks something like this. It would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals. And so this might be the direction you wanna go at. And what's neat is once you fit a linear regression, it's not difficult to mathematically unwind from your linear model back to an exponential one. So the big takeaway here is is that the tools of linear regression can be useful even when the underlying relationship between x and y are nonlinear. And the way that we do that is by transforming the data."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "She randomized 50 workdays between a treatment group and a control group. For each day from the treatment group, she took bus A, and for each day from the control group, she took bus B. Each day, she timed the length of her drive. This is really interesting what she did. It's very important. She randomized the 50 workdays. It's important she did this instead of just kind of waking up in the morning and just deciding on her own which bus to take because humans are infamously bad at being random."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is really interesting what she did. It's very important. She randomized the 50 workdays. It's important she did this instead of just kind of waking up in the morning and just deciding on her own which bus to take because humans are infamously bad at being random. Even when we think we're being random, we're actually not that random. She might inadvertently be taking bus A earlier in the week where maybe the commute times are shorter, or maybe she inadvertently takes bus A when the weather is better, when there's less traffic. Remember, there's a natural tendency for human beings to want to confirm their hypothesis."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's important she did this instead of just kind of waking up in the morning and just deciding on her own which bus to take because humans are infamously bad at being random. Even when we think we're being random, we're actually not that random. She might inadvertently be taking bus A earlier in the week where maybe the commute times are shorter, or maybe she inadvertently takes bus A when the weather is better, when there's less traffic. Remember, there's a natural tendency for human beings to want to confirm their hypothesis. If she thinks that bus A is faster, maybe she'll want to pick the days where she'll get data to confirm her hypothesis. It's really important that she randomized the 50 workdays. What I can imagine she did is maybe she wrote each of the workdays, the dates, on a piece of paper."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, there's a natural tendency for human beings to want to confirm their hypothesis. If she thinks that bus A is faster, maybe she'll want to pick the days where she'll get data to confirm her hypothesis. It's really important that she randomized the 50 workdays. What I can imagine she did is maybe she wrote each of the workdays, the dates, on a piece of paper. She would have 50 pieces of paper, and then she turned them all upside down, or maybe she closed her eyes, and then she moved them all over her table. Then with her eyes closed, she randomly moved them to either the left or the right of the table. If they move to the left of the table, then those are the days she'll take bus A."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I can imagine she did is maybe she wrote each of the workdays, the dates, on a piece of paper. She would have 50 pieces of paper, and then she turned them all upside down, or maybe she closed her eyes, and then she moved them all over her table. Then with her eyes closed, she randomly moved them to either the left or the right of the table. If they move to the left of the table, then those are the days she'll take bus A. If she moves them to the right of the table, those are the days she takes bus B. That's how she can make sure that this is truly random. All right."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "If they move to the left of the table, then those are the days she'll take bus A. If she moves them to the right of the table, those are the days she takes bus B. That's how she can make sure that this is truly random. All right. Then they tell us, the results, this is important, the results of the experiment show that the median travel duration for bus A is eight minutes less than the median travel duration for bus B. Or one way to think about it, if we said the treatment group median minus the control group median, what would we get? Well, the treatment group is eight minutes less than the control group."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right. Then they tell us, the results, this is important, the results of the experiment show that the median travel duration for bus A is eight minutes less than the median travel duration for bus B. Or one way to think about it, if we said the treatment group median minus the control group median, what would we get? Well, the treatment group is eight minutes less than the control group. This is A, this is B. If this is eight less than this, then this is going to be equal to negative eight. This is just another way of restating what I have underlined right over here."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the treatment group is eight minutes less than the control group. This is A, this is B. If this is eight less than this, then this is going to be equal to negative eight. This is just another way of restating what I have underlined right over here. Someone's car alarm went off. I hope you're not hearing that. Anyway, I'll try to pay attention while it's going off."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is just another way of restating what I have underlined right over here. Someone's car alarm went off. I hope you're not hearing that. Anyway, I'll try to pay attention while it's going off. To test whether the results could be explained by random chance, she created the table below, which summarizes the results of 1,000 re-randomizations of the data, with differences between medians rounded to the nearest five minutes. What is going on over here? You might say, well, look, she got her result that she wanted to get."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Anyway, I'll try to pay attention while it's going off. To test whether the results could be explained by random chance, she created the table below, which summarizes the results of 1,000 re-randomizations of the data, with differences between medians rounded to the nearest five minutes. What is going on over here? You might say, well, look, she got her result that she wanted to get. She sees this data seems to confirm that bus A gets her to work faster. What's all this other business with re-randomizations she's doing? Well, the important thing to realize is, and she realizes this, is that she might have just gotten this data that I underlined by random chance."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "You might say, well, look, she got her result that she wanted to get. She sees this data seems to confirm that bus A gets her to work faster. What's all this other business with re-randomizations she's doing? Well, the important thing to realize is, and she realizes this, is that she might have just gotten this data that I underlined by random chance. There's some chance maybe A and B are completely similar in terms of how long they take in reality, and she just happened to pick bus A on days where bus A got to work faster. Maybe bus B is faster, but she just happened to take bus A on the days that it was faster, the days that just happened to have less traffic. What she's doing here is she re-randomized the data, and she wants to see, well, with all this re-randomized data, out of these 1,000 re-randomizations, what fraction of them do I get a result like this?"}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the important thing to realize is, and she realizes this, is that she might have just gotten this data that I underlined by random chance. There's some chance maybe A and B are completely similar in terms of how long they take in reality, and she just happened to pick bus A on days where bus A got to work faster. Maybe bus B is faster, but she just happened to take bus A on the days that it was faster, the days that just happened to have less traffic. What she's doing here is she re-randomized the data, and she wants to see, well, with all this re-randomized data, out of these 1,000 re-randomizations, what fraction of them do I get a result like this? Do I get a result where A is eight minutes or more faster, I guess, or you could say that the median travel duration for bus A is eight minutes less or even less than that, than the median travel for bus B. If it was nine minutes less, or 10 minutes less, or 15 minutes less, those are all the interesting ones. Those are the ones that confirm our hypothesis that bus A gets to work faster."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "What she's doing here is she re-randomized the data, and she wants to see, well, with all this re-randomized data, out of these 1,000 re-randomizations, what fraction of them do I get a result like this? Do I get a result where A is eight minutes or more faster, I guess, or you could say that the median travel duration for bus A is eight minutes less or even less than that, than the median travel for bus B. If it was nine minutes less, or 10 minutes less, or 15 minutes less, those are all the interesting ones. Those are the ones that confirm our hypothesis that bus A gets to work faster. Let's look at this table. It's not below. It's actually to the right."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Those are the ones that confirm our hypothesis that bus A gets to work faster. Let's look at this table. It's not below. It's actually to the right. Let's just remind ourselves what she did here because the first time you try to process this, it can seem a little bit daunting. In her experiment, let me write this down, experiment, the car alarm outside, which you probably, hopefully, are not hearing. It's actually a surprisingly pleasant-sounding car alarm."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually to the right. Let's just remind ourselves what she did here because the first time you try to process this, it can seem a little bit daunting. In her experiment, let me write this down, experiment, the car alarm outside, which you probably, hopefully, are not hearing. It's actually a surprisingly pleasant-sounding car alarm. It sounds like a slightly obnoxious bird. Anyway, her experiment is, the way I described it, 25 days she would take bus A, 25 days she took bus B, and she would record all the travel times. Let's say that I just have 25 data points in each column."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's actually a surprisingly pleasant-sounding car alarm. It sounds like a slightly obnoxious bird. Anyway, her experiment is, the way I described it, 25 days she would take bus A, 25 days she took bus B, and she would record all the travel times. Let's say that I just have 25 data points in each column. Let's say you get 12 minutes, 20 minutes, 25 minutes, and you just keep going. There's 25 data points. Let's just say that there are 12 data points less than 20 minutes and 12 data points more than 20 minutes."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say that I just have 25 data points in each column. Let's say you get 12 minutes, 20 minutes, 25 minutes, and you just keep going. There's 25 data points. Let's just say that there are 12 data points less than 20 minutes and 12 data points more than 20 minutes. In this circumstance, her median time for bus A would be 20 minutes. I just made this number up. In order for this to be eight minutes less than the median time for bus B, the median for bus B, we would have to be 28."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's just say that there are 12 data points less than 20 minutes and 12 data points more than 20 minutes. In this circumstance, her median time for bus A would be 20 minutes. I just made this number up. In order for this to be eight minutes less than the median time for bus B, the median for bus B, we would have to be 28. Maybe you have data points here. Maybe this is 18. You have 12 more that are less than 28."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "In order for this to be eight minutes less than the median time for bus B, the median for bus B, we would have to be 28. Maybe you have data points here. Maybe this is 18. You have 12 more that are less than 28. Then you have 12 more that are greater than 28. The median time for bus B would be 28. Once again, I just made this data up."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "You have 12 more that are less than 28. Then you have 12 more that are greater than 28. The median time for bus B would be 28. Once again, I just made this data up. If you took treatment group median, and I'll just write TGM for short, TGM minus control group median, what do you get? 20 minus 28 is negative eight. This is the actual results of, these are theoretical or potential results, hypothetical results, for her actual experiment."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Once again, I just made this data up. If you took treatment group median, and I'll just write TGM for short, TGM minus control group median, what do you get? 20 minus 28 is negative eight. This is the actual results of, these are theoretical or potential results, hypothetical results, for her actual experiment. What's all this business over here? What she did is she took these times and she said, you know what, let's just imagine a world where I could have gotten any of these times randomly on either bus. She just randomly resorted them between A and B."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the actual results of, these are theoretical or potential results, hypothetical results, for her actual experiment. What's all this business over here? What she did is she took these times and she said, you know what, let's just imagine a world where I could have gotten any of these times randomly on either bus. She just randomly resorted them between A and B. She did that a thousand times. The first time, the second time, the third time, and she does this 1,000 times. I'm assuming she used some type of a computer program to do it."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "She just randomly resorted them between A and B. She did that a thousand times. The first time, the second time, the third time, and she does this 1,000 times. I'm assuming she used some type of a computer program to do it. Each time, once again, she just took the data that she had and she just rearranged it. She just reshuffled it. Maybe A on one day, maybe it gets this 18, maybe it gets the 25, maybe it gets a 30."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm assuming she used some type of a computer program to do it. Each time, once again, she just took the data that she had and she just rearranged it. She just reshuffled it. Maybe A on one day, maybe it gets this 18, maybe it gets the 25, maybe it gets a 30. Once again, so we've got the 18, the 25, and the 30, and maybe B gets the, and she's reshuffling all this other data points that I just have with dots, and maybe B, let's see, she got the 18, 25, and 30, maybe 12, 20, and 28. In this circumstance, this random reshuffling, and she keeps doing it over and over again, in this random reshuffling, the treatment group median minus the control group median is going to be what? It's going to be equal to positive five."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe A on one day, maybe it gets this 18, maybe it gets the 25, maybe it gets a 30. Once again, so we've got the 18, the 25, and the 30, and maybe B gets the, and she's reshuffling all this other data points that I just have with dots, and maybe B, let's see, she got the 18, 25, and 30, maybe 12, 20, and 28. In this circumstance, this random reshuffling, and she keeps doing it over and over again, in this random reshuffling, the treatment group median minus the control group median is going to be what? It's going to be equal to positive five. In this random shuffling, this hypothetical scenario, bus A's median would have been five minutes more, longer than bus B's. If she gets this result with this random resorting, this would have been, and this is actually a, she would have had a column here for five, and then she would have notched, put one notch right over here, but it looks like she classified things, or maybe she didn't even get the data, but she classified them by multiples of two. But then if she got this again, then she would have put a two here, and then she should have said, okay, in how many of these random reshufflings am I getting a scenario where there's a five-minute difference, or where the treatment group is five minutes longer?"}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be equal to positive five. In this random shuffling, this hypothetical scenario, bus A's median would have been five minutes more, longer than bus B's. If she gets this result with this random resorting, this would have been, and this is actually a, she would have had a column here for five, and then she would have notched, put one notch right over here, but it looks like she classified things, or maybe she didn't even get the data, but she classified them by multiples of two. But then if she got this again, then she would have put a two here, and then she should have said, okay, in how many of these random reshufflings am I getting a scenario where there's a five-minute difference, or where the treatment group is five minutes longer? So what is this saying? So for example, this is saying that 18 out of the 1,000 reshufflings, which she just randomly reshuffled the data, 18 out of those 1,000 times, she found a scenario where her treatment group median was 10 minutes longer than her control group, where bus A's median was, this hypothetical re-randomization, where the treatment group is 10 minutes slower than the control group. There were 159 times where the treatment group, once again, in her random reshuffling, these aren't based on observations, these are random reshufflings, there's 159 times where her treatment group is four minutes slower than her control group."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "But then if she got this again, then she would have put a two here, and then she should have said, okay, in how many of these random reshufflings am I getting a scenario where there's a five-minute difference, or where the treatment group is five minutes longer? So what is this saying? So for example, this is saying that 18 out of the 1,000 reshufflings, which she just randomly reshuffled the data, 18 out of those 1,000 times, she found a scenario where her treatment group median was 10 minutes longer than her control group, where bus A's median was, this hypothetical re-randomization, where the treatment group is 10 minutes slower than the control group. There were 159 times where the treatment group, once again, in her random reshuffling, these aren't based on observations, these are random reshufflings, there's 159 times where her treatment group is four minutes slower than her control group. So the whole reason for doing this is she says, okay, what's the probability of getting a result like this or better? And I say better is, you know, I guess one that even more confirms her hypothesis, that the treatment group is faster than the control group. Well, this scenario is this one right over here, and then another one where the treatment group is even faster is this right over here."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "There were 159 times where the treatment group, once again, in her random reshuffling, these aren't based on observations, these are random reshufflings, there's 159 times where her treatment group is four minutes slower than her control group. So the whole reason for doing this is she says, okay, what's the probability of getting a result like this or better? And I say better is, you know, I guess one that even more confirms her hypothesis, that the treatment group is faster than the control group. Well, this scenario is this one right over here, and then another one where the treatment group is even faster is this right over here. Here the treatment group median is 10 less than the control group median. So in how many of these scenarios out of the 1,000 is this occurring? Well, this one occurs 85 times, this one occurs eight, so if you add these two together, 93 out of the 1,000 times out of her re-randomization, or I guess you could say 9.3% of the time, if the data, 9.3% of the randomized, in the 1,000 re-randomizations, 9.3% of the time she got data that was as validating of a hypothesis or more than the actual experiment."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this scenario is this one right over here, and then another one where the treatment group is even faster is this right over here. Here the treatment group median is 10 less than the control group median. So in how many of these scenarios out of the 1,000 is this occurring? Well, this one occurs 85 times, this one occurs eight, so if you add these two together, 93 out of the 1,000 times out of her re-randomization, or I guess you could say 9.3% of the time, if the data, 9.3% of the randomized, in the 1,000 re-randomizations, 9.3% of the time she got data that was as validating of a hypothesis or more than the actual experiment. So one way to think about this is the probability of randomly getting the results from her experiment, or better results from her experiment, are 9.3%. So they're low, it's a reasonably low probability that this happened purely by chance. Now, a question is, well, what's the threshold?"}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, this one occurs 85 times, this one occurs eight, so if you add these two together, 93 out of the 1,000 times out of her re-randomization, or I guess you could say 9.3% of the time, if the data, 9.3% of the randomized, in the 1,000 re-randomizations, 9.3% of the time she got data that was as validating of a hypothesis or more than the actual experiment. So one way to think about this is the probability of randomly getting the results from her experiment, or better results from her experiment, are 9.3%. So they're low, it's a reasonably low probability that this happened purely by chance. Now, a question is, well, what's the threshold? If it was a 50%, you say, okay, this was, you know, very likely to happen by chance. If this was a 25%, you're like, okay, it's less likely to happen by chance, but it could happen. 9.3%, I mean, it's roughly 10%, you know, for every 10 people who do an experiment like she did, one person, even if it was random, one person would get data like this."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, a question is, well, what's the threshold? If it was a 50%, you say, okay, this was, you know, very likely to happen by chance. If this was a 25%, you're like, okay, it's less likely to happen by chance, but it could happen. 9.3%, I mean, it's roughly 10%, you know, for every 10 people who do an experiment like she did, one person, even if it was random, one person would get data like this. So what typically happens among statisticians is they draw a threshold, and the threshold for statistical significance is usually 5%. So one way to think about the probability of her getting this result by chance, or this result, or a more extreme result, one that more confirms her hypothesis by chance, is 9.3%. Now, if your cutoff for significance is 5%, if you say, okay, this has to be 5% or less, then you say, okay, this is not statistically significant."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "9.3%, I mean, it's roughly 10%, you know, for every 10 people who do an experiment like she did, one person, even if it was random, one person would get data like this. So what typically happens among statisticians is they draw a threshold, and the threshold for statistical significance is usually 5%. So one way to think about the probability of her getting this result by chance, or this result, or a more extreme result, one that more confirms her hypothesis by chance, is 9.3%. Now, if your cutoff for significance is 5%, if you say, okay, this has to be 5% or less, then you say, okay, this is not statistically significant. There's a more than a 5% chance that I could have gotten this result purely through random chance. Now, once again, that just depends on where you have that threshold. So when we go back, I think we've already answered the final question."}, {"video_title": "Statistical significance on bus speeds   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, if your cutoff for significance is 5%, if you say, okay, this has to be 5% or less, then you say, okay, this is not statistically significant. There's a more than a 5% chance that I could have gotten this result purely through random chance. Now, once again, that just depends on where you have that threshold. So when we go back, I think we've already answered the final question. According to the simulations, what is the probability of the treatment group's median being lower than the control group's median by eight minutes or more? Which, once again, eight minutes or more, that would be negative eight and negative 10, and we just figured that out. That was 93 out of the 1,000 re-randomizations, so it's a 9.3% chance."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "Her test statistic, I can never say that right, was t is equal to negative 1.9. Assume that the conditions for inference were met. What is the approximate p-value for Miriam's test? So pause this video and see if you can figure this out on your own. All right, well I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "So pause this video and see if you can figure this out on your own. All right, well I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18. So to test that null hypothesis, Miriam takes a sample. Sample size is equal to seven. From that, she would calculate her sample mean and her sample standard deviation."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "The alternative is that it's less than 18. So to test that null hypothesis, Miriam takes a sample. Sample size is equal to seven. From that, she would calculate her sample mean and her sample standard deviation. And from that, she would calculate this t-statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean. And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "From that, she would calculate her sample mean and her sample standard deviation. And from that, she would calculate this t-statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean. And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us. This is equal to negative 1.9. And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean?"}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us. This is equal to negative 1.9. And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9. So it's this area right there. And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9. So it's this area right there. And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. So let's go there, that's the number six right there, click Enter. And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number. Our upper bound would be negative 1.9, negative 1.9."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. So let's go there, that's the number six right there, click Enter. And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number. Our upper bound would be negative 1.9, negative 1.9. And then our degrees of freedom, that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six, and so there we have it. And then so this would be, our p-value would be approximately 0.053."}, {"video_title": "Using TI calculator for P-value from t statistic   AP Statistics   Khan Academy.mp3", "Sentence": "Our upper bound would be negative 1.9, negative 1.9. And then our degrees of freedom, that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six, and so there we have it. And then so this would be, our p-value would be approximately 0.053. So our p-value would be approximately 0.053. And then what Miriam would do is, would compare this p-value to her preset significance level, to alpha. If this is below alpha, then she would reject her null hypothesis, which would suggest the alternative."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And this is in the context of significance testing. So just as a little bit of review, in order to do a significance test, we first come up with a null and an alternative hypothesis. And we'll do this on some population in question. These will say some hypotheses about a true parameter for this population. And the null hypothesis tends to be kind of what was always assumed or the status quo, while the alternative hypothesis, hey, there's news here, there's something alternative here. And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample. We take a sample from this population."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "These will say some hypotheses about a true parameter for this population. And the null hypothesis tends to be kind of what was always assumed or the status quo, while the alternative hypothesis, hey, there's news here, there's something alternative here. And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample. We take a sample from this population. Using that sample, we calculate a statistic, we calculate a statistic that's trying to estimate the parameter in question. And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true. And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "We take a sample from this population. Using that sample, we calculate a statistic, we calculate a statistic that's trying to estimate the parameter in question. And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true. And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. Let me write this down. So this right over here, this is our p-value. This should be all we review, we introduce it in other videos."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. Let me write this down. So this right over here, this is our p-value. This should be all we review, we introduce it in other videos. We have seen in other videos if our p-value is less than our significance level, then we reject, reject our null hypothesis. And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis. And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "This should be all we review, we introduce it in other videos. We have seen in other videos if our p-value is less than our significance level, then we reject, reject our null hypothesis. And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis. And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. And the reason why this makes sense is if the probability of getting this statistic from a sample of a certain size, if we assume that the null hypothesis is true, is reasonably low, if it's below a threshold, maybe this threshold is 5%, if the probability of that happening was less than 5%, then hey, maybe it's reasonable to reject it. But we might be wrong in either of these scenarios, and that's where these errors come into play. Let's make a grid to make this clear."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. And the reason why this makes sense is if the probability of getting this statistic from a sample of a certain size, if we assume that the null hypothesis is true, is reasonably low, if it's below a threshold, maybe this threshold is 5%, if the probability of that happening was less than 5%, then hey, maybe it's reasonable to reject it. But we might be wrong in either of these scenarios, and that's where these errors come into play. Let's make a grid to make this clear. So there's the reality. Let me put reality up here. So the reality is there's two possible scenarios in reality."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Let's make a grid to make this clear. So there's the reality. Let me put reality up here. So the reality is there's two possible scenarios in reality. One is is that the null hypothesis is true, and the other is that the null hypothesis is false. And then based on our significance test, there's two things that we might do. We might reject the null hypothesis, or we might fail to reject the null hypothesis."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "So the reality is there's two possible scenarios in reality. One is is that the null hypothesis is true, and the other is that the null hypothesis is false. And then based on our significance test, there's two things that we might do. We might reject the null hypothesis, or we might fail to reject the null hypothesis. And so let's put a little grid here to think about the different combinations, the different scenarios here. So in a scenario where the null hypothesis is true, but we reject it, that feels like an error. We shouldn't reject something that is true, and that indeed is a type one error."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "We might reject the null hypothesis, or we might fail to reject the null hypothesis. And so let's put a little grid here to think about the different combinations, the different scenarios here. So in a scenario where the null hypothesis is true, but we reject it, that feels like an error. We shouldn't reject something that is true, and that indeed is a type one error. Type one error. You shouldn't reject the null hypothesis if it was true. You should reject the null hypothesis if it was true."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "We shouldn't reject something that is true, and that indeed is a type one error. Type one error. You shouldn't reject the null hypothesis if it was true. You should reject the null hypothesis if it was true. And you could even figure out what is the probability of getting a type one error? Well, that's gonna be your significance level. Because if your null hypothesis is true, let's say that your significance level is 5%."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "You should reject the null hypothesis if it was true. And you could even figure out what is the probability of getting a type one error? Well, that's gonna be your significance level. Because if your null hypothesis is true, let's say that your significance level is 5%. Well, 5% of the time, even if your null hypothesis is true, you're going to get a statistic that's going to make you reject the null hypothesis. So one way to think about the probability of a type one error is your significance level. Now, if your null hypothesis is true and you fail to reject it, well, that's good."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Because if your null hypothesis is true, let's say that your significance level is 5%. Well, 5% of the time, even if your null hypothesis is true, you're going to get a statistic that's going to make you reject the null hypothesis. So one way to think about the probability of a type one error is your significance level. Now, if your null hypothesis is true and you fail to reject it, well, that's good. This, we could write this as, this is a correct, correct conclusion. The good thing just happened to happen this time. Now, if your null hypothesis is false and you reject it, that's also good."}, {"video_title": "Introduction to Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Now, if your null hypothesis is true and you fail to reject it, well, that's good. This, we could write this as, this is a correct, correct conclusion. The good thing just happened to happen this time. Now, if your null hypothesis is false and you reject it, that's also good. That is the correct, correct conclusion. But if your null hypothesis is false and you fail to reject it, well, then that is a type two error. That is a type two error."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So let's go back to an example that we've seen before. We're rolling a fair six-sided die. And so there's six possibilities. We could get a one, a two, a three, a four, a five, or a six. Now let's say we ask ourselves, what is the probability of rolling, of rolling a number that is less than or equal to, less than or equal to two? What is this going to be? Well, there are six equally likely possibilities."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "We could get a one, a two, a three, a four, a five, or a six. Now let's say we ask ourselves, what is the probability of rolling, of rolling a number that is less than or equal to, less than or equal to two? What is this going to be? Well, there are six equally likely possibilities. And rolling less than or equal to two, well, that means I'm either rolling a one or a two. So two, one, two, of the six equally likely possibilities meet my constraints. So there is a 2 6th probability of rolling a number less than or equal to two."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, there are six equally likely possibilities. And rolling less than or equal to two, well, that means I'm either rolling a one or a two. So two, one, two, of the six equally likely possibilities meet my constraints. So there is a 2 6th probability of rolling a number less than or equal to two. Or I could just rewrite that as an equivalent fraction. I could say there's a 1 3rd probability. I could go either way."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So there is a 2 6th probability of rolling a number less than or equal to two. Or I could just rewrite that as an equivalent fraction. I could say there's a 1 3rd probability. I could go either way. Now let's ask ourselves another question. What is the probability of rolling a number greater than or equal, greater than or equal to three? Well, once again, there are six equally likely possibilities."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "I could go either way. Now let's ask ourselves another question. What is the probability of rolling a number greater than or equal, greater than or equal to three? Well, once again, there are six equally likely possibilities. And how many of them involve rolling greater than or equal to three? Let's see, one, two, three, four. These possibilities right over here."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, once again, there are six equally likely possibilities. And how many of them involve rolling greater than or equal to three? Let's see, one, two, three, four. These possibilities right over here. Roll a three, a four, a five, or a six. So four out of the six equally likely possibilities. Or I could rewrite this as an equivalent fraction as 2 3rds."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "These possibilities right over here. Roll a three, a four, a five, or a six. So four out of the six equally likely possibilities. Or I could rewrite this as an equivalent fraction as 2 3rds. So what's more likely? Rolling a number that's less than or equal to two or rolling a number that's greater than or equal to three? Well, you can see right over here."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Or I could rewrite this as an equivalent fraction as 2 3rds. So what's more likely? Rolling a number that's less than or equal to two or rolling a number that's greater than or equal to three? Well, you can see right over here. The probability of rolling greater than or equal to three is 2 3rds, while the probability of rolling less than or equal to two is only 1 3rd. This number is greater. So this has a greater probability."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, you can see right over here. The probability of rolling greater than or equal to three is 2 3rds, while the probability of rolling less than or equal to two is only 1 3rd. This number is greater. So this has a greater probability. Or another way of thinking about it, rolling greater than or equal to three is more likely than rolling less than or equal to two. In fact, not only is it more likely, you see that 2 3rds is twice 1 3rd. This right over here is twice as likely."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So this has a greater probability. Or another way of thinking about it, rolling greater than or equal to three is more likely than rolling less than or equal to two. In fact, not only is it more likely, you see that 2 3rds is twice 1 3rd. This right over here is twice as likely. You're twice as likely to roll a number greater than or equal to three than you are to roll a number less than or equal to two. And you can even see right over here that you have twice as many possibilities of the six equally likely ones, four versus two. Four versus two here."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "This right over here is twice as likely. You're twice as likely to roll a number greater than or equal to three than you are to roll a number less than or equal to two. And you can even see right over here that you have twice as many possibilities of the six equally likely ones, four versus two. Four versus two here. And so you say, okay, I get it, Sal. You know, if the probability is a larger number, the event is more likely. It makes sense."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Four versus two here. And so you say, okay, I get it, Sal. You know, if the probability is a larger number, the event is more likely. It makes sense. And in this case, it's twice, the number is twice as large, so it's twice as likely. But what's the range of possible probabilities? How low can a probability get?"}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "It makes sense. And in this case, it's twice, the number is twice as large, so it's twice as likely. But what's the range of possible probabilities? How low can a probability get? And how high can a probability get? So let's think about the first question. So how low, how low can a probability go?"}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "How low can a probability get? And how high can a probability get? So let's think about the first question. So how low, how low can a probability go? How low? So what's the lowest probability that you can imagine for anything? Well, let's give ourselves, let's give ourselves a little bit of a experiment."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So how low, how low can a probability go? How low? So what's the lowest probability that you can imagine for anything? Well, let's give ourselves, let's give ourselves a little bit of a experiment. Let's ask ourselves the probability of rolling, of rolling a seven. Well, once, and pause the video and try to figure it out on your own. Well, there are six equally likely possibilities."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, let's give ourselves, let's give ourselves a little bit of a experiment. Let's ask ourselves the probability of rolling, of rolling a seven. Well, once, and pause the video and try to figure it out on your own. Well, there are six equally likely possibilities. And how many of them involve rolling a seven? Well, none of them. It's impossible to roll a seven."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, there are six equally likely possibilities. And how many of them involve rolling a seven? Well, none of them. It's impossible to roll a seven. So none of the six. So we could say this probability is zero. And if you see a probability of zero, someone says the probability of that thing happening is zero."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "It's impossible to roll a seven. So none of the six. So we could say this probability is zero. And if you see a probability of zero, someone says the probability of that thing happening is zero. That means it's impossible. That means in no, in no world can that happen if it's exactly zero. So this right here, the probability is zero."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And if you see a probability of zero, someone says the probability of that thing happening is zero. That means it's impossible. That means in no, in no world can that happen if it's exactly zero. So this right here, the probability is zero. That means it is impossible. It is impossible. Now how high can a probability get?"}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So this right here, the probability is zero. That means it is impossible. It is impossible. Now how high can a probability get? So how, how high can a probability get? Well, let's think about, let's say probability of rolling, rolling any number, any number from one to six. Well, I have six equally, I have six, I have six equally likely possibilities."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now how high can a probability get? So how, how high can a probability get? Well, let's think about, let's say probability of rolling, rolling any number, any number from one to six. Well, I have six equally, I have six, I have six equally likely possibilities. And any one of those six meets this constraint. I would have rolled a number, any number from one to six, including one and six. So there's six equally likely possibilities."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, I have six equally, I have six, I have six equally likely possibilities. And any one of those six meets this constraint. I would have rolled a number, any number from one to six, including one and six. So there's six equally likely possibilities. And so the probability is one. And so if someone says the probability is zero, it's impossible. And if someone says the probability is one, that means it's definitely going to happen."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "So there's six equally likely possibilities. And so the probability is one. And so if someone says the probability is zero, it's impossible. And if someone says the probability is one, that means it's definitely going to happen. Going, going, going, it's definitely going to happen. So the maximum probability for anything is one. The minimum probability is zero."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And if someone says the probability is one, that means it's definitely going to happen. Going, going, going, it's definitely going to happen. So the maximum probability for anything is one. The minimum probability is zero. You don't have negative probabilities, and you don't have probabilities greater than one. And you might be thinking, wait, wait, you know, I've seen things that, you know, they look like larger numbers than one. And you're probably thinking of seeing this as a percentage."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "The minimum probability is zero. You don't have negative probabilities, and you don't have probabilities greater than one. And you might be thinking, wait, wait, you know, I've seen things that, you know, they look like larger numbers than one. And you're probably thinking of seeing this as a percentage. So one as a percentage, you could also write this as 100%. This right over here as a percentage is 100%. But 100% is the same thing as one."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "And you're probably thinking of seeing this as a percentage. So one as a percentage, you could also write this as 100%. This right over here as a percentage is 100%. But 100% is the same thing as one. You can't have a probability at 110%. 110% would be the same thing as 1.1. Now this is really interesting, because you'd often see someone say, hey, you know, something for sure is going to happen, or something is impossible."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "But 100% is the same thing as one. You can't have a probability at 110%. 110% would be the same thing as 1.1. Now this is really interesting, because you'd often see someone say, hey, you know, something for sure is going to happen, or something is impossible. But even a lot of the things that we think for sure are going to happen, there's some probability, or some chance that they don't happen. So for example, you might hear someone say, well, what's the probability that the sun will rise tomorrow? Well, you might say it's going to happen for sure."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Now this is really interesting, because you'd often see someone say, hey, you know, something for sure is going to happen, or something is impossible. But even a lot of the things that we think for sure are going to happen, there's some probability, or some chance that they don't happen. So for example, you might hear someone say, well, what's the probability that the sun will rise tomorrow? Well, you might say it's going to happen for sure. But you gotta remember, you know, some type of weird cosmological event might occur, some kind of strange, huge, planet-sized object in space might come and knock the Earth out of its rotation. Who knows what could happen. All of these have a very low, a very low likelihood, very, very, very, very, very low likelihood."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Well, you might say it's going to happen for sure. But you gotta remember, you know, some type of weird cosmological event might occur, some kind of strange, huge, planet-sized object in space might come and knock the Earth out of its rotation. Who knows what could happen. All of these have a very low, a very low likelihood, very, very, very, very, very low likelihood. But it's hard to say it's exactly one. If I'd said the probability that the sun will rise, sun will rise tomorrow, tomorrow, instead of saying one, I would probably say it's 0.999, and I would throw a lot of nines over here. I wouldn't say it's 0.9 repeating forever, and actually there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "All of these have a very low, a very low likelihood, very, very, very, very, very low likelihood. But it's hard to say it's exactly one. If I'd said the probability that the sun will rise, sun will rise tomorrow, tomorrow, instead of saying one, I would probably say it's 0.999, and I would throw a lot of nines over here. I wouldn't say it's 0.9 repeating forever, and actually there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive. But I would say there's a very high probability, but even something that's such a high probability, it's going to be close to one, but I won't say it's exactly one because there could be some kind of quasar that blasts us with gamma rays, or who knows what might happen. But it's a very, very high probability. Same thing, you know, the probability, the probability here, probability that my pet, my pet gopher, my pet gopher could compose, could write the next great novel, writes, writes a novel, and actually not just a novel, a great novel."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "I wouldn't say it's 0.9 repeating forever, and actually there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive. But I would say there's a very high probability, but even something that's such a high probability, it's going to be close to one, but I won't say it's exactly one because there could be some kind of quasar that blasts us with gamma rays, or who knows what might happen. But it's a very, very high probability. Same thing, you know, the probability, the probability here, probability that my pet, my pet gopher, my pet gopher could compose, could write the next great novel, writes, writes a novel, and actually not just a novel, a great novel. Just a novel wouldn't be that impressive for a gopher, but writes a great novel, well, once again, you know, this gopher sitting there typing at a keyboard, it would seem somewhat random, but there is some probability that it actually does it. There's some chance it does it, so I would put this at a very low one. I would say it's exactly zero."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "Same thing, you know, the probability, the probability here, probability that my pet, my pet gopher, my pet gopher could compose, could write the next great novel, writes, writes a novel, and actually not just a novel, a great novel. Just a novel wouldn't be that impressive for a gopher, but writes a great novel, well, once again, you know, this gopher sitting there typing at a keyboard, it would seem somewhat random, but there is some probability that it actually does it. There's some chance it does it, so I would put this at a very low one. I would say it's exactly zero. If we had an infinite number of gophers doing this forever, who knows? Maybe one of them might write that great novel. In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "I would say it's exactly zero. If we had an infinite number of gophers doing this forever, who knows? Maybe one of them might write that great novel. In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would. But just one gopher trying to write a novel, what's the probability they write a great novel? Well, I would say it's pretty close to zero. I would throw a lot of zeros."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would. But just one gopher trying to write a novel, what's the probability they write a great novel? Well, I would say it's pretty close to zero. I would throw a lot of zeros. I would throw a lot of zeros here, and at some point, you might have something like this. So once again, not absolutely impossible, but pretty close to, pretty, pretty close to impossible. And so, big takeaways, higher probability, more likely."}, {"video_title": "Intuitive sense of probabilities   Statistics and probability   7th grade   Khan Academy.mp3", "Sentence": "I would throw a lot of zeros. I would throw a lot of zeros here, and at some point, you might have something like this. So once again, not absolutely impossible, but pretty close to, pretty, pretty close to impossible. And so, big takeaways, higher probability, more likely. The lowest probability you can get to, zero. Highest probability is one. If your probability is more, if, you know, when you talk about coin flipping, if you say the probability of heads for a fair coin, you say, well, that's 1.5, that means it's equally likely to happen or not happen."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "In the next few videos, I'm going to embark on something that will just result in a formula that's pretty straightforward to apply. And in most statistics classes, you'll just see that end product. But I actually want to show how to get there. But I just want to warn you right now. It's going to be a lot of hairy math, most of it hairy algebra, and then we're actually going to have to do a little bit of calculus near the end. We're going to have to do a few partial derivatives. So if any of that sounds daunting or sounds like something that will discourage you in some way, you don't have to watch it."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "But I just want to warn you right now. It's going to be a lot of hairy math, most of it hairy algebra, and then we're actually going to have to do a little bit of calculus near the end. We're going to have to do a few partial derivatives. So if any of that sounds daunting or sounds like something that will discourage you in some way, you don't have to watch it. You could skip to the end and just get the formula that we're going to derive. But I at least find it pretty satisfying to actually derive it. So what we're going to think about here is, let's say we have n points on a coordinate plane."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if any of that sounds daunting or sounds like something that will discourage you in some way, you don't have to watch it. You could skip to the end and just get the formula that we're going to derive. But I at least find it pretty satisfying to actually derive it. So what we're going to think about here is, let's say we have n points on a coordinate plane. And they all don't have to be in the first quadrant. But just for simplicity or visualization, I'll draw them all in the first quadrant. So let's say I have this point right over here."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what we're going to think about here is, let's say we have n points on a coordinate plane. And they all don't have to be in the first quadrant. But just for simplicity or visualization, I'll draw them all in the first quadrant. So let's say I have this point right over here. Let me do them in different colors. Let's say I have this point right over here. And that coordinate is x1, y1."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say I have this point right over here. Let me do them in different colors. Let's say I have this point right over here. And that coordinate is x1, y1. And then let's say I have another point over here. I have, let me do that in a different color. Say I have another point over here."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that coordinate is x1, y1. And then let's say I have another point over here. I have, let me do that in a different color. Say I have another point over here. The coordinates there are x2, y2. And then I could keep adding points and I could keep drawing them. We'd just have a ton of points there and there and there."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Say I have another point over here. The coordinates there are x2, y2. And then I could keep adding points and I could keep drawing them. We'd just have a ton of points there and there and there. And we go all the way to the nth point. All the way to the actual nth point. Maybe it's over here."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We'd just have a ton of points there and there and there. And we go all the way to the nth point. All the way to the actual nth point. Maybe it's over here. The nth point is over here. And we're just going to call that xn, yn. So we have n points here."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe it's over here. The nth point is over here. And we're just going to call that xn, yn. So we have n points here. I haven't drawn all of the actual points. But what I want to do is find a line that minimizes the squared distances to these different points. So let's think about it."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have n points here. I haven't drawn all of the actual points. But what I want to do is find a line that minimizes the squared distances to these different points. So let's think about it. Let's visualize that line for a second. So there's going to be some line. And I'm going to try to draw a line that kind of approximates what these points are doing."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about it. Let's visualize that line for a second. So there's going to be some line. And I'm going to try to draw a line that kind of approximates what these points are doing. So let me draw this line here. So maybe the line might look something like this. I'm going to try my best to approximate it."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm going to try to draw a line that kind of approximates what these points are doing. So let me draw this line here. So maybe the line might look something like this. I'm going to try my best to approximate it. So maybe it looks something like that. Actually, let me draw it a little bit different. Maybe it looks something like that."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to try my best to approximate it. So maybe it looks something like that. Actually, let me draw it a little bit different. Maybe it looks something like that. I don't even know what it looks like right now. What we want to do is minimize the squared error from each of these points to the line. So let's think about what that means."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe it looks something like that. I don't even know what it looks like right now. What we want to do is minimize the squared error from each of these points to the line. So let's think about what that means. So if the equation of this line right here is y is equal to mx plus b. And this just comes straight out of algebra 1. This is the slope of the line."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about what that means. So if the equation of this line right here is y is equal to mx plus b. And this just comes straight out of algebra 1. This is the slope of the line. And this is the y-intercept. This is actually the point 0b right here. What I want to do is I want to find, and that's what the topic of the next few videos are going to be, I want to find an m and a b."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the slope of the line. And this is the y-intercept. This is actually the point 0b right here. What I want to do is I want to find, and that's what the topic of the next few videos are going to be, I want to find an m and a b. So I want to find these two things that define this line so that it minimizes the squared error. So let me define what the error even is. So for each of these points, the error between it and the line is the vertical distance."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "What I want to do is I want to find, and that's what the topic of the next few videos are going to be, I want to find an m and a b. So I want to find these two things that define this line so that it minimizes the squared error. So let me define what the error even is. So for each of these points, the error between it and the line is the vertical distance. So this right here we can call error 1. And then this right here would be error 2. It would be the vertical distance between that point and the line."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for each of these points, the error between it and the line is the vertical distance. So this right here we can call error 1. And then this right here would be error 2. It would be the vertical distance between that point and the line. Or you could think of it the y value of this point and the y value of the line. And you just keep going all the way to the end point between the y value of this point and the y value of the line. So this error right here, error 1, if you think about it, it is this value right here, this y value."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It would be the vertical distance between that point and the line. Or you could think of it the y value of this point and the y value of the line. And you just keep going all the way to the end point between the y value of this point and the y value of the line. So this error right here, error 1, if you think about it, it is this value right here, this y value. It's equal to y1 minus this y value. Well, what's this y value going to be? Well, over here we have x is equal to x1."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this error right here, error 1, if you think about it, it is this value right here, this y value. It's equal to y1 minus this y value. Well, what's this y value going to be? Well, over here we have x is equal to x1. And this point is the point mx1 plus b. You take x1 into this equation of the line, and you're going to get this point right over here. So that's literally going to be equal to mx1 plus b."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, over here we have x is equal to x1. And this point is the point mx1 plus b. You take x1 into this equation of the line, and you're going to get this point right over here. So that's literally going to be equal to mx1 plus b. That's that first error. We can keep doing it with all of the points. This error right over here is going to be y2 minus mx2 plus b, so y2."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's literally going to be equal to mx1 plus b. That's that first error. We can keep doing it with all of the points. This error right over here is going to be y2 minus mx2 plus b, so y2. And then this right here, this point right here, is mx2 plus b, the value when you take x2 into this line. And then we keep going all the way to our nth point. This error right here is going to be yn minus mxn plus b."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This error right over here is going to be y2 minus mx2 plus b, so y2. And then this right here, this point right here, is mx2 plus b, the value when you take x2 into this line. And then we keep going all the way to our nth point. This error right here is going to be yn minus mxn plus b. Now, what we want to do, so if we wanted to just take the straight up sum of the errors, we could just sum these things up. But what we want to do is minimize the square of the error between each of these points, each of these end points in the line. So let me define, I'll do this in a new color."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This error right here is going to be yn minus mxn plus b. Now, what we want to do, so if we wanted to just take the straight up sum of the errors, we could just sum these things up. But what we want to do is minimize the square of the error between each of these points, each of these end points in the line. So let me define, I'll do this in a new color. Let me define the squared error against this line as being equal to the sum of these squared errors. So this error right here, or error 1 we could call it, is y1 minus mx1 plus b. And we're going to square it."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me define, I'll do this in a new color. Let me define the squared error against this line as being equal to the sum of these squared errors. So this error right here, or error 1 we could call it, is y1 minus mx1 plus b. And we're going to square it. So this is the error 1 squared. And then we're going to go to error 2 squared. Error 2 squared is y2 minus mx2 plus b."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we're going to square it. So this is the error 1 squared. And then we're going to go to error 2 squared. Error 2 squared is y2 minus mx2 plus b. And then we're going to square that error. We're squaring this error. And then we keep going."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Error 2 squared is y2 minus mx2 plus b. And then we're going to square that error. We're squaring this error. And then we keep going. We're going to keep going. We're going to go n spaces, or n points I should say. We keep going all the way to this nth error."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we keep going. We're going to keep going. We're going to go n spaces, or n points I should say. We keep going all the way to this nth error. The nth error is going to be yn minus mxn plus b. And then we're going to square it. And then we are going to square it."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "We keep going all the way to this nth error. The nth error is going to be yn minus mxn plus b. And then we're going to square it. And then we are going to square it. So this is the squared error of a line. And I want to find, and what we're going to do over the next few videos, is I want to find the m and b that minimizes this value. That minimizes the squared error of this line right here."}, {"video_title": "Squared error of regression line   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we are going to square it. So this is the squared error of a line. And I want to find, and what we're going to do over the next few videos, is I want to find the m and b that minimizes this value. That minimizes the squared error of this line right here. So if you view this as the best metric for how good a fit a line is, we're going to try to find the best fitting line for these points. And I'll continue in the next video, because I find that with these very hairy math problems, it's good to kind of just deliver one concept at a time. And it also minimizes my probability of making a mistake."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What are the odds of making 10 free throws in a row? Here's my good friend Sal with the answer. That's a great question LeBron and I think the answer might surprise you. So I looked up your career free throw percentage and you're right at around 75% which is a little bit higher than my free throw percentage. And one way to interpret that, if we have a million LeBron James, as you can imagine any large number of LeBron James is taking a free throw. So let's say that this line represents all of the LeBron James that take that first free throw. Let's call that free throw number one."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I looked up your career free throw percentage and you're right at around 75% which is a little bit higher than my free throw percentage. And one way to interpret that, if we have a million LeBron James, as you can imagine any large number of LeBron James is taking a free throw. So let's say that this line represents all of the LeBron James that take that first free throw. Let's call that free throw number one. We would expect on average that 75% of them would make that first free throw. So let me draw 75%. So this is about halfway."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's call that free throw number one. We would expect on average that 75% of them would make that first free throw. So let me draw 75%. So this is about halfway. This would be 25. This would get us to 75. So we would expect 75% of them would make that first free throw."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is about halfway. This would be 25. This would get us to 75. So we would expect 75% of them would make that first free throw. And then the other 25% we would expect on average would miss that first free throw. Now what we care about are the ones that keep making the free throws. We want 10 in a row."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we would expect 75% of them would make that first free throw. And then the other 25% we would expect on average would miss that first free throw. Now what we care about are the ones that keep making the free throws. We want 10 in a row. So let's just focus on the 75% that made the first one. Some of these 25% might make some free throws going forward but we don't care about them anymore. They're kind of out of the game."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want 10 in a row. So let's just focus on the 75% that made the first one. Some of these 25% might make some free throws going forward but we don't care about them anymore. They're kind of out of the game. So let's go to free throw number two. Free throw number two. What percentage of the folks who made of the LeBron Jameses that made that first free throw, what percentage would we expect to make the second one?"}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "They're kind of out of the game. So let's go to free throw number two. Free throw number two. What percentage of the folks who made of the LeBron Jameses that made that first free throw, what percentage would we expect to make the second one? And we're going to assume that whether or not you made the first one has no bearing on the probability of you making the second. This continues to be the probability of a LeBron James making a given free throw. So we would expect 75% of these LeBron Jameses to also make the second one."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What percentage of the folks who made of the LeBron Jameses that made that first free throw, what percentage would we expect to make the second one? And we're going to assume that whether or not you made the first one has no bearing on the probability of you making the second. This continues to be the probability of a LeBron James making a given free throw. So we would expect 75% of these LeBron Jameses to also make the second one. So we're going to take 75% of 75%. So this is about half of that 75%. This would be a quarter."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we would expect 75% of these LeBron Jameses to also make the second one. So we're going to take 75% of 75%. So this is about half of that 75%. This would be a quarter. This would be 3 4ths, which is exactly 75%. So right over here. So this represents of the ones that made the first one, how many also made the second one."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This would be a quarter. This would be 3 4ths, which is exactly 75%. So right over here. So this represents of the ones that made the first one, how many also made the second one. So you could say the percentage of the LeBron Jameses that we would expect on average to make the first two free throws. So this length right over here is 75% of this 75% right over there. And I think you might begin to see a pattern emerging."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this represents of the ones that made the first one, how many also made the second one. So you could say the percentage of the LeBron Jameses that we would expect on average to make the first two free throws. So this length right over here is 75% of this 75% right over there. And I think you might begin to see a pattern emerging. Let's go to the third free throw. Free throw number three. So what percentage of these folks are going to make the third one?"}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I think you might begin to see a pattern emerging. Let's go to the third free throw. Free throw number three. So what percentage of these folks are going to make the third one? Well 75% of them are going to make the third one. So 75% are going to make the third one. So what is this going to be?"}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what percentage of these folks are going to make the third one? Well 75% of them are going to make the third one. So 75% are going to make the third one. So what is this going to be? This is going to be 75% of this number, of this length, which is 75% of 75%. And if you were to go all the way to free throw number 10, and I think you see the pattern here. If we were to go all the way to free throw number 10, so I'm just skipping a bunch, we're going to get some very, very, very small fraction that have made all 10."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what is this going to be? This is going to be 75% of this number, of this length, which is 75% of 75%. And if you were to go all the way to free throw number 10, and I think you see the pattern here. If we were to go all the way to free throw number 10, so I'm just skipping a bunch, we're going to get some very, very, very small fraction that have made all 10. It's essentially going to be 75% times 75% times 75%, 10 times. 75% being multiplied repeatedly 10 times. So this is going to be what we're left off with, is going to be 75% times 75%."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we were to go all the way to free throw number 10, so I'm just skipping a bunch, we're going to get some very, very, very small fraction that have made all 10. It's essentially going to be 75% times 75% times 75%, 10 times. 75% being multiplied repeatedly 10 times. So this is going to be what we're left off with, is going to be 75% times 75%. And let me copy and paste this, just so it doesn't take forever. So copy and then paste it. So times out, but the multiplication signs later."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is going to be what we're left off with, is going to be 75% times 75%. And let me copy and paste this, just so it doesn't take forever. So copy and then paste it. So times out, but the multiplication signs later. So that's 4, that's 6, that's 8, and then that is 10. Right over there, and let me throw the multiplication signs in there. So times, times, times, times."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So times out, but the multiplication signs later. So that's 4, that's 6, that's 8, and then that is 10. Right over there, and let me throw the multiplication signs in there. So times, times, times, times. So this little fraction that made all 10 of them is going to be equal to this value right over here. 75%. So that's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So times, times, times, times. So this little fraction that made all 10 of them is going to be equal to this value right over here. 75%. So that's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 75% being repeatedly multiplied 10 times. Now this would obviously take me forever to do it by hand, and even on a calculator if I were to punch all of this in I might make a mistake. But lucky for us, there is a mathematical operator that is essentially repeated multiplication, and that's taking an exponent."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 75% being repeatedly multiplied 10 times. Now this would obviously take me forever to do it by hand, and even on a calculator if I were to punch all of this in I might make a mistake. But lucky for us, there is a mathematical operator that is essentially repeated multiplication, and that's taking an exponent. So another way of writing that right over there, we could write that as 75% to the 10th power, repeatedly multiplying 75% 10 times. These are the same expression. And 75%, the word percent, literally means per hundred."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "But lucky for us, there is a mathematical operator that is essentially repeated multiplication, and that's taking an exponent. So another way of writing that right over there, we could write that as 75% to the 10th power, repeatedly multiplying 75% 10 times. These are the same expression. And 75%, the word percent, literally means per hundred. You might recognize the root word cent from things like century, 100 years in a century, 100 cents in a dollar. So this literally means per hundred. So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And 75%, the word percent, literally means per hundred. You might recognize the root word cent from things like century, 100 years in a century, 100 cents in a dollar. So this literally means per hundred. So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power. Now let's get our calculator out and see what this evaluates to. So 0.75 to the 10th power gets us to 0.056. And I'll just round to the nearest hundredths."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power. Now let's get our calculator out and see what this evaluates to. So 0.75 to the 10th power gets us to 0.056. And I'll just round to the nearest hundredths. So if we round to the nearest hundredths, it gets us to 0.06. So this is roughly equal to, if we round to the nearest hundredths, 0.06, which is equal to roughly, when we round, a 6% probability of making 10 free throws in a row, which even though you have quite a high free throw percentage, this is not that high of a probability. It's a little bit better than a 1 in 20 chance."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll just round to the nearest hundredths. So if we round to the nearest hundredths, it gets us to 0.06. So this is roughly equal to, if we round to the nearest hundredths, 0.06, which is equal to roughly, when we round, a 6% probability of making 10 free throws in a row, which even though you have quite a high free throw percentage, this is not that high of a probability. It's a little bit better than a 1 in 20 chance. Now what I want to throw out there for everyone else watching this is to think about how we can make a general statement about anybody. If anybody has some free throw percentage and they want to say, what's the probability of making 10 in a row? How can we say that?"}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's a little bit better than a 1 in 20 chance. Now what I want to throw out there for everyone else watching this is to think about how we can make a general statement about anybody. If anybody has some free throw percentage and they want to say, what's the probability of making 10 in a row? How can we say that? Well, I think you saw the pattern right over here. The probability of making, let's call it n, where n is the number of free throws we care about, n free throws in a row for somebody, and we're not just talking about LeBron here, it's going to be their free throw percentage, in this case LeBron's was 75%, to the number of free throws that we want to get in a row, so to the nth power. So for example, you might want to play around with your own free throw percentage."}, {"video_title": "Free throwing probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "How can we say that? Well, I think you saw the pattern right over here. The probability of making, let's call it n, where n is the number of free throws we care about, n free throws in a row for somebody, and we're not just talking about LeBron here, it's going to be their free throw percentage, in this case LeBron's was 75%, to the number of free throws that we want to get in a row, so to the nth power. So for example, you might want to play around with your own free throw percentage. If your free throw percentage, let's say it's 60%, which is the same thing as 0.6. So let's say you have a 60% free throw percentage and you want to see your probability of getting 5 in a row, you would take that to the 5th power. And you'd get what looks like, if you round to the nearest hundredths, it would be about 8%."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "The first graph shows the relationship between test grades and the amount of time the student spent studying. So this is study time on this axis, and this is the test grade on this axis. And the second graph shows the relationship between test grades and shoe size. So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that. And it makes sense that there would be."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that. And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply. There's a negative linear relationship between study time and score."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply. There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship. The more you study, the better your score would be."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship. The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score. That's right."}, {"video_title": "Studying, shoe size, and test scores scatter plots   Probability and Statistics   Khan Academy.mp3", "Sentence": "And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score. That's right. And no relationship between shoe size and score. Well, I'm going to go with that one. Both graphs show positive linear trends of approximately equal strength."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "What I want to do in this video is review much of what we've already talked about and then hopefully build some of the intuition on why we divide by n minus 1 if we want to have an unbiased estimate of the population variance when we're calculating the sample variance. So let's think about a population. So let's say this is the population right over here and it is of size capital N. And we also have a sample of that population. So a sample of that population and at its size we have lowercase n data points. So let's think about all of the parameters and statistics that we know about so far. So the first is the idea of the mean. So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So a sample of that population and at its size we have lowercase n data points. So let's think about all of the parameters and statistics that we know about so far. So the first is the idea of the mean. So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? Well, when we're trying to calculate it on the population, we are calculating a parameter. So let me write this down. So this is going to be, so for the population, we are calculating a parameter."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? Well, when we're trying to calculate it on the population, we are calculating a parameter. So let me write this down. So this is going to be, so for the population, we are calculating a parameter. And when we attempt to calculate something for a sample, we would call that a statistic. So how do we think about the mean for a population? Well, first of all, we denote it with the Greek letter mu."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So this is going to be, so for the population, we are calculating a parameter. And when we attempt to calculate something for a sample, we would call that a statistic. So how do we think about the mean for a population? Well, first of all, we denote it with the Greek letter mu. And we essentially take every data point in our population. So we take the sum of every data point. So we start at the first data point and we go all the way to the capital Nth data point."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "Well, first of all, we denote it with the Greek letter mu. And we essentially take every data point in our population. So we take the sum of every data point. So we start at the first data point and we go all the way to the capital Nth data point. So every data point we add up. So this is the ith data point. So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So we start at the first data point and we go all the way to the capital Nth data point. So every data point we add up. So this is the ith data point. So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean, we do a very similar thing with the sample. And we denote it with an X with a bar over it."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean, we do a very similar thing with the sample. And we denote it with an X with a bar over it. And that's going to be taking every data point in the sample, so going up to lowercase n, adding them up. So these are the sum of all the data points in our sample. And then dividing by the number of data points that we actually had."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "And we denote it with an X with a bar over it. And that's going to be taking every data point in the sample, so going up to lowercase n, adding them up. So these are the sum of all the data points in our sample. And then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much the data points vary from the mean. So let's write variance right over here. And how do we denote and calculate variance for a population?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "And then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much the data points vary from the mean. So let's write variance right over here. And how do we denote and calculate variance for a population? Well, for a population, we'd say that the variance, we use the Greek letter sigma squared, is equal to, and you could view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so I equal 1 all the way to N, we take that data point, subtract from it the population mean. So if you want to calculate this, you'd want to figure this out."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "And how do we denote and calculate variance for a population? Well, for a population, we'd say that the variance, we use the Greek letter sigma squared, is equal to, and you could view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so I equal 1 all the way to N, we take that data point, subtract from it the population mean. So if you want to calculate this, you'd want to figure this out. Or that's one way to do it. We'll see there's other ways to do it, where you can kind of calculate them at the same time. But you would, the easiest or the most intuitive, calculate this first."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So if you want to calculate this, you'd want to figure this out. Or that's one way to do it. We'll see there's other ways to do it, where you can kind of calculate them at the same time. But you would, the easiest or the most intuitive, calculate this first. And for each of the data points, take the data point and subtract it from that. Subtract the mean from that. Square it."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "But you would, the easiest or the most intuitive, calculate this first. And for each of the data points, take the data point and subtract it from that. Subtract the mean from that. Square it. And then divide by the total number of data points you have. Now we get to the interesting part, sample variance. There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "Square it. And then divide by the total number of data points you have. Now we get to the interesting part, sample variance. There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. One way is the biased sample variance, the non-unbiased estimator of the population variance. And that's denoted, usually denoted, by S with a subscript N. And what is the biased estimator? How would we calculate it?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. One way is the biased sample variance, the non-unbiased estimator of the population variance. And that's denoted, usually denoted, by S with a subscript N. And what is the biased estimator? How would we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here, but we would do it for our sample, not our population. So for every data point in our sample, so we have N of them, we take that data point and from it we subtract our sample mean, we subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "How would we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here, but we would do it for our sample, not our population. So for every data point in our sample, so we have N of them, we take that data point and from it we subtract our sample mean, we subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger, this is going to be larger, this is going to be smaller."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger, this is going to be larger, this is going to be smaller. And this one we refer to the unbiased estimate. And this one we refer to the biased estimate. If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So this is going to be larger, this is going to be larger, this is going to be smaller. And this one we refer to the unbiased estimate. And this one we refer to the biased estimate. If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. So you'd probably divide by N minus 1. But let's think about why this estimate would be biased, and why we might want to have an estimate like this that is larger, and then maybe in the future we could have a computer program or something that really makes us feel better that dividing by N minus 1 gives us a better estimate of the true population variance. So let's imagine all of the data in a population, and I'm just going to plot them on a number line."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. So you'd probably divide by N minus 1. But let's think about why this estimate would be biased, and why we might want to have an estimate like this that is larger, and then maybe in the future we could have a computer program or something that really makes us feel better that dividing by N minus 1 gives us a better estimate of the true population variance. So let's imagine all of the data in a population, and I'm just going to plot them on a number line. All the data. So this is my number line, this is my number line, and let me plot all of the data points in my population. So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So let's imagine all of the data in a population, and I'm just going to plot them on a number line. All the data. So this is my number line, this is my number line, and let me plot all of the data points in my population. So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. So these are just points on the number line. Now let's say I take a sample of this. So this is my entire population."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. So these are just points on the number line. Now let's say I take a sample of this. So this is my entire population. So let's see how many I have. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So in this case, what would be my big N?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So this is my entire population. So let's see how many I have. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So in this case, what would be my big N? My big N would be 14. Now let's say I take a sample, a lowercase N of, let's say my sample size is 3. I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So in this case, what would be my big N? My big N would be 14. Now let's say I take a sample, a lowercase N of, let's say my sample size is 3. I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. So the way I drew it, and I'm not going to calculate it exactly, it looks like the mean might sit someplace roughly right over here. So the mean, the true population mean, the parameter is going to sit right over here. Now let's think about what happens when we sample."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. So the way I drew it, and I'm not going to calculate it exactly, it looks like the mean might sit someplace roughly right over here. So the mean, the true population mean, the parameter is going to sit right over here. Now let's think about what happens when we sample. And I'm going to do just a very small sample size just to give us the intuition, but this is true of any sample size. So let's say we have sample size of 3. So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "Now let's think about what happens when we sample. And I'm going to do just a very small sample size just to give us the intuition, but this is true of any sample size. So let's say we have sample size of 3. So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. So for example, if we sample to that point, that point, and that point, I could imagine our sample mean might actually sit pretty close to our population mean. But there's a distinct possibility that maybe when I take a sample, I sample that, that, and that. And the key idea here is when you take a sample, your sample mean is always going to sit within your sample."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. So for example, if we sample to that point, that point, and that point, I could imagine our sample mean might actually sit pretty close to our population mean. But there's a distinct possibility that maybe when I take a sample, I sample that, that, and that. And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. And so there is a possibility that when you take your sample, your mean could even be outside of the sample. And so in this situation, and this is just to give you an intuition, so here your sample mean is going to be sitting someplace in there. And so if you were to just calculate the distance from each of these points to the sample mean, so this distance, that distance, and you square it, and you were to divide by the number of data points you have, this is going to be a much lower estimate than the true variance from the actual population mean, where these things are much, much, much further."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample   Khan Academy.mp3", "Sentence": "And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. And so there is a possibility that when you take your sample, your mean could even be outside of the sample. And so in this situation, and this is just to give you an intuition, so here your sample mean is going to be sitting someplace in there. And so if you were to just calculate the distance from each of these points to the sample mean, so this distance, that distance, and you square it, and you were to divide by the number of data points you have, this is going to be a much lower estimate than the true variance from the actual population mean, where these things are much, much, much further. Now you're always not going to have the true population mean outside of your sample, but it's possible that you do. So in general, when you just take your points, find the square to distance to your sample mean, which is always going to sit inside of your data, even though the true population mean could be outside of it, or it could be at one end of your data, however you might want to think about it, you are likely to be underestimating the true population variance. So this right over here is an underestimate."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "Find the probability of getting exactly two heads when flipping three coins. So let's think about the sample space here. Let's think about all of the possible outcomes. So I could get all heads. So on flip one I get a head, flip two I get a head, flip three I get a head. I could get two heads and then a tail. I could get heads, tail, heads."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I could get all heads. So on flip one I get a head, flip two I get a head, flip three I get a head. I could get two heads and then a tail. I could get heads, tail, heads. Or I could get heads, tails, tails. I could get tails, heads, heads. I could get tail, heads, tails."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could get heads, tail, heads. Or I could get heads, tails, tails. I could get tails, heads, heads. I could get tail, heads, tails. I could get tails, tails, heads. Or I could get tails, tails, and tails. These are all the different ways that I could flip three coins."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could get tail, heads, tails. I could get tails, tails, heads. Or I could get tails, tails, and tails. These are all the different ways that I could flip three coins. And you could maybe say that this is the first flip, the second flip, and the third flip. Now, so this right over here is the sample space. There's eight possible outcomes."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are all the different ways that I could flip three coins. And you could maybe say that this is the first flip, the second flip, and the third flip. Now, so this right over here is the sample space. There's eight possible outcomes. Let me write this. The probability of exactly two heads. I'll say h is there for short."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's eight possible outcomes. Let me write this. The probability of exactly two heads. I'll say h is there for short. The probability of exactly two heads. Well, what is the size of our sample space? I have eight possible outcomes."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll say h is there for short. The probability of exactly two heads. Well, what is the size of our sample space? I have eight possible outcomes. So eight, this is possible outcomes, or the size of our sample space, possible outcomes. And how many of those possible outcomes are associated with this event? You could call this a compound event, because there's more than one outcome that's associated with this."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "I have eight possible outcomes. So eight, this is possible outcomes, or the size of our sample space, possible outcomes. And how many of those possible outcomes are associated with this event? You could call this a compound event, because there's more than one outcome that's associated with this. Well, so let's think about exactly two heads. This is three heads, so it's not exactly two heads. This is exactly two heads right over here."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could call this a compound event, because there's more than one outcome that's associated with this. Well, so let's think about exactly two heads. This is three heads, so it's not exactly two heads. This is exactly two heads right over here. This is exactly two heads right over here. There's only one head. This is exactly two heads."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is exactly two heads right over here. This is exactly two heads right over here. There's only one head. This is exactly two heads. This is only one head, only one head, no heads. So you have one, two, three of the possible outcomes are associated with this event. So you have three possible outcomes."}, {"video_title": "Example All the ways you can flip a coin   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is exactly two heads. This is only one head, only one head, no heads. So you have one, two, three of the possible outcomes are associated with this event. So you have three possible outcomes. Three outcomes associated with the event. Three outcomes satisfy this event, or are associated with this event. So the probability of getting exactly two heads when flipping three coins is three outcomes satisfying this event over eight possible outcomes."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And our alternative hypothesis is that the drug just has an effect. We didn't say whether the drug would lower the response time or raise the response time. We just said the drug has an effect that it will not, the mean when you have the drug will not be the same thing as the population mean. And then the null hypothesis is no, your mean with the drug is going to be the same thing as the population mean. It has no effect. In this situation, where we're really just testing to see if it had an effect, whether an extreme positive effect or an extreme negative effect would have both been considered an effect, we did something called a two-tailed test. This is called a two-tailed test."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the null hypothesis is no, your mean with the drug is going to be the same thing as the population mean. It has no effect. In this situation, where we're really just testing to see if it had an effect, whether an extreme positive effect or an extreme negative effect would have both been considered an effect, we did something called a two-tailed test. This is called a two-tailed test. Because frankly, a super high response time, if you had a response time that was more than 3 standard deviations, that would have also made us likely to reject the null hypothesis. So we were dealing with kind of both tails. You could have done a similar type of hypothesis test with the same experiment, where you only have a one-tailed test."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is called a two-tailed test. Because frankly, a super high response time, if you had a response time that was more than 3 standard deviations, that would have also made us likely to reject the null hypothesis. So we were dealing with kind of both tails. You could have done a similar type of hypothesis test with the same experiment, where you only have a one-tailed test. And the way we could have done that is we still could have had the null hypothesis, we still could have had the null hypothesis be that the drug has no effect. Drug has no effect. Or that the mean with the drug is still going to be 1.2 seconds, our mean response time."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could have done a similar type of hypothesis test with the same experiment, where you only have a one-tailed test. And the way we could have done that is we still could have had the null hypothesis, we still could have had the null hypothesis be that the drug has no effect. Drug has no effect. Or that the mean with the drug is still going to be 1.2 seconds, our mean response time. Now if we wanted to do a one-tailed test, and for some reason we already had maybe a view that this drug would lower response times, then our alternative hypothesis, and just so you get familiar with different types of notation, some books or teachers will write the alternative hypothesis as H1, sometimes they write it as H alternative, either one is fine. If you want to do a one-tailed test, you could say that the drug lowers response time. Or that the mean with the drug is less than 1.2 seconds."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or that the mean with the drug is still going to be 1.2 seconds, our mean response time. Now if we wanted to do a one-tailed test, and for some reason we already had maybe a view that this drug would lower response times, then our alternative hypothesis, and just so you get familiar with different types of notation, some books or teachers will write the alternative hypothesis as H1, sometimes they write it as H alternative, either one is fine. If you want to do a one-tailed test, you could say that the drug lowers response time. Or that the mean with the drug is less than 1.2 seconds. Now if you do a one-tailed test like this, what we're thinking about is, what we want to look at is, we have our sampling distribution. Actually I can just use the drawing that I had up here. You had your sampling distribution of the sample mean."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or that the mean with the drug is less than 1.2 seconds. Now if you do a one-tailed test like this, what we're thinking about is, what we want to look at is, we have our sampling distribution. Actually I can just use the drawing that I had up here. You had your sampling distribution of the sample mean. We know what the mean of that was, it's 1.2 seconds, same as the population mean. We were able to estimate its standard deviation using our sample standard deviation, and that was reasonable because it has a sample size of greater than 30, so we can still deal with a normal distribution for the sampling distribution. And using that, we saw that the result, the sample mean that we got, the 1.05 seconds, is three standard deviations below the mean."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "You had your sampling distribution of the sample mean. We know what the mean of that was, it's 1.2 seconds, same as the population mean. We were able to estimate its standard deviation using our sample standard deviation, and that was reasonable because it has a sample size of greater than 30, so we can still deal with a normal distribution for the sampling distribution. And using that, we saw that the result, the sample mean that we got, the 1.05 seconds, is three standard deviations below the mean. So if we look at it, let me just redraw it with our new hypothesis test. So this is the sampling distribution. It has a mean right over here at 1.2 seconds."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And using that, we saw that the result, the sample mean that we got, the 1.05 seconds, is three standard deviations below the mean. So if we look at it, let me just redraw it with our new hypothesis test. So this is the sampling distribution. It has a mean right over here at 1.2 seconds. And the result we got was three standard deviations below the mean. One, two, three standard deviations below the mean. That was what our 1.05 seconds were."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It has a mean right over here at 1.2 seconds. And the result we got was three standard deviations below the mean. One, two, three standard deviations below the mean. That was what our 1.05 seconds were. So when you set it up like this, where you're not just saying that the drug has an effect, in that case, and that was the last video, you'd look at both tails. But here we're saying we only care, does the drug lower our response time? And just like we did before, you say, okay, let's say the drug doesn't lower our response time."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "That was what our 1.05 seconds were. So when you set it up like this, where you're not just saying that the drug has an effect, in that case, and that was the last video, you'd look at both tails. But here we're saying we only care, does the drug lower our response time? And just like we did before, you say, okay, let's say the drug doesn't lower our response time. If the drug doesn't lower our response time, what was the probability, or what is the probability of getting a lowering this extreme or more extreme? So here it will only be one of the tails that we consider when we set our alternative hypothesis like that, that we think it lowers. So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And just like we did before, you say, okay, let's say the drug doesn't lower our response time. If the drug doesn't lower our response time, what was the probability, or what is the probability of getting a lowering this extreme or more extreme? So here it will only be one of the tails that we consider when we set our alternative hypothesis like that, that we think it lowers. So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here. Let me put it this way. More extreme than 1.05 seconds, or let me say lower, because in the last video we cared about more extreme, because even a really high result would have said, okay, the mean is definitely not 1.2 seconds. But in this case we care about means that are lower."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here. Let me put it this way. More extreme than 1.05 seconds, or let me say lower, because in the last video we cared about more extreme, because even a really high result would have said, okay, the mean is definitely not 1.2 seconds. But in this case we care about means that are lower. So now we care about the probability of a result lower than 1.05 seconds. That's the same thing as getting a sample from the sampling distribution that's more than three standard deviations below the mean. And in this case we are only going to consider the area in this one tail."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But in this case we care about means that are lower. So now we care about the probability of a result lower than 1.05 seconds. That's the same thing as getting a sample from the sampling distribution that's more than three standard deviations below the mean. And in this case we are only going to consider the area in this one tail. So this right here would be a one-tail test, where we only care about one direction below the mean. And if you look at the one-tail test, this area over here, we saw last time that both of these areas combined are 0.3%. But if you're only considering one of these areas, if you're only considering this one over here, it's going to be half of that, because the normal distribution is symmetric."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And in this case we are only going to consider the area in this one tail. So this right here would be a one-tail test, where we only care about one direction below the mean. And if you look at the one-tail test, this area over here, we saw last time that both of these areas combined are 0.3%. But if you're only considering one of these areas, if you're only considering this one over here, it's going to be half of that, because the normal distribution is symmetric. So it's going to be 0.13%. So this one right here is going to be 0.15%. Or if you were to express it as a decimal, this is going to be 0.0015."}, {"video_title": "One-tailed and two-tailed tests   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "But if you're only considering one of these areas, if you're only considering this one over here, it's going to be half of that, because the normal distribution is symmetric. So it's going to be 0.13%. So this one right here is going to be 0.15%. Or if you were to express it as a decimal, this is going to be 0.0015. So once again, if you set up your hypotheses like this, you would have said, if your null hypothesis is correct, there would have only been a 0.15% chance of getting a result lower than the result we got. So that would be very unlikely, so we will reject the null hypothesis and go with the alternative. And in this situation, your p-value is going to be the 0.0015."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So there's going to be two bins of balls. So you're gonna have two bins of balls. One of them's gonna have 56 balls in it. So 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're gonna do is they're gonna pick five balls from this bin right over here."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're gonna do is they're gonna pick five balls from this bin right over here. And you have to get the exact numbers of those five balls. It can be in any order. So let me just draw them."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so what they're gonna do is they're gonna pick five balls from this bin right over here. And you have to get the exact numbers of those five balls. It can be in any order. So let me just draw them. So it's one ball. I'll shade it so it looks like a ball. Two balls."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let me just draw them. So it's one ball. I'll shade it so it looks like a ball. Two balls. Three balls. Four balls. And five balls that they're going to pick."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Two balls. Three balls. Four balls. And five balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. From a bin of 56."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And five balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. From a bin of 56. And then you have to get the mega ball right. And then they're gonna just pick one ball from there, which they call the mega ball. They're gonna pick one ball from there."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "From a bin of 56. And then you have to get the mega ball right. And then they're gonna just pick one ball from there, which they call the mega ball. They're gonna pick one ball from there. And obviously this is just going to be picked, this is gonna be one of 46. So from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "They're gonna pick one ball from there. And obviously this is just going to be picked, this is gonna be one of 46. So from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. So essentially all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. So essentially all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. So this is literally, you could view this as 56 choose five. Or if you wanna think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. So this is literally, you could view this as 56 choose five. Or if you wanna think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities. 52, because I've already picked four balls out of that. Now this number right over here, when you multiply it out, this is the number of permutations if I cared about order. So if I got that exact combination."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then the ball after that, there's going to be 52 possibilities. 52, because I've already picked four balls out of that. Now this number right over here, when you multiply it out, this is the number of permutations if I cared about order. So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. So what you wanna do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. So what you wanna do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left. Then the one after that will have three positions left. One after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "Then the next one will have four positions left. Then the one after that will have three positions left. One after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four. So it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then the fifth one will be completely determined because you've already placed the other four. So it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that. So just the white balls, we have 55, sorry, 56 times 55 times 54 times 53 times 52. And we're gonna divide that by five times four times three times two. We don't have to multiply by one, but I'll just do that just to show what we're doing."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so let's calculate that. So just the white balls, we have 55, sorry, 56 times 55 times 54 times 53 times 52. And we're gonna divide that by five times four times three times two. We don't have to multiply by one, but I'll just do that just to show what we're doing. And then that gives us about 3.8 million. So let me actually take that, let me put that off screen. So let me write that number down."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "We don't have to multiply by one, but I'll just do that just to show what we're doing. And then that gives us about 3.8 million. So let me actually take that, let me put that off screen. So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange ball. So you're gonna multiply that times 46. And so that's going to get you, so when you multiply it times 46, bring the calculator back."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange ball. So you're gonna multiply that times 46. And so that's going to get you, so when you multiply it times 46, bring the calculator back. So we're gonna multiply our previous answer times 46. And it just means my previous answer times 46. I get a little under 176 million."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so that's going to get you, so when you multiply it times 46, bring the calculator back. So we're gonna multiply our previous answer times 46. And it just means my previous answer times 46. I get a little under 176 million. A little under 176 million. So that is, let me write that number down. So that gives us 175,711,536."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "I get a little under 176 million. A little under 176 million. So that is, let me write that number down. So that gives us 175,711,536. So your odds of winning it with one entry, because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities, your odds of winning is going to be one over this. And to put this in a little bit of context, I looked it up on the internet, what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So that gives us 175,711,536. So your odds of winning it with one entry, because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities, your odds of winning is going to be one over this. And to put this in a little bit of context, I looked it up on the internet, what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. One in 10,000 chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be one in 10,000 times one in 10,000. And so that will get you one in, and we're going to have now eight zeros."}, {"video_title": "Mega millions jackpot probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. One in 10,000 chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be one in 10,000 times one in 10,000. And so that will get you one in, and we're going to have now eight zeros. One, two, three, four, five, six, seven, eight. So that gives you one in 100 million. So you're actually twice almost, this is very rough, you're roughly twice as likely to get struck by lightning twice in your life than to win the mega jackpot."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Well, this looks pretty much like a binomial random variable. In fact, I'm pretty confident it is a binomial random variable, and we could just go down the checklist. The outcome of each trial can be a success or failure. So trial, outcome, success, or failure. It's either gonna go either way. The result of each trial is independent from the other ones. Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So trial, outcome, success, or failure. It's either gonna go either way. The result of each trial is independent from the other ones. Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial. So result, let me write this trial, I'll just do a shorthand trial, results, results, independent. Independent, that's an important condition. Let's see, there are a fixed number of trials, fixed number of trials."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial. So result, let me write this trial, I'll just do a shorthand trial, results, results, independent. Independent, that's an important condition. Let's see, there are a fixed number of trials, fixed number of trials. In this case, we're gonna have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Let's see, there are a fixed number of trials, fixed number of trials. In this case, we're gonna have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success. Probability on each trial. So yes, indeed, this met all the conditions for being a binomial, binomial random, random variable. And this was all just a little bit of review about things that we have talked about in other videos."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Same probability of success. Probability on each trial. So yes, indeed, this met all the conditions for being a binomial, binomial random, random variable. And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y? So this says the number of rolls until we get a six on a fair die. So this one strikes us as a little bit different, but let's see where it is actually different."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y? So this says the number of rolls until we get a six on a fair die. So this one strikes us as a little bit different, but let's see where it is actually different. So does it meet that the trial outcomes, that there's a clear success or failure for each trial? Well, yeah, we're just gonna keep rolling. So each time we roll, it's a trial."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this one strikes us as a little bit different, but let's see where it is actually different. So does it meet that the trial outcomes, that there's a clear success or failure for each trial? Well, yeah, we're just gonna keep rolling. So each time we roll, it's a trial. And success is when we get a six, failure is when we don't get a six. So the outcome of each trial can be classified as either a success or failure. So it meets, and let me put the checks right over here, it meets this first constraint."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So each time we roll, it's a trial. And success is when we get a six, failure is when we don't get a six. So the outcome of each trial can be classified as either a success or failure. So it meets, and let me put the checks right over here, it meets this first constraint. Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll or the fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So we have the independence."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it meets, and let me put the checks right over here, it meets this first constraint. Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll or the fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So we have the independence. And we also have the same probability of success on each trial. In every case, it's a 1 6th probability that I get a six. So this stays constant."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So we have the independence. And we also have the same probability of success on each trial. In every case, it's a 1 6th probability that I get a six. So this stays constant. And I skipped this third condition for a reason. Because we clearly don't have a fixed number of trials. Over here, we could roll 50 times until we get a six."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this stays constant. And I skipped this third condition for a reason. Because we clearly don't have a fixed number of trials. Over here, we could roll 50 times until we get a six. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six. In fact, think about what the minimum value of y is and what the maximum value of y is. So the minimum value that this random variable can take, I'll just call it min y, is equal to what?"}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Over here, we could roll 50 times until we get a six. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six. In fact, think about what the minimum value of y is and what the maximum value of y is. So the minimum value that this random variable can take, I'll just call it min y, is equal to what? Well, it's gonna take at least one roll, so that's the minimum value. But what is the maximum value for y? And I'll let you think about that."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So the minimum value that this random variable can take, I'll just call it min y, is equal to what? Well, it's gonna take at least one roll, so that's the minimum value. But what is the maximum value for y? And I'll let you think about that. Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And I'll let you think about that. Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going. So this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials. In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success?"}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going. So this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials. In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? Maybe that's a general way of framing this type of random variable. How many trials until success, while the binomial random variable was, how many trials, or how many successes, I should say, how many successes in finite number of trials? So if you see this general form and it meets these conditions, you can feel good it's a binomial random variable, but if you're meeting this condition, clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials, we're talking about how many trials until success, then this type of random variable is called a geometric, geometric random variable."}, {"video_title": "Geometric random variables introduction   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? Maybe that's a general way of framing this type of random variable. How many trials until success, while the binomial random variable was, how many trials, or how many successes, I should say, how many successes in finite number of trials? So if you see this general form and it meets these conditions, you can feel good it's a binomial random variable, but if you're meeting this condition, clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials, we're talking about how many trials until success, then this type of random variable is called a geometric, geometric random variable. And we will see why in future videos it is called geometric because the math that involves the probabilities of various outcomes looks a lot like geometric growth or geometric sequences and series that we look at in other types of mathematics. And in case I forgot to mention, the reason why we call binomial random variables is because when you think about the probabilities of different outcomes, you have these things called binomial coefficients based on combinatorics, and those come out of things like Pascal's triangle and when you take a binomial to ever-increasing powers. So that's where those words come from."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so the oldest sibling right over here, he decides, well look, I'll just put all of our names into a bowl, and then I'll just randomly pick one of our names out of the bowl each night, and then that person is going to be, so this is the bowl right over here, and I'm just gonna put four sheets of paper in there, each of them's gonna have one of their names, and then he's just going to randomly pick it out each night and then that's the person who's going to do their dishes. So they all say, well, you know, that seems like a reasonably fair thing to do, and so they start that process. So let's say that after the first three nights, that he, the oldest brother here, and let's call him Bill, let's say after three nights, Bill has not had to do the dishes. So at that point, the rest of the siblings are starting to think maybe, just maybe something fishy is happening. So what I wanna think about is, what is the probability of that happening? What's the probability of three nights in a row, Bill does not get picked? If we assume that we were randomly taking, if Bill was truly randomly taking these things out of the bowl and not cheating in some way, what's the probability that that would happen, that three nights in a row, Bill would not be picked?"}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So at that point, the rest of the siblings are starting to think maybe, just maybe something fishy is happening. So what I wanna think about is, what is the probability of that happening? What's the probability of three nights in a row, Bill does not get picked? If we assume that we were randomly taking, if Bill was truly randomly taking these things out of the bowl and not cheating in some way, what's the probability that that would happen, that three nights in a row, Bill would not be picked? I encourage you to pause the video and think about that. Well, let's think about the probability that Bill's not picked on a given night. If it's truly random, so we're going to assume, we're going to assume that Bill's not cheating."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we assume that we were randomly taking, if Bill was truly randomly taking these things out of the bowl and not cheating in some way, what's the probability that that would happen, that three nights in a row, Bill would not be picked? I encourage you to pause the video and think about that. Well, let's think about the probability that Bill's not picked on a given night. If it's truly random, so we're going to assume, we're going to assume that Bill's not cheating. So assume, assume truly random, truly random and that each of the sheets of paper have a one in four chance of being picked. What's the probability that Bill does not get picked? Well, there's, so let me, the probability that, I guess I'm gonna write this, Bill not picked on a night, on a night, well, there's four equally likely outcomes and three of them result in Bill not getting picked."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's truly random, so we're going to assume, we're going to assume that Bill's not cheating. So assume, assume truly random, truly random and that each of the sheets of paper have a one in four chance of being picked. What's the probability that Bill does not get picked? Well, there's, so let me, the probability that, I guess I'm gonna write this, Bill not picked on a night, on a night, well, there's four equally likely outcomes and three of them result in Bill not getting picked. So there's a 3 4th probability that Bill is not picked on a given night. Well, what's the probability that Bill's not picked three nights in a row? Let me write that down."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, there's, so let me, the probability that, I guess I'm gonna write this, Bill not picked on a night, on a night, well, there's four equally likely outcomes and three of them result in Bill not getting picked. So there's a 3 4th probability that Bill is not picked on a given night. Well, what's the probability that Bill's not picked three nights in a row? Let me write that down. So the probability Bill not picked three nights in a row, well, that's the probability that he's not picked on the first night times the probability that he's not picked on the second night times the probability that he's not picked on the third night. So that's going to be three to the third power or three times three times three. So that's 27 over four to the third power."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write that down. So the probability Bill not picked three nights in a row, well, that's the probability that he's not picked on the first night times the probability that he's not picked on the second night times the probability that he's not picked on the third night. So that's going to be three to the third power or three times three times three. So that's 27 over four to the third power. Four times four times four is 64. And if we want to express that as a decimal, so that is 27, let me get my calculator out, that is 27 divided by 64 is equal to, and I'll just round to the nearest hundredth right here, 0.42. So that is equal to 0.42."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's 27 over four to the third power. Four times four times four is 64. And if we want to express that as a decimal, so that is 27, let me get my calculator out, that is 27 divided by 64 is equal to, and I'll just round to the nearest hundredth right here, 0.42. So that is equal to 0.42. And so this doesn't seem that unlikely. It's a little less likely than kind of even odds, but you wouldn't question someone's credibility if there's a 42%, roughly a 42% chance that three nights in a row Bill would not be picked. So this seems like if you're assuming truly random that it's a reasonable, your hypothesis that it's truly random, there's a good chance that you're right."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that is equal to 0.42. And so this doesn't seem that unlikely. It's a little less likely than kind of even odds, but you wouldn't question someone's credibility if there's a 42%, roughly a 42% chance that three nights in a row Bill would not be picked. So this seems like if you're assuming truly random that it's a reasonable, your hypothesis that it's truly random, there's a good chance that you're right. There's a 42% chance you would have the outcome you saw if your assumption is true. But let's say you keep doing this and you trust your older brother. You know, why would he want to cheat out his younger siblings?"}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this seems like if you're assuming truly random that it's a reasonable, your hypothesis that it's truly random, there's a good chance that you're right. There's a 42% chance you would have the outcome you saw if your assumption is true. But let's say you keep doing this and you trust your older brother. You know, why would he want to cheat out his younger siblings? But let's say that Bill's not picked 12 nights in a row. So then everyone's starting to get a little bit, everyone's starting to get a little bit suspicious with Bill right over here. And so they say, well, you know, well, we're gonna give him the benefit of the doubt."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "You know, why would he want to cheat out his younger siblings? But let's say that Bill's not picked 12 nights in a row. So then everyone's starting to get a little bit, everyone's starting to get a little bit suspicious with Bill right over here. And so they say, well, you know, well, we're gonna give him the benefit of the doubt. Assuming that he's being completely honest, that this is a completely random process, what is the probability that he would not be picked 12 nights in a row? Well, just write that down. So the probability Bill, and it's really the same stuff that I just wrote up here."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so they say, well, you know, well, we're gonna give him the benefit of the doubt. Assuming that he's being completely honest, that this is a completely random process, what is the probability that he would not be picked 12 nights in a row? Well, just write that down. So the probability Bill, and it's really the same stuff that I just wrote up here. I'll just say Bill not picked 12 nights in a row. Well, that's going to be three, you're gonna take 12 3 4ths and multiply them together. It's going to be 3 4ths to the 12th power."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability Bill, and it's really the same stuff that I just wrote up here. I'll just say Bill not picked 12 nights in a row. Well, that's going to be three, you're gonna take 12 3 4ths and multiply them together. It's going to be 3 4ths to the 12th power. And what is this going to be equal to? Well, let's see, if you take, well, 3 4ths is, I'll just write three divided by three, divided by four, which is gonna be 0.75, to the 12th power. Now, this is a much smaller, this is now, if we actually, this is going to be 0.3, I guess we could go to one more decimal place, 0.32, or we could say, so this is 0.032, I should say, which is equal to, so this is approximately equal to, let me write that, which is equal to 3.2%."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be 3 4ths to the 12th power. And what is this going to be equal to? Well, let's see, if you take, well, 3 4ths is, I'll just write three divided by three, divided by four, which is gonna be 0.75, to the 12th power. Now, this is a much smaller, this is now, if we actually, this is going to be 0.3, I guess we could go to one more decimal place, 0.32, or we could say, so this is 0.032, I should say, which is equal to, so this is approximately equal to, let me write that, which is equal to 3.2%. So now you have every right to start thinking that something is getting fishy. You could say, well, look, if there was, and this is what statisticians actually do, they often just define a threshold, hey, you know, if the probability of this happening purely by chance is more than 5%, then I'll say, well, maybe it was happening by chance. But if the probability of this happening purely by chance was, you know, and this is the threshold that statisticians often use is 5%, but that's somewhat arbitrarily defined, but this is a fairly low probability that it would happen fairly by chance."}, {"video_title": "Simple hypothesis testing   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, this is a much smaller, this is now, if we actually, this is going to be 0.3, I guess we could go to one more decimal place, 0.32, or we could say, so this is 0.032, I should say, which is equal to, so this is approximately equal to, let me write that, which is equal to 3.2%. So now you have every right to start thinking that something is getting fishy. You could say, well, look, if there was, and this is what statisticians actually do, they often just define a threshold, hey, you know, if the probability of this happening purely by chance is more than 5%, then I'll say, well, maybe it was happening by chance. But if the probability of this happening purely by chance was, you know, and this is the threshold that statisticians often use is 5%, but that's somewhat arbitrarily defined, but this is a fairly low probability that it would happen fairly by chance. So you might be tempted to reject the hypothesis, to reject the hypothesis that it was truly random, that Bill is cheating in some way. And you could imagine, if it wasn't 12 in a row, if it was 20 in a row, then this probability becomes really, really, really, really, really small. And so your hypothesis that it's truly random starts to really come into doubt."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So it says here I have a 0.35 probability of making a free throw. What is the probability of making four out of seven free throws? Well, this is a classic binomial random variable question. If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal to four? Now what we're going to see is we can use a function on our TI-84 named binomec, or binomepdf I should say, binomepdf, which is short for binomial probability distribution function. And what you're going to want to do here is use three arguments."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal to four? Now what we're going to see is we can use a function on our TI-84 named binomec, or binomepdf I should say, binomepdf, which is short for binomial probability distribution function. And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And once again, if you're taking the test, you should mark that. That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes?"}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And once again, if you're taking the test, you should mark that. That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing calculator out. So there's a couple of ways to input this."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing calculator out. So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here, click Enter."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the probability of success in each trial is 0.35. And now when I type in four here, it doesn't mean what is the probability that I make exactly four free throws. It is the probability that I make zero, one, two, three, or four free throws."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the probability of success in each trial is 0.35. And now when I type in four here, it doesn't mean what is the probability that I make exactly four free throws. It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94."}, {"video_title": "Binompdf and binomcdf functions   Random variables   AP Statistics   Khan Academy.mp3", "Sentence": "This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94. So this is approximately 0.94. So hopefully you found that helpful. These calculators can be very useful, especially on something like an AP Stats exam."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So let's see what's going on here. The horizontal axis here, they say years since 1965. So this point right over here, this is zero years since 1965, so this really represents 1965. And we see it looks like around, let's see if I were to eyeball it, it looks like it's around 42% of Americans, just looking at this graph, I know that's not an exact number, roughly 41 or 42% of Americans smoked in 1965 based on this graph. And then five years later, this would be 1970, 10 years later, that would be 1975. And they don't sample the data, or we don't have data from every given year. This is just from some of the years that we happen to have."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And we see it looks like around, let's see if I were to eyeball it, it looks like it's around 42% of Americans, just looking at this graph, I know that's not an exact number, roughly 41 or 42% of Americans smoked in 1965 based on this graph. And then five years later, this would be 1970, 10 years later, that would be 1975. And they don't sample the data, or we don't have data from every given year. This is just from some of the years that we happen to have. But what is clear, it looks like we have a negative linear relationship right over here, that it would not be difficult to fit a line, so let me try to do that. So I'm just gonna eyeball it and try to fit a line to this data. So our line might look something like that."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "This is just from some of the years that we happen to have. But what is clear, it looks like we have a negative linear relationship right over here, that it would not be difficult to fit a line, so let me try to do that. So I'm just gonna eyeball it and try to fit a line to this data. So our line might look something like that. So it looks like a pretty strong negative linear relationship. When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing. So that's what makes it a negative relationship."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So our line might look something like that. So it looks like a pretty strong negative linear relationship. When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing. So that's what makes it a negative relationship. Now what are they asking? They want us to estimate the percentage of American adults who smoked in 1945. Well 1945 would be to the left of zero."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So that's what makes it a negative relationship. Now what are they asking? They want us to estimate the percentage of American adults who smoked in 1945. Well 1945 would be to the left of zero. So we could even think of it as if 1945 is 20 years before 1965. So let me see if I can draw that. So 20 years before 1965."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Well 1945 would be to the left of zero. So we could even think of it as if 1945 is 20 years before 1965. So let me see if I can draw that. So 20 years before 1965. Let's see, this would be five years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as negative 20 right over here. Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So 20 years before 1965. Let's see, this would be five years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as negative 20 right over here. Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. And one thing that we could do is very roughly just try to extend this negative linear relationship backwards, and they allow us to do that by saying assuming the trend shown in the data has been consistent. So the trend has been consistent. This line represents the trend."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. And one thing that we could do is very roughly just try to extend this negative linear relationship backwards, and they allow us to do that by saying assuming the trend shown in the data has been consistent. So the trend has been consistent. This line represents the trend. So let's just keep going backwards. Keep going backwards at the same rate. So something like that."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "This line represents the trend. So let's just keep going backwards. Keep going backwards at the same rate. So something like that. I wanna make sure that it looks at the same, looks like it's the same rate right over here. And you could just try to eyeball it. You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So something like that. I wanna make sure that it looks at the same, looks like it's the same rate right over here. And you could just try to eyeball it. You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. This seems like we're about 52% right over here. Another way to think about it would be to actually try to calculate the rate of decline. And let's say we do it over every 20 years because that'll be useful because we're going 20 years back."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. This seems like we're about 52% right over here. Another way to think about it would be to actually try to calculate the rate of decline. And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. So if we go 20 years from this point, so this is 1965, you go 20 years in the future. So that is 10 years and then that is 20 years. So my change in the horizontal is 20 years."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. So if we go 20 years from this point, so this is 1965, you go 20 years in the future. So that is 10 years and then that is 20 years. So my change in the horizontal is 20 years. What's the change in the vertical? Well it looks like we have a decrease of a little bit more than 10%. Looks like it's 11 or 12% decrease."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So my change in the horizontal is 20 years. What's the change in the vertical? Well it looks like we have a decrease of a little bit more than 10%. Looks like it's 11 or 12% decrease. So I'll just say minus 11% right there. And let's see if that's consistent. If we were to go another 20 years."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "Looks like it's 11 or 12% decrease. So I'll just say minus 11% right there. And let's see if that's consistent. If we were to go another 20 years. So if we go another 20 years, it looks like once again, we've gone down by about 10%. So that looks like roughly 10%. If we're following the line, it should actually be the same number."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "If we were to go another 20 years. So if we go another 20 years, it looks like once again, we've gone down by about 10%. So that looks like roughly 10%. If we're following the line, it should actually be the same number. So let me write it this way. It's approximately down 10%. So that little squiggly line, I'm just saying approximately negative 10% every 20 years."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "If we're following the line, it should actually be the same number. So let me write it this way. It's approximately down 10%. So that little squiggly line, I'm just saying approximately negative 10% every 20 years. Negative 10% every 20 years. So if you go back 20 years, you should increase your percentage by 20%. So this should go up by, or you should increase your percentage by 10% I should say."}, {"video_title": "Smoking in 1945   Data and modeling   8th grade   Khan Academy.mp3", "Sentence": "So that little squiggly line, I'm just saying approximately negative 10% every 20 years. Negative 10% every 20 years. So if you go back 20 years, you should increase your percentage by 20%. So this should go up by, or you should increase your percentage by 10% I should say. So if we started at 41 or 42, once again, this is what we saw when we just eyeballed it. You should get to 51 or 52%. So my estimate of the percentage of American adults who smoked in 1945 would be 51 or 52%."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "This is from ck12.org's open source flexbook, their AP Statistics flexbook. And I've taken the problems from their normal distribution chapter, so you could go to their site and actually look up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what they have here."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So this is unusual in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like?"}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "You would start touching the sun. There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "It might look something like that. It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here. And likewise, in this distribution, your median, your middle value, might be someplace like this. But because it's right skewed, and for the most part it only has one big hump, this hump won't change things too much because it's small, your mean is going to be to the right of it."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions   Khan Academy.mp3", "Sentence": "So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here. And likewise, in this distribution, your median, your middle value, might be someplace like this. But because it's right skewed, and for the most part it only has one big hump, this hump won't change things too much because it's small, your mean is going to be to the right of it. So that's another reason why it's called a right skewed or positively skewed distribution. So to answer the question, these are my feelings about all of them, but I would say, the other choices explain why you believe they would not follow, well they said, which of the following data sets is most likely to be normally distributed? Well, I would say choice A, but it's really a matter of opinion, at least in this question."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "If all four numbers match the four winning numbers, regardless of order, the player wins. What is the probability that the winning numbers are 3, 15, 46, and 49? So the way to think about this problem, they say that we're going to choose four numbers from 60. So one way to think about it is how many different outcomes are there if we choose four numbers out of 60? Now, this is equivalent to saying how many combinations are there if we have 60 items? In this case, we have 60 numbers, and we are going to choose four. And we don't care about the order."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So one way to think about it is how many different outcomes are there if we choose four numbers out of 60? Now, this is equivalent to saying how many combinations are there if we have 60 items? In this case, we have 60 numbers, and we are going to choose four. And we don't care about the order. That's why we're dealing with combinations, not permutations. We don't care about the order. How many different groups of four can we pick out of 60?"}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And we don't care about the order. That's why we're dealing with combinations, not permutations. We don't care about the order. How many different groups of four can we pick out of 60? We don't care what order we pick them in. We've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but then we'll think about what it's actually saying."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "How many different groups of four can we pick out of 60? We don't care what order we pick them in. We've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but then we'll think about what it's actually saying. This is 60 factorial over 60 minus 4 factorial divided also by 4 factorial, or in the denominator, multiplied by 4 factorial. This is the formula right here. What this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59 times 58 times 57."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "I'll write the formula here, but then we'll think about what it's actually saying. This is 60 factorial over 60 minus 4 factorial divided also by 4 factorial, or in the denominator, multiplied by 4 factorial. This is the formula right here. What this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59 times 58 times 57. That's what this expression right here is. And if you think about it, the first number you pick, there's one of 60 numbers, that number is kind of out of the game, then you can pick from one of 59, then from one of 58, then of one of 57. So if you cared about order, this is the number of permutations you could pick four items out of 60 without replacing them."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "What this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59 times 58 times 57. That's what this expression right here is. And if you think about it, the first number you pick, there's one of 60 numbers, that number is kind of out of the game, then you can pick from one of 59, then from one of 58, then of one of 57. So if you cared about order, this is the number of permutations you could pick four items out of 60 without replacing them. Now, this is when you cared about order, but you're kind of over-counting because it's counting different permutations that are essentially the same combination, the same set of four numbers, and that's why we're dividing by 4 factorial here. Because 4 factorial is essentially the number of ways that four numbers can be arranged in four places. The first number can be in one of four slots, the second in one of three, then two, then one."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So if you cared about order, this is the number of permutations you could pick four items out of 60 without replacing them. Now, this is when you cared about order, but you're kind of over-counting because it's counting different permutations that are essentially the same combination, the same set of four numbers, and that's why we're dividing by 4 factorial here. Because 4 factorial is essentially the number of ways that four numbers can be arranged in four places. The first number can be in one of four slots, the second in one of three, then two, then one. That's why you're dividing by 4 factorial. But anyway, let's just evaluate this. This will tell us how many possible outcomes are there for the lottery game."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "The first number can be in one of four slots, the second in one of three, then two, then one. That's why you're dividing by 4 factorial. But anyway, let's just evaluate this. This will tell us how many possible outcomes are there for the lottery game. So this is equal to, we already said the blue part is equivalent to 60 times 59 times 58 times 57. So that's literally 60 factorial divided by essentially 56 factorial. And then you have your 4 factorial over here, which is 4 times 3 times 2 times 1."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "This will tell us how many possible outcomes are there for the lottery game. So this is equal to, we already said the blue part is equivalent to 60 times 59 times 58 times 57. So that's literally 60 factorial divided by essentially 56 factorial. And then you have your 4 factorial over here, which is 4 times 3 times 2 times 1. And we could simplify it a little bit just before we break out the calculator. 60 divided by 4 is 15. And then let's see, 15 divided by 3 is 5."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then you have your 4 factorial over here, which is 4 times 3 times 2 times 1. And we could simplify it a little bit just before we break out the calculator. 60 divided by 4 is 15. And then let's see, 15 divided by 3 is 5. And let's see, we have a 58 divided by 2. 58 divided by 2 is 29. So our answer is going to be 5 times 59 times 29 times 57."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "And then let's see, 15 divided by 3 is 5. And let's see, we have a 58 divided by 2. 58 divided by 2 is 29. So our answer is going to be 5 times 59 times 29 times 57. This isn't going to be our answer. This is going to be the number of combinations we can get if we choose four numbers out of 60 and we don't care about order. So let's take the calculator out now."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So our answer is going to be 5 times 59 times 29 times 57. This isn't going to be our answer. This is going to be the number of combinations we can get if we choose four numbers out of 60 and we don't care about order. So let's take the calculator out now. So we have 5 times 59 times 29 times 57 is equal to 487,635. So let me write that down. That is 487,635 combinations."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "So let's take the calculator out now. So we have 5 times 59 times 29 times 57 is equal to 487,635. So let me write that down. That is 487,635 combinations. If you're picking four numbers, you're choosing four numbers out of 60 or 60 choose 4. Now, the question they say is, what is the probability that the winning numbers are 3, 15, 46, and 49? Well, this is just one particular of the combinations."}, {"video_title": "Example Lottery probability   Probability and combinatorics   Precalculus   Khan Academy.mp3", "Sentence": "That is 487,635 combinations. If you're picking four numbers, you're choosing four numbers out of 60 or 60 choose 4. Now, the question they say is, what is the probability that the winning numbers are 3, 15, 46, and 49? Well, this is just one particular of the combinations. This is just one of the 487,635 possible outcomes. So the probability of 3, 15, 46, 49 winning is just equal to, well, this is just one of the outcomes out of 487,635. So that right there is your probability of winning."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "The scatter plot below shows the relationship between how many hours students spent studying and their score on the test. A line was fit to the data to model the relationship. They don't tell us how the line was fit, but this actually looks like a pretty good fit if I just eyeball it. Which of these linear equations best describes the given model? So this point right over here, this shows that some student, at least self-reported, they studied a little bit more than half an hour, and they didn't actually do that well on the test. Looks like they scored a 43 or a 44 on the test. This right over here shows, or like this one over here is a student who says they studied two hours, and it looks like they scored about a 64 or 65 on the test."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "Which of these linear equations best describes the given model? So this point right over here, this shows that some student, at least self-reported, they studied a little bit more than half an hour, and they didn't actually do that well on the test. Looks like they scored a 43 or a 44 on the test. This right over here shows, or like this one over here is a student who says they studied two hours, and it looks like they scored about a 64 or 65 on the test. And this over here, or this over here, looks like a student who studied over four hours, or they reported that, and they got, looks like a 95 or a 96 on the exam. And so then, and these are all the different students, each of these points represents a student, and then they fit a line. And when they say which of these linear equations best describes the given model, they're really saying which of these linear equations describes or is being plotted right over here by this line that's trying to fit to the data."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "This right over here shows, or like this one over here is a student who says they studied two hours, and it looks like they scored about a 64 or 65 on the test. And this over here, or this over here, looks like a student who studied over four hours, or they reported that, and they got, looks like a 95 or a 96 on the exam. And so then, and these are all the different students, each of these points represents a student, and then they fit a line. And when they say which of these linear equations best describes the given model, they're really saying which of these linear equations describes or is being plotted right over here by this line that's trying to fit to the data. So essentially, we just wanna figure out what is the equation of this line? Well, it looks like the y-intercept right over here is 20, and it looks like all of these choices here have a y-intercept of 20, so that doesn't help us much. But let's think about what the slope is."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "And when they say which of these linear equations best describes the given model, they're really saying which of these linear equations describes or is being plotted right over here by this line that's trying to fit to the data. So essentially, we just wanna figure out what is the equation of this line? Well, it looks like the y-intercept right over here is 20, and it looks like all of these choices here have a y-intercept of 20, so that doesn't help us much. But let's think about what the slope is. When we increase by one, when we increase along our x-axis by one, so change in x is one, what is our change in y? Our change in y looks like, let's see, we went from 20 to 40. It looks like we got, went up by 20."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "But let's think about what the slope is. When we increase by one, when we increase along our x-axis by one, so change in x is one, what is our change in y? Our change in y looks like, let's see, we went from 20 to 40. It looks like we got, went up by 20. So our change in y over change in x for this model, for this line that's trying to fit to the data, is 20 over one. So this is going to be our slope. And if we look at all of these choices, only this one has a slope of 20."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "It looks like we got, went up by 20. So our change in y over change in x for this model, for this line that's trying to fit to the data, is 20 over one. So this is going to be our slope. And if we look at all of these choices, only this one has a slope of 20. So it would be this choice right over here. Based on this equation, estimate the score for a student that spent 3.8 hours studying. So we would go to 3.8, which is right around, let's see, this would be, 3.8 would be right around here."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "And if we look at all of these choices, only this one has a slope of 20. So it would be this choice right over here. Based on this equation, estimate the score for a student that spent 3.8 hours studying. So we would go to 3.8, which is right around, let's see, this would be, 3.8 would be right around here. So let's estimate that score. So if I go straight up, where do we intersect our model? Where do we intersect our line?"}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "So we would go to 3.8, which is right around, let's see, this would be, 3.8 would be right around here. So let's estimate that score. So if I go straight up, where do we intersect our model? Where do we intersect our line? So it looks like they would get a pretty high score. Let's see, if I were to take it to the vertical axis, it looks like they would get about a 97. So I would write, my estimate is that they would get a 97 based on this model."}, {"video_title": "Example estimating from regression line.mp3", "Sentence": "Where do we intersect our line? So it looks like they would get a pretty high score. Let's see, if I were to take it to the vertical axis, it looks like they would get about a 97. So I would write, my estimate is that they would get a 97 based on this model. And once again, this is only a model. It's not a guarantee that if someone studies 3.8 hours, they're gonna get a 97, but it could give an indication of what maybe, might be reasonable to expect, assuming that the time studying is the variable that matters. But you also have to be careful with these models, because it might imply, if you kept going, that if you study for nine hours, you're gonna get a 200 on the exam, even though something like that is impossible."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "It never hurts to get a bit more practice. So this is problem number five from the normal distribution chapter from ck12.org's AP Statistics Flexbook. So they're saying the 2007 AP Statistics examination scores were not normally distributed with a mean of 2.8 and a standard deviation of 1.34. They cite some College Board stuff here. I didn't copy and paste that. What is the approximate z-score? Remember, z-score is just how many standard deviations you are away from the mean."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "They cite some College Board stuff here. I didn't copy and paste that. What is the approximate z-score? Remember, z-score is just how many standard deviations you are away from the mean. What is the approximate z-score that corresponds to an exam score of 5? So we really just have to figure out, this is a pretty straightforward problem, we just need to figure out how many standard deviations is 5 from the mean? You just take 5 minus 2.8."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, z-score is just how many standard deviations you are away from the mean. What is the approximate z-score that corresponds to an exam score of 5? So we really just have to figure out, this is a pretty straightforward problem, we just need to figure out how many standard deviations is 5 from the mean? You just take 5 minus 2.8. The mean is 2.8. Let me be very clear. Mean is 2.8."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "You just take 5 minus 2.8. The mean is 2.8. Let me be very clear. Mean is 2.8. They give us that. Didn't even have to calculate it. So the mean is 2.8."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "Mean is 2.8. They give us that. Didn't even have to calculate it. So the mean is 2.8. So 5 minus 2.8 is equal to 2.2. So we're 2.2 above the mean. And if we want that in terms of standard deviations, we just divide by our standard deviation."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the mean is 2.8. So 5 minus 2.8 is equal to 2.2. So we're 2.2 above the mean. And if we want that in terms of standard deviations, we just divide by our standard deviation. We divide by 1.34. I'll take out the calculator for this. So we have 2.2 divided by 1.34 is equal to 1.64."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "And if we want that in terms of standard deviations, we just divide by our standard deviation. We divide by 1.34. I'll take out the calculator for this. So we have 2.2 divided by 1.34 is equal to 1.64. So this is equal to 1.64. And that's choice C. So this was actually very straightforward. We just had to see how far away we are from the mean if we get a score of 5, which hopefully you will get if you're taking the AP Statistics exam after watching these videos."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have 2.2 divided by 1.34 is equal to 1.64. So this is equal to 1.64. And that's choice C. So this was actually very straightforward. We just had to see how far away we are from the mean if we get a score of 5, which hopefully you will get if you're taking the AP Statistics exam after watching these videos. And then you divide by the standard deviation to say how many standard deviations away from the mean is the score of 5, it's 1.64. I think the only tricky thing here might have been you might have been tempted to pick choice E, which says a z-score cannot be calculated because the distribution is not normal. And I think the reason why you might have had that temptation is because we've been using z-scores within the context of a normal distribution."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "We just had to see how far away we are from the mean if we get a score of 5, which hopefully you will get if you're taking the AP Statistics exam after watching these videos. And then you divide by the standard deviation to say how many standard deviations away from the mean is the score of 5, it's 1.64. I think the only tricky thing here might have been you might have been tempted to pick choice E, which says a z-score cannot be calculated because the distribution is not normal. And I think the reason why you might have had that temptation is because we've been using z-scores within the context of a normal distribution. But a z-score literally just means how many standard deviations you are away from the mean. It could apply to any distribution that you can calculate a mean and a standard deviation for. So E is not the correct answer."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I think the reason why you might have had that temptation is because we've been using z-scores within the context of a normal distribution. But a z-score literally just means how many standard deviations you are away from the mean. It could apply to any distribution that you can calculate a mean and a standard deviation for. So E is not the correct answer. A z-score can apply to a non-normal distribution. So the answer is C. And I guess that's a good point of clarification to get out of the way. I thought I would do two problems in this video just because that one was pretty short."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "So E is not the correct answer. A z-score can apply to a non-normal distribution. So the answer is C. And I guess that's a good point of clarification to get out of the way. I thought I would do two problems in this video just because that one was pretty short. So problem number six, the heights of fifth grade boys in the United States is approximately normally distributed. That's good to know. With a mean height of 143.5 centimeters, so it's a mean of 143.5 centimeters, and a standard deviation of about 7.1 centimeters, standard deviation of 7.1 centimeters, what is the probability that a randomly chosen fifth grade boy would be taller than 157.7 centimeters?"}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "I thought I would do two problems in this video just because that one was pretty short. So problem number six, the heights of fifth grade boys in the United States is approximately normally distributed. That's good to know. With a mean height of 143.5 centimeters, so it's a mean of 143.5 centimeters, and a standard deviation of about 7.1 centimeters, standard deviation of 7.1 centimeters, what is the probability that a randomly chosen fifth grade boy would be taller than 157.7 centimeters? Let's just draw out this distribution like we've done in a bunch of problems. So far, they're just asking us one question. So we can mark this distribution up a good bit."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "With a mean height of 143.5 centimeters, so it's a mean of 143.5 centimeters, and a standard deviation of about 7.1 centimeters, standard deviation of 7.1 centimeters, what is the probability that a randomly chosen fifth grade boy would be taller than 157.7 centimeters? Let's just draw out this distribution like we've done in a bunch of problems. So far, they're just asking us one question. So we can mark this distribution up a good bit. Let's say that's our distribution. And the mean here, the mean they told us is 143.5. They're asking us taller than 157.7."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we can mark this distribution up a good bit. Let's say that's our distribution. And the mean here, the mean they told us is 143.5. They're asking us taller than 157.7. So we're going the upwards direction. So one standard deviation above the mean will take us right there. And we just have to add 7.1 to this number right here."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "They're asking us taller than 157.7. So we're going the upwards direction. So one standard deviation above the mean will take us right there. And we just have to add 7.1 to this number right here. We're going up by 7.1. So 143.5 plus 7.1 is what? 150.6."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we just have to add 7.1 to this number right here. We're going up by 7.1. So 143.5 plus 7.1 is what? 150.6. That's one standard deviation. If we were to go another standard deviation, we go 7.1 more, what's 7.1 plus 150.6? It's 157.7, which just happens to be the exact number they asked for."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "150.6. That's one standard deviation. If we were to go another standard deviation, we go 7.1 more, what's 7.1 plus 150.6? It's 157.7, which just happens to be the exact number they asked for. They're asking for heights, the probability of getting a height higher than that. So they want to know what's the probability that we fall under this area right here. Or essentially, more than two standard deviations from the mean, or above two standard deviations."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's 157.7, which just happens to be the exact number they asked for. They're asking for heights, the probability of getting a height higher than that. So they want to know what's the probability that we fall under this area right here. Or essentially, more than two standard deviations from the mean, or above two standard deviations. We can't count this left tail right there. So we can use the empirical rule. If we do our standard deviations to the left, that's one standard deviation, two standard deviations."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or essentially, more than two standard deviations from the mean, or above two standard deviations. We can't count this left tail right there. So we can use the empirical rule. If we do our standard deviations to the left, that's one standard deviation, two standard deviations. We know what this whole area is. The area, let me pick a different color. So we know what this area is."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "If we do our standard deviations to the left, that's one standard deviation, two standard deviations. We know what this whole area is. The area, let me pick a different color. So we know what this area is. The area within two standard deviations. The empirical rule tells us, or even better, the 68, 95, 99.7 rule tells us that this area, because it's within two standard deviations, is 95% or 0.95, or it's 95% of the area under the normal distribution. Which tells us that what's left over, this tail that we care about, and this left tail right here has to make up the rest of it, or 5%."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we know what this area is. The area within two standard deviations. The empirical rule tells us, or even better, the 68, 95, 99.7 rule tells us that this area, because it's within two standard deviations, is 95% or 0.95, or it's 95% of the area under the normal distribution. Which tells us that what's left over, this tail that we care about, and this left tail right here has to make up the rest of it, or 5%. So those two combined have to be 5%. And these are symmetrical. We've done this before."}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "Which tells us that what's left over, this tail that we care about, and this left tail right here has to make up the rest of it, or 5%. So those two combined have to be 5%. And these are symmetrical. We've done this before. This is actually a little redundant from other problems we've done. But if these are combined 5% and they're the same, then each of these are 2.5%. So to the answer to the question, what is the probability that a randomly chosen fifth grade boy would be taller than 157.7 centimeters?"}, {"video_title": "ck12.org More empirical rule and z-score practice   Probability and Statistics   Khan Academy.mp3", "Sentence": "We've done this before. This is actually a little redundant from other problems we've done. But if these are combined 5% and they're the same, then each of these are 2.5%. So to the answer to the question, what is the probability that a randomly chosen fifth grade boy would be taller than 157.7 centimeters? Well, that's literally just the area under this right green part. Maybe I'll do it in a different color. This magenta part that I'm coloring right now, that's just that area."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "We are told a large nationwide poll recently showed an unemployment rate of 9% in the United States. The mayor of a local town wonders if this national result holds true for her town. So she plans on taking a sample of her residents to see if the unemployment rate is significantly different than 9% in her town. Let P represent the unemployment rate in her town. Here are the hypotheses she'll use. So her null hypothesis is that, hey, the unemployment rate in her town is the same as for the country. And her alternative hypothesis is that it is not the same."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Let P represent the unemployment rate in her town. Here are the hypotheses she'll use. So her null hypothesis is that, hey, the unemployment rate in her town is the same as for the country. And her alternative hypothesis is that it is not the same. Under which of the following conditions would the mayor commit a type one error? So pause this video and see if you can figure it out on your own. Now let's work through this together."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And her alternative hypothesis is that it is not the same. Under which of the following conditions would the mayor commit a type one error? So pause this video and see if you can figure it out on your own. Now let's work through this together. So let's just remind ourselves what a type one error even is. This is a situation where we reject the null hypothesis even though it is true. Reject null hypothesis even though, even though our null hypothesis is true."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's work through this together. So let's just remind ourselves what a type one error even is. This is a situation where we reject the null hypothesis even though it is true. Reject null hypothesis even though, even though our null hypothesis is true. And in general, if you're committing either a type one or a type two error, you're doing the wrong thing. You're doing something that somehow contradicts reality even though you didn't intend to. And so in this case, that would be rejecting the hypothesis that the unemployment rate is 9% in this town even though it actually is 9% in this town."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Reject null hypothesis even though, even though our null hypothesis is true. And in general, if you're committing either a type one or a type two error, you're doing the wrong thing. You're doing something that somehow contradicts reality even though you didn't intend to. And so in this case, that would be rejecting the hypothesis that the unemployment rate is 9% in this town even though it actually is 9% in this town. So let's see which of these choices match up to that. She concludes the town's unemployment rate is not 9% when it actually is. Yeah, in this situation, in order to conclude that the unemployment rate is not 9%, she would have to reject the null hypothesis even though the null hypothesis is actually true, even though the unemployment rate actually is 9%."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And so in this case, that would be rejecting the hypothesis that the unemployment rate is 9% in this town even though it actually is 9% in this town. So let's see which of these choices match up to that. She concludes the town's unemployment rate is not 9% when it actually is. Yeah, in this situation, in order to conclude that the unemployment rate is not 9%, she would have to reject the null hypothesis even though the null hypothesis is actually true, even though the unemployment rate actually is 9%. So I'm liking this choice. But let's read the other ones just to make sure. She concludes the town's unemployment rate is not 9% when it actually is not."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Yeah, in this situation, in order to conclude that the unemployment rate is not 9%, she would have to reject the null hypothesis even though the null hypothesis is actually true, even though the unemployment rate actually is 9%. So I'm liking this choice. But let's read the other ones just to make sure. She concludes the town's unemployment rate is not 9% when it actually is not. Well, this wouldn't be an error. If the null hypothesis isn't true, it's not a problem to reject it. So this one wouldn't be an error."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "She concludes the town's unemployment rate is not 9% when it actually is not. Well, this wouldn't be an error. If the null hypothesis isn't true, it's not a problem to reject it. So this one wouldn't be an error. She concludes the town's unemployment rate is 9% when it actually is. Well, once again, this would not be an error. This would be failing to reject the null hypothesis when the null hypothesis is actually true, not an error."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "So this one wouldn't be an error. She concludes the town's unemployment rate is 9% when it actually is. Well, once again, this would not be an error. This would be failing to reject the null hypothesis when the null hypothesis is actually true, not an error. Choice D, she concludes the town's unemployment rate is 9% when it actually is not. So this is a situation where she fails to reject the null hypothesis even though the null hypothesis is not true. So this one right over here, this one would actually be, this is an error, this is an error, but this is a type two error."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "This would be failing to reject the null hypothesis when the null hypothesis is actually true, not an error. Choice D, she concludes the town's unemployment rate is 9% when it actually is not. So this is a situation where she fails to reject the null hypothesis even though the null hypothesis is not true. So this one right over here, this one would actually be, this is an error, this is an error, but this is a type two error. So one way to think about it, first you say, okay, am I making an error? Am I rejecting something that's true or am I failing to reject something that's false? And the rejecting something that is true, that's type one, and failing to reject something that is false, that is type two."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "So this one right over here, this one would actually be, this is an error, this is an error, but this is a type two error. So one way to think about it, first you say, okay, am I making an error? Am I rejecting something that's true or am I failing to reject something that's false? And the rejecting something that is true, that's type one, and failing to reject something that is false, that is type two. And so with that in mind, let's do another example. A large university is curious if they should build another cafeteria. They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40%, in which case, they will consider building a new cafeteria."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And the rejecting something that is true, that's type one, and failing to reject something that is false, that is type two. And so with that in mind, let's do another example. A large university is curious if they should build another cafeteria. They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40%, in which case, they will consider building a new cafeteria. Let P represent the proportion of students interested in a meal plan. Here are the hypotheses they'll use. So the null hypothesis is that 40% or fewer of the students are interested in a meal plan, while the alternative hypothesis is that more than 40% are interested."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40%, in which case, they will consider building a new cafeteria. Let P represent the proportion of students interested in a meal plan. Here are the hypotheses they'll use. So the null hypothesis is that 40% or fewer of the students are interested in a meal plan, while the alternative hypothesis is that more than 40% are interested. What would be the consequence of a type two error in this context? So once again, pause this video and try to answer this for yourself. Okay, now let's do it together."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "So the null hypothesis is that 40% or fewer of the students are interested in a meal plan, while the alternative hypothesis is that more than 40% are interested. What would be the consequence of a type two error in this context? So once again, pause this video and try to answer this for yourself. Okay, now let's do it together. Let's just remind ourselves what a type two error is. We just talked about it. So failing, failing to reject, in this case, our null hypothesis, even, even though it is false."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Okay, now let's do it together. Let's just remind ourselves what a type two error is. We just talked about it. So failing, failing to reject, in this case, our null hypothesis, even, even though it is false. So this would be a scenario where this is false, which would mean that more than 40% actually do want a meal plan, but you fail to reject this. So what would happen is is that you wouldn't build another cafeteria because you'd say, hey, no, there are not that many people who are interested in the meal plan, but you wouldn't, but actually, there are a lot of people who are interested in the meal plan, and so you probably won't have enough cafeteria space. And so this says, they don't consider building a new cafeteria when they should."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "So failing, failing to reject, in this case, our null hypothesis, even, even though it is false. So this would be a scenario where this is false, which would mean that more than 40% actually do want a meal plan, but you fail to reject this. So what would happen is is that you wouldn't build another cafeteria because you'd say, hey, no, there are not that many people who are interested in the meal plan, but you wouldn't, but actually, there are a lot of people who are interested in the meal plan, and so you probably won't have enough cafeteria space. And so this says, they don't consider building a new cafeteria when they should. Yeah, this is exactly right. They don't consider building a new cafeteria when they shouldn't. Well, this would just be a correct conclusion."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "And so this says, they don't consider building a new cafeteria when they should. Yeah, this is exactly right. They don't consider building a new cafeteria when they shouldn't. Well, this would just be a correct conclusion. They consider building a new cafeteria when they shouldn't. And so this is a scenario where they do reject the null hypothesis even though the null hypothesis is true. So this right over here would be a type one error."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "Well, this would just be a correct conclusion. They consider building a new cafeteria when they shouldn't. And so this is a scenario where they do reject the null hypothesis even though the null hypothesis is true. So this right over here would be a type one error. Type one error. Because if they're considering building a new cafeteria, that means they rejected the null hypothesis. Even when they shouldn't, that means that the null hypothesis was true, so type one."}, {"video_title": "Examples identifying Type I and Type II errors   AP Statistics   Khan Academy.mp3", "Sentence": "So this right over here would be a type one error. Type one error. Because if they're considering building a new cafeteria, that means they rejected the null hypothesis. Even when they shouldn't, that means that the null hypothesis was true, so type one. They consider building a new cafeteria when they should. Well, once again, this wouldn't be an error at all. This would be a correct conclusion."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I have a bag, and in that bag I am going to put some green cubes in that bag. And in particular, I am going to put 8 green cubes. I'm also going to put some spheres in that bag. Let's say I'm going to put 9 spheres, and these are the green spheres. I'm also going to put some yellow cubes in that bag. Some yellow cubes. I'm going to put 5 of those."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say I'm going to put 9 spheres, and these are the green spheres. I'm also going to put some yellow cubes in that bag. Some yellow cubes. I'm going to put 5 of those. And I'm also going to put some yellow spheres in this bag. Yellow spheres. And let's say I put 7 of those."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to put 5 of those. And I'm also going to put some yellow spheres in this bag. Yellow spheres. And let's say I put 7 of those. I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag. And what I want to think about in this video is what are the probabilities of getting different types of objects. So for example, what is the probability of getting a cube?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's say I put 7 of those. I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag. And what I want to think about in this video is what are the probabilities of getting different types of objects. So for example, what is the probability of getting a cube? A cube of any color. What is the probability of getting a cube? Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for example, what is the probability of getting a cube? A cube of any color. What is the probability of getting a cube? Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag. Did I do that right? This is 14, yep, 29 objects. So let's draw all of the possible objects."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "There are 29 objects in the bag. Did I do that right? This is 14, yep, 29 objects. So let's draw all of the possible objects. And I'll represent it as this big area right over here. So these are all the possible objects. There are 29 possible objects."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's draw all of the possible objects. And I'll represent it as this big area right over here. So these are all the possible objects. There are 29 possible objects. So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first. And how many of them meet our constraint of being a cube? Well, I have 8 green cubes, and I have 5 yellow cubes."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "There are 29 possible objects. So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first. And how many of them meet our constraint of being a cube? Well, I have 8 green cubes, and I have 5 yellow cubes. So there are a total of 13 cubes. So let me draw that set of cubes. So there's 13 cubes."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, I have 8 green cubes, and I have 5 yellow cubes. So there are a total of 13 cubes. So let me draw that set of cubes. So there's 13 cubes. I could draw it like this. There are 13 cubes. This right here is the set of cubes."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's 13 cubes. I could draw it like this. There are 13 cubes. This right here is the set of cubes. This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes. So the probability of getting a cube is the number of events that meet our criterion. So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This right here is the set of cubes. This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes. So the probability of getting a cube is the number of events that meet our criterion. So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29. That includes the cubes and the spheres. Now let's ask a different question. What is the probability of getting a yellow object, either a cube or a sphere?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29. That includes the cubes and the spheres. Now let's ask a different question. What is the probability of getting a yellow object, either a cube or a sphere? So once again, how many things meet our conditions here? We have 5 plus 7. There's 12 yellow objects in the bag."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of getting a yellow object, either a cube or a sphere? So once again, how many things meet our conditions here? We have 5 plus 7. There's 12 yellow objects in the bag. So we have 29 equally likely possibilities. I'll do that in the same color. We have 29 equally likely possibilities, and of those, 12 meet our criterion."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's 12 yellow objects in the bag. So we have 29 equally likely possibilities. I'll do that in the same color. We have 29 equally likely possibilities, and of those, 12 meet our criterion. So let me draw 12 right over here. I'll do my best attempt. It looks something like the set of yellow objects."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 29 equally likely possibilities, and of those, 12 meet our criterion. So let me draw 12 right over here. I'll do my best attempt. It looks something like the set of yellow objects. There are 12 objects that are yellow. So the 12 that meet our conditions are 12 over all the possibilities, 29. So the probability of getting a cube, 13 29ths."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "It looks something like the set of yellow objects. There are 12 objects that are yellow. So the 12 that meet our conditions are 12 over all the possibilities, 29. So the probability of getting a cube, 13 29ths. Probability of getting a yellow, 12 29ths. Now let's ask something a little bit more interesting. What is the probability of getting a yellow cube?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of getting a cube, 13 29ths. Probability of getting a yellow, 12 29ths. Now let's ask something a little bit more interesting. What is the probability of getting a yellow cube? I'll put it in yellow, so we care about the color now. So this thing is yellow. What is the probability of getting a yellow cube?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of getting a yellow cube? I'll put it in yellow, so we care about the color now. So this thing is yellow. What is the probability of getting a yellow cube? Well, there's 29 equally likely possibilities. And of those 29 equally likely possibilities, 5 of those are yellow cubes, or yellow cubes. 5 of them."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of getting a yellow cube? Well, there's 29 equally likely possibilities. And of those 29 equally likely possibilities, 5 of those are yellow cubes, or yellow cubes. 5 of them. So the probability is 5 29ths. And where would we see that on this Venn diagram that I've drawn? This is just a way to visualize the different probabilities."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "5 of them. So the probability is 5 29ths. And where would we see that on this Venn diagram that I've drawn? This is just a way to visualize the different probabilities. And they become interesting when you start thinking about where sets overlap, or even where they don't overlap. So here we're thinking about things that are members of the set yellow. So they're in this set, and they are cubes."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is just a way to visualize the different probabilities. And they become interesting when you start thinking about where sets overlap, or even where they don't overlap. So here we're thinking about things that are members of the set yellow. So they're in this set, and they are cubes. So this area right over here, that's the overlap of these two sets. So this area right over here, this represents things that are both yellow and cubes, because they are inside both circles. So this right over here, let me write it over here."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they're in this set, and they are cubes. So this area right over here, that's the overlap of these two sets. So this area right over here, this represents things that are both yellow and cubes, because they are inside both circles. So this right over here, let me write it over here. So there's 5 objects that are both yellow and cubes. Now let's ask, and this is probably the most interesting thing to ask, what is the probability of getting something that is yellow or a cube? A cube of any color."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this right over here, let me write it over here. So there's 5 objects that are both yellow and cubes. Now let's ask, and this is probably the most interesting thing to ask, what is the probability of getting something that is yellow or a cube? A cube of any color. Well we still know that the denominator here is going to be 29. These are all of the equally likely possibilities that might jump out of the bag. But what are the possibilities that meet our conditions?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "A cube of any color. Well we still know that the denominator here is going to be 29. These are all of the equally likely possibilities that might jump out of the bag. But what are the possibilities that meet our conditions? Well one way to think about it is, well there's 12 things that would meet the yellow condition. So that would be this entire circle right over here. 12 things that meet the yellow condition."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "But what are the possibilities that meet our conditions? Well one way to think about it is, well there's 12 things that would meet the yellow condition. So that would be this entire circle right over here. 12 things that meet the yellow condition. So this right over here is 12. This is the number of yellow, that is 12. And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "12 things that meet the yellow condition. So this right over here is 12. This is the number of yellow, that is 12. And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5. These 5 are counted as part of this 12. One way to think about it is, there are 7 yellow objects that are not cubes. Those are the spheres."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5. These 5 are counted as part of this 12. One way to think about it is, there are 7 yellow objects that are not cubes. Those are the spheres. There are 5 yellow objects that are cubes. And then there are 8 cubes that are not yellow. That's one way to think about it."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Those are the spheres. There are 5 yellow objects that are cubes. And then there are 8 cubes that are not yellow. That's one way to think about it. So when we counted this 12, the number of yellow, we counted all of this. So we can't just add the number of cubes to it, because then we would count this middle part again. So then we have to essentially count cubes, the number of cubes, which is 13."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's one way to think about it. So when we counted this 12, the number of yellow, we counted all of this. So we can't just add the number of cubes to it, because then we would count this middle part again. So then we have to essentially count cubes, the number of cubes, which is 13. So number of cubes. And we'll have to subtract out this middle section right over here. Let me do this."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So then we have to essentially count cubes, the number of cubes, which is 13. So number of cubes. And we'll have to subtract out this middle section right over here. Let me do this. So subtract out the middle section right over here. So minus 5. So this is the number of yellow cubes."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me do this. So subtract out the middle section right over here. So minus 5. So this is the number of yellow cubes. It feels weird to write the word yellow in green. The number of yellow cubes. Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what?"}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the number of yellow cubes. It feels weird to write the word yellow in green. The number of yellow cubes. Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what? It's 20. Did I do that right? 12 minus, yep, it's 20."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what? It's 20. Did I do that right? 12 minus, yep, it's 20. So that's one way you just get this is equal to 20 over 29. But the more interesting thing than even the answer of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video. So let's think about this a little bit."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "12 minus, yep, it's 20. So that's one way you just get this is equal to 20 over 29. But the more interesting thing than even the answer of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video. So let's think about this a little bit. We can rewrite this fraction right over here. We can rewrite this as 12 over 29 plus 13 over 29 minus 5 over 29. And this was the number of yellow over the total possibilities."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about this a little bit. We can rewrite this fraction right over here. We can rewrite this as 12 over 29 plus 13 over 29 minus 5 over 29. And this was the number of yellow over the total possibilities. So this right over here was the probability of getting a yellow. And this right over here was the number of cubes over the total possibilities. So this is plus the probability of getting a cube."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this was the number of yellow over the total possibilities. So this right over here was the probability of getting a yellow. And this right over here was the number of cubes over the total possibilities. So this is plus the probability of getting a cube. And this right over here was the number of yellow cubes over the total possibilities. So this right over here was minus the probability of yellow and a cube. Let me write it that way."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is plus the probability of getting a cube. And this right over here was the number of yellow cubes over the total possibilities. So this right over here was minus the probability of yellow and a cube. Let me write it that way. Minus the probability of yellow. I'll write yellow in yellow. Yellow and getting a cube."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write it that way. Minus the probability of yellow. I'll write yellow in yellow. Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0."}, {"video_title": "Addition rule for probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0. But if things are not mutually exclusive, you would have to subtract out the overlap. Probably the best way to think about it is to just always realize that you have to subtract out the overlap. And obviously, if something is mutually exclusive, the probability of getting A and B is going to be 0."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe it's the mean of a population, the mean height of all the people in the city. And we've determined that it's unpractical, or there's no way for us to know the true population parameter, but we could try to estimate it by taking a sample size. So we take n samples, and then we calculate a statistic based on that. We've also seen that not only can we calculate the statistic which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. And so that confidence interval would look something like this. It would be the value of the statistic that we have just calculated, plus or minus some margin of error. And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "We've also seen that not only can we calculate the statistic which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. And so that confidence interval would look something like this. It would be the value of the statistic that we have just calculated, plus or minus some margin of error. And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic. Now what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic. Now what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate? Well, in that world, the statistic is the sample proportion. So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate? Well, in that world, the statistic is the sample proportion. So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size. If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size. If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size. But we often will not know this."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size. But we often will not know this. In fact, it's very unusual to know this. And so sometimes you will say, okay, if we don't know this, let's just figure out the sample standard deviation of our sample here. So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "But we often will not know this. In fact, it's very unusual to know this. And so sometimes you will say, okay, if we don't know this, let's just figure out the sample standard deviation of our sample here. So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. But it turns out that this is not, not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level. And so that's why statisticians have invented another statistic."}, {"video_title": "Introduction to t statistics   Confidence intervals   AP Statistics   Khan Academy.mp3", "Sentence": "So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. But it turns out that this is not, not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level. And so that's why statisticians have invented another statistic. Instead of using z, they call it t. Instead of using a z table, they use a t table. And we're going to see this in future videos. And so if you are actually trying to construct a confidence interval for a sample mean, and you don't know the true standard deviation of your population, which is normally the case, instead of doing this, what we're going to do is, we're gonna take our sample mean, plus or minus, and our critical value, we'll call that t star, times our sample standard deviation, which we can calculate, divided by the square root of n. And so the real functional difference is that this actually is going to give us the confidence interval that actually has the level of confidence that we want."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "For a senior project, Richard is researching how much money a college graduate can expect to earn based on his or her major. He finds the following interesting facts. Basketball superstar Michael Jordan was a geology major at the University of North Carolina. There were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. Of the 35 finance majors from Wesleyan University, 32 got high-paying consulting jobs, and the other three were unemployed. For geology majors from the University of North Carolina, the median income will likely be, and we have some options here, less than, equal to, or greater than the mean."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "There were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. Of the 35 finance majors from Wesleyan University, 32 got high-paying consulting jobs, and the other three were unemployed. For geology majors from the University of North Carolina, the median income will likely be, and we have some options here, less than, equal to, or greater than the mean. And then we have to answer the same question for civil engineering majors from Montana. The median income, oh actually, this is, we're both about median. The median income will be, and we compare it against the mean, and then for finance majors from Wesleyan, we're gonna compare the median income to the mean."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "For geology majors from the University of North Carolina, the median income will likely be, and we have some options here, less than, equal to, or greater than the mean. And then we have to answer the same question for civil engineering majors from Montana. The median income, oh actually, this is, we're both about median. The median income will be, and we compare it against the mean, and then for finance majors from Wesleyan, we're gonna compare the median income to the mean. So to visualize this a little bit more, I've copy and pasted this exact same problem onto my scratch pad, so here it is. I can now write on this. So let's think about each of these."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The median income will be, and we compare it against the mean, and then for finance majors from Wesleyan, we're gonna compare the median income to the mean. So to visualize this a little bit more, I've copy and pasted this exact same problem onto my scratch pad, so here it is. I can now write on this. So let's think about each of these. For geology majors from UNC, the median income will likely be, how will that compare to the mean? Well, what do they tell us about UNC? They tell us that Michael Jordan was a geology major at the University of North Carolina."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about each of these. For geology majors from UNC, the median income will likely be, how will that compare to the mean? Well, what do they tell us about UNC? They tell us that Michael Jordan was a geology major at the University of North Carolina. So what will the distribution of salaries probably look like? So if we're thinking about the University of North Carolina, it probably looks something like this. And I'm gonna do a very rough, a very rough distribution right over here."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "They tell us that Michael Jordan was a geology major at the University of North Carolina. So what will the distribution of salaries probably look like? So if we're thinking about the University of North Carolina, it probably looks something like this. And I'm gonna do a very rough, a very rough distribution right over here. And let's say that this salary, this would be a salary of zero, and let's say that this is a salary of, I don't know, let me put a salary of 50K here. I'll do this in thousands. Let's say this is 100,000 right over here."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm gonna do a very rough, a very rough distribution right over here. And let's say that this salary, this would be a salary of zero, and let's say that this is a salary of, I don't know, let me put a salary of 50K here. I'll do this in thousands. Let's say this is 100,000 right over here. And then you have Michael Jordan, who is, actually I'll do a little gap here because he's so far up. I don't know exactly what he was making, but it was definitely in the tens of millions of dollars a year. So Michael Jordan is way, way, way, way up here."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say this is 100,000 right over here. And then you have Michael Jordan, who is, actually I'll do a little gap here because he's so far up. I don't know exactly what he was making, but it was definitely in the tens of millions of dollars a year. So Michael Jordan is way, way, way, way up here. So if you were to make a histogram or a plot of all of the salaries, you could say, okay, well, you know, maybe we have, if you put all of the folks from geology majors at University of North Carolina, well, there's probably, especially right when they graduated, there's probably, you know, one, two, three, I could keep doing it. A bunch of people, maybe making 50K, maybe some people making a little bit more, maybe some people up here, maybe some people there, some people there, some people there, like there. Maybe someone's making 100K, maybe someone, so maybe it's a couple of people up there."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So Michael Jordan is way, way, way, way up here. So if you were to make a histogram or a plot of all of the salaries, you could say, okay, well, you know, maybe we have, if you put all of the folks from geology majors at University of North Carolina, well, there's probably, especially right when they graduated, there's probably, you know, one, two, three, I could keep doing it. A bunch of people, maybe making 50K, maybe some people making a little bit more, maybe some people up here, maybe some people there, some people there, some people there, like there. Maybe someone's making 100K, maybe someone, so maybe it's a couple of people up there. Maybe someone isn't making anything. Maybe they weren't able to find a job. And then, of course, you have Michael Jordan up here making $10 million or $20 million or something like that."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Maybe someone's making 100K, maybe someone, so maybe it's a couple of people up there. Maybe someone isn't making anything. Maybe they weren't able to find a job. And then, of course, you have Michael Jordan up here making $10 million or $20 million or something like that. So when you have a situation like this, where you have this outlier of Michael Jordan, it's going to put, one way I think about it, it kind of tugs on the mean. It wouldn't affect the median, because remember, the median is the middle value. So it doesn't matter how high this number is."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then, of course, you have Michael Jordan up here making $10 million or $20 million or something like that. So when you have a situation like this, where you have this outlier of Michael Jordan, it's going to put, one way I think about it, it kind of tugs on the mean. It wouldn't affect the median, because remember, the median is the middle value. So it doesn't matter how high this number is. You could make this a trillion dollars. It's not going to change what the middle value is. The middle value is still going to be the same middle value."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it doesn't matter how high this number is. You could make this a trillion dollars. It's not going to change what the middle value is. The middle value is still going to be the same middle value. You could move this anywhere around in this range. It's just not going to change the median. But the mean will change."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "The middle value is still going to be the same middle value. You could move this anywhere around in this range. It's just not going to change the median. But the mean will change. If this becomes really, really astronomically high, it will distort the actual mean here. Actually, it would distort it a good bit. So for geology majors from UNC, the median income is going to be lower than the mean, because Michael Jordan is pulling the mean up."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the mean will change. If this becomes really, really astronomically high, it will distort the actual mean here. Actually, it would distort it a good bit. So for geology majors from UNC, the median income is going to be lower than the mean, because Michael Jordan is pulling the mean up. So let me fill that in. So for geology majors from UNC, the median will be less than the mean. Now let's think about the other ones."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for geology majors from UNC, the median income is going to be lower than the mean, because Michael Jordan is pulling the mean up. So let me fill that in. So for geology majors from UNC, the median will be less than the mean. Now let's think about the other ones. For civil engineering majors from Montana, the median income will be blank the mean. Well, they tell us there were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now let's think about the other ones. For civil engineering majors from Montana, the median income will be blank the mean. Well, they tell us there were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. So let's say all three of them earned $50,000. Let's say that's their salary. So if you were to calculate the mean, it would be 50 plus 50 plus 50 over three, which of course is 50."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "They all took the exact same job at the same company, earning the same salary. So let's say all three of them earned $50,000. Let's say that's their salary. So if you were to calculate the mean, it would be 50 plus 50 plus 50 over three, which of course is 50. That would be the mean. If you wanted the median, you list the salaries in order, and then you take the middle one. Well, the middle one is 50."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you were to calculate the mean, it would be 50 plus 50 plus 50 over three, which of course is 50. That would be the mean. If you wanted the median, you list the salaries in order, and then you take the middle one. Well, the middle one is 50. So in this case, the median is equal to the mean. So let's fill that in. Median is equal to the mean."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the middle one is 50. So in this case, the median is equal to the mean. So let's fill that in. Median is equal to the mean. And then finally, let me go back to my scratch pad. Whoops, let me go back to my scratch pad here. For finance majors from Wesleyan, the median income will be blank the mean."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "Median is equal to the mean. And then finally, let me go back to my scratch pad. Whoops, let me go back to my scratch pad here. For finance majors from Wesleyan, the median income will be blank the mean. So let's think about this distribution here. So here, we have out of the 35, 32 got high-paying consulting jobs. So let's say that they were making six figures."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "For finance majors from Wesleyan, the median income will be blank the mean. So let's think about this distribution here. So here, we have out of the 35, 32 got high-paying consulting jobs. So let's say that they were making six figures. So the distribution might look something like this, where if this is zero, and let's say this is 50K, and let's say that this right over here is $100,000 a year. So 32 got high-paying consulting jobs. You might have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that they were making six figures. So the distribution might look something like this, where if this is zero, and let's say this is 50K, and let's say that this right over here is $100,000 a year. So 32 got high-paying consulting jobs. You might have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. So the distribution for the people who got the jobs might look something like that. But there were three people who were unemployed. So let's say they got no income."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "You might have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. So the distribution for the people who got the jobs might look something like that. But there were three people who were unemployed. So let's say they got no income. So you have one, two, three. So this is now, you have three outliers, like the Michael Jordan situation, but instead of them being very high, they are very low. So they're going to pull the mean lower."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say they got no income. So you have one, two, three. So this is now, you have three outliers, like the Michael Jordan situation, but instead of them being very high, they are very low. So they're going to pull the mean lower. They're not going to, if these were zero, or if these were 50, or if these were over here, they're not going to affect the median. The middle number is still going to be the same. But they are going to pull down the mean."}, {"video_title": "Means and medians of different distributions   Probability and Statistics   Khan Academy.mp3", "Sentence": "So they're going to pull the mean lower. They're not going to, if these were zero, or if these were 50, or if these were over here, they're not going to affect the median. The middle number is still going to be the same. But they are going to pull down the mean. So here, I would say that the median income will likely be higher, will likely be greater than the mean, because the mean is going to get pulled down by these outliers, these three people not making anything. So let's fill that out. For finance majors from Wesleyan, the median income will likely be greater than the mean."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "I can assume that's pronounced Ted-eff. And what it allows us to do is give us an intuition as to why we divide by n minus one when we calculate our sample variance and why that gives us an unbiased estimate of population variance. So the way this starts off, and I encourage you to go try this out yourself, is that you can construct a distribution. It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point. So I've constructed a population."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point. So I've constructed a population. I can throw out some random points up here. So this is our population. And as you saw while I was doing that, I was calculating parameters for the population."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "So I've constructed a population. I can throw out some random points up here. So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared. These are these numbers squared."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared. These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. Anything less than negative one, if we did negative n minus 1.05, or n minus 1.5, we start overestimating the variance. Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance. And you can do this for samples of different sizes."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate   Khan Academy.mp3", "Sentence": "When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. Anything less than negative one, if we did negative n minus 1.05, or n minus 1.5, we start overestimating the variance. Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance. And you can do this for samples of different sizes. Let me try a sample size six. And here you go, once again, as I press, I'm just keeping generate sample pressed down. As we generate more and more and more samples, and for all of the a's, we essentially take the average across those samples for the variance, depending on how we calculate it, you'll see that, once again, our best estimate is pretty darn close, is pretty darn close to negative one."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "In a local teaching district, a technology grant is available to teachers in order to install a cluster of four computers in their classroom. From the 6,250 teachers in the district, 250 were randomly selected and asked if they felt that computers were an essential teaching tool for their classroom. Of those selected, 142 teachers felt that computers were an essential teaching tool. Then they asked us, calculate a 99% confidence interval for the proportion of teachers who felt that computers are an essential teaching tool. Let's think about the entire population. We weren't able to survey all of them, but the entire population, some of them fall into the bucket and we'll define that as 1. They thought it was a good tool."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Then they asked us, calculate a 99% confidence interval for the proportion of teachers who felt that computers are an essential teaching tool. Let's think about the entire population. We weren't able to survey all of them, but the entire population, some of them fall into the bucket and we'll define that as 1. They thought it was a good tool. They thought that the computers were a good tool. We'll just define a zero value as a teacher that says not good. Some proportion of the total teachers think that it is a good learning tool."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "They thought it was a good tool. They thought that the computers were a good tool. We'll just define a zero value as a teacher that says not good. Some proportion of the total teachers think that it is a good learning tool. That proportion is P. The rest of them think it's a bad learning tool. 1 minus P. We have a Bernoulli distribution right over here. We know that the mean of this distribution, or the expected value of this distribution, is actually going to be P. It's actually going to be a value."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Some proportion of the total teachers think that it is a good learning tool. That proportion is P. The rest of them think it's a bad learning tool. 1 minus P. We have a Bernoulli distribution right over here. We know that the mean of this distribution, or the expected value of this distribution, is actually going to be P. It's actually going to be a value. It's neither 0 or 1, so not an actual value that you could actually get out of a teacher if you were to ask them. They cannot say something in between good and not good, but the actual expected value is something in between. It is P. What we do is, we're taking a sample of those 250 teachers, and we got that 142 felt that computers were an essential teaching tool."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that the mean of this distribution, or the expected value of this distribution, is actually going to be P. It's actually going to be a value. It's neither 0 or 1, so not an actual value that you could actually get out of a teacher if you were to ask them. They cannot say something in between good and not good, but the actual expected value is something in between. It is P. What we do is, we're taking a sample of those 250 teachers, and we got that 142 felt that computers were an essential teaching tool. In our survey, we had 250 sampled, and we got 142 said that it is good. We'll say that this is a 1. We got 142 1's, or we sampled 1 142 times from this distribution."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It is P. What we do is, we're taking a sample of those 250 teachers, and we got that 142 felt that computers were an essential teaching tool. In our survey, we had 250 sampled, and we got 142 said that it is good. We'll say that this is a 1. We got 142 1's, or we sampled 1 142 times from this distribution. Then the rest of the times, what's left over? There's another 108 who said that it's not good. 108 said not good, or you could view them as you were sampling a 0."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We got 142 1's, or we sampled 1 142 times from this distribution. Then the rest of the times, what's left over? There's another 108 who said that it's not good. 108 said not good, or you could view them as you were sampling a 0. 108 plus 142 is 250. What is our sample mean here? We have 1 times 142 plus 0 times 108 divided by our total number of samples, divided by 250."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "108 said not good, or you could view them as you were sampling a 0. 108 plus 142 is 250. What is our sample mean here? We have 1 times 142 plus 0 times 108 divided by our total number of samples, divided by 250. It is equal to 142 over 250. You could even view this as the sample proportion of teachers who thought that the computers were a good teaching tool. Let me get a calculator out to calculate this."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have 1 times 142 plus 0 times 108 divided by our total number of samples, divided by 250. It is equal to 142 over 250. You could even view this as the sample proportion of teachers who thought that the computers were a good teaching tool. Let me get a calculator out to calculate this. We have 142 divided by 250 is equal to 0.568. Our sample proportion is 0.568, or 56.8% either one. 0.568."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me get a calculator out to calculate this. We have 142 divided by 250 is equal to 0.568. Our sample proportion is 0.568, or 56.8% either one. 0.568. Let's also figure out our sample variance, because we can use it later for building our confidence interval. Our sample variance here, we're going to take the weighted sum of the squared differences from the mean and divide by this minus 1. We can get the best estimator of the true variance."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "0.568. Let's also figure out our sample variance, because we can use it later for building our confidence interval. Our sample variance here, we're going to take the weighted sum of the squared differences from the mean and divide by this minus 1. We can get the best estimator of the true variance. It's 1 times 142 samples that were 1 minus 0.568 away from our sample mean. We're this far from the sample mean 142 times, and we're going to square those distances. Plus the other 108 times, we got a 0, so we were 0 minus 0.568 away from the sample mean."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We can get the best estimator of the true variance. It's 1 times 142 samples that were 1 minus 0.568 away from our sample mean. We're this far from the sample mean 142 times, and we're going to square those distances. Plus the other 108 times, we got a 0, so we were 0 minus 0.568 away from the sample mean. Then we are going to divide that by the total number of samples minus 1. That minus 1 is our adjuster so that we don't underestimate. 250 minus 1, let's get our calculator out again."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Plus the other 108 times, we got a 0, so we were 0 minus 0.568 away from the sample mean. Then we are going to divide that by the total number of samples minus 1. That minus 1 is our adjuster so that we don't underestimate. 250 minus 1, let's get our calculator out again. We have 142 times 1 minus 0.568 squared plus 108 times 0 minus 0.568 squared, and then all of that divided by 250 minus 1 is 249. Our sample variance is 0.246. If you were to take the square root of that, our actual sample standard deviation is 0.496."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "250 minus 1, let's get our calculator out again. We have 142 times 1 minus 0.568 squared plus 108 times 0 minus 0.568 squared, and then all of that divided by 250 minus 1 is 249. Our sample variance is 0.246. If you were to take the square root of that, our actual sample standard deviation is 0.496. I'll just round that up to 0.50. That is our sample standard deviation. Let's think of it this way."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you were to take the square root of that, our actual sample standard deviation is 0.496. I'll just round that up to 0.50. That is our sample standard deviation. Let's think of it this way. We are sampling from some sampling distribution of the sample mean. It looks like this over here. It looks like that over there."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's think of it this way. We are sampling from some sampling distribution of the sample mean. It looks like this over here. It looks like that over there. It has some mean. The sampling distribution of the sample mean is actually going to be the same thing as this mean over here. It's going to be the same mean value, which is the same thing as our population proportion."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It looks like that over there. It has some mean. The sampling distribution of the sample mean is actually going to be the same thing as this mean over here. It's going to be the same mean value, which is the same thing as our population proportion. We've seen this multiple times. The sampling distribution's standard deviation, so the standard deviation of the sampling distribution, we could view that as one standard deviation right over there. The standard deviation of the sampling distribution we've seen multiple times is equal to the standard deviation of our original population divided by the square root of the number of samples."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's going to be the same mean value, which is the same thing as our population proportion. We've seen this multiple times. The sampling distribution's standard deviation, so the standard deviation of the sampling distribution, we could view that as one standard deviation right over there. The standard deviation of the sampling distribution we've seen multiple times is equal to the standard deviation of our original population divided by the square root of the number of samples. We do not know this right over here. We do not know the actual standard deviation in our population. Our best estimate of that, and that's why we call it confident, we're confident that the real population proportion is going to be in this interval."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The standard deviation of the sampling distribution we've seen multiple times is equal to the standard deviation of our original population divided by the square root of the number of samples. We do not know this right over here. We do not know the actual standard deviation in our population. Our best estimate of that, and that's why we call it confident, we're confident that the real population proportion is going to be in this interval. We're confident, but we're not 100% sure because we're going to estimate this over here. If we're estimating this, we're really estimating that over there. If this can be estimated, it's going to be estimated by this sample standard deviation."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Our best estimate of that, and that's why we call it confident, we're confident that the real population proportion is going to be in this interval. We're confident, but we're not 100% sure because we're going to estimate this over here. If we're estimating this, we're really estimating that over there. If this can be estimated, it's going to be estimated by this sample standard deviation. Then we can say this is going to be approximately, or if we didn't get a weird, completely skewed sample, it actually might not even be approximately if we just had a really strange sample. Maybe we should write confident that. Confident that the standard deviation of our sampling distribution is going to be around, instead of using this, we can use our standard deviation of our sample, our sample standard deviation."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If this can be estimated, it's going to be estimated by this sample standard deviation. Then we can say this is going to be approximately, or if we didn't get a weird, completely skewed sample, it actually might not even be approximately if we just had a really strange sample. Maybe we should write confident that. Confident that the standard deviation of our sampling distribution is going to be around, instead of using this, we can use our standard deviation of our sample, our sample standard deviation. 0.50 divided by the square root of 250, and what's that going to be? That is going to be, so we have this value right over here, actually I don't have to round it, divided by the square root of 250. We get 0.031, so this is equal to 0.031 over here."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Confident that the standard deviation of our sampling distribution is going to be around, instead of using this, we can use our standard deviation of our sample, our sample standard deviation. 0.50 divided by the square root of 250, and what's that going to be? That is going to be, so we have this value right over here, actually I don't have to round it, divided by the square root of 250. We get 0.031, so this is equal to 0.031 over here. That's one standard deviation. Now, they want a 99% confidence interval. The way I think about it is, if I randomly pick a sample from the sampling distribution, what's the 99% chance, or how many, let me think of it this way, how many standard deviations away from the mean do we have to be that we can be 99% confident that any sample from the sampling distribution will be in that interval?"}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We get 0.031, so this is equal to 0.031 over here. That's one standard deviation. Now, they want a 99% confidence interval. The way I think about it is, if I randomly pick a sample from the sampling distribution, what's the 99% chance, or how many, let me think of it this way, how many standard deviations away from the mean do we have to be that we can be 99% confident that any sample from the sampling distribution will be in that interval? Another way to think about it, think about how many standard deviations we need to be away from the mean. We're going to be a certain number of standard deviations away from the mean, such that any sample, any mean that we sample from here, any sample from this distribution has a 99% chance of being with plus or minus that many standard deviations. It might be from there to there."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "The way I think about it is, if I randomly pick a sample from the sampling distribution, what's the 99% chance, or how many, let me think of it this way, how many standard deviations away from the mean do we have to be that we can be 99% confident that any sample from the sampling distribution will be in that interval? Another way to think about it, think about how many standard deviations we need to be away from the mean. We're going to be a certain number of standard deviations away from the mean, such that any sample, any mean that we sample from here, any sample from this distribution has a 99% chance of being with plus or minus that many standard deviations. It might be from there to there. That's what we want. We want a 99% chance that if we pick a sample from the sampling distribution of the sample mean, it will be within this many standard deviations of the actual mean. To figure that out, let's look at an actual z-table."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "It might be from there to there. That's what we want. We want a 99% chance that if we pick a sample from the sampling distribution of the sample mean, it will be within this many standard deviations of the actual mean. To figure that out, let's look at an actual z-table. We want 99% confidence. Another way to think about it, if we want 99% confidence, if we just look at the upper half right over here, that orange area should be 0.475, because if this is 0.475, then this other part is going to be 0.475. We want to get to 99%, so it's not going to be 0.475."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "To figure that out, let's look at an actual z-table. We want 99% confidence. Another way to think about it, if we want 99% confidence, if we just look at the upper half right over here, that orange area should be 0.475, because if this is 0.475, then this other part is going to be 0.475. We want to get to 99%, so it's not going to be 0.475. We're going to have to go to 0.495 if we want 99% confidence. This area has to be 0.495 over here, because if that is, that over here will also be, so that their sum will be 99% of the area. This is 0.495, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this, is going to be 0.5."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We want to get to 99%, so it's not going to be 0.475. We're going to have to go to 0.495 if we want 99% confidence. This area has to be 0.495 over here, because if that is, that over here will also be, so that their sum will be 99% of the area. This is 0.495, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this, is going to be 0.5. It's going to be 0.5 plus 0.495. It's going to be 0.995. Let me make sure I got that right, 0.995."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 0.495, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this, is going to be 0.5. It's going to be 0.5 plus 0.495. It's going to be 0.995. Let me make sure I got that right, 0.995. Let's look at our z-table. Where do we get 0.995 on our z-table? 0.995 is pretty close, just to have a little error."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me make sure I got that right, 0.995. Let's look at our z-table. Where do we get 0.995 on our z-table? 0.995 is pretty close, just to have a little error. It will be right over here. This is 0.9951. Another way to think about it is 99, so this value right here gives us the whole cumulative area, up to that, up to our mean."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "0.995 is pretty close, just to have a little error. It will be right over here. This is 0.9951. Another way to think about it is 99, so this value right here gives us the whole cumulative area, up to that, up to our mean. If you look at the entire distribution like this, this is the mean right over here. This tells us that at 2.5 standard deviations above the mean, so this is 2.5 times the standard deviation of the sampling distribution, if you look at this whole area over here, if you look at the z-table, is going to be 0.9951, which tells us that just this area right over here is going to be 0.4951, which tells us that this area plus a symmetric area of that many standard deviations below the mean, if you combine them, 0.4951 times 2 gets us to 99.2. This whole area right here is 99.992."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Another way to think about it is 99, so this value right here gives us the whole cumulative area, up to that, up to our mean. If you look at the entire distribution like this, this is the mean right over here. This tells us that at 2.5 standard deviations above the mean, so this is 2.5 times the standard deviation of the sampling distribution, if you look at this whole area over here, if you look at the z-table, is going to be 0.9951, which tells us that just this area right over here is going to be 0.4951, which tells us that this area plus a symmetric area of that many standard deviations below the mean, if you combine them, 0.4951 times 2 gets us to 99.2. This whole area right here is 99.992. If we look at the area of 2.5 standard deviations above and below the mean, let me be careful, this isn't just 2.5, we have to add another digit of precision. This is 2.5, and the next digit of precision is given by this column over here. We have to look all the way up into the second to last column, and we have to add a digit of 8 here."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "This whole area right here is 99.992. If we look at the area of 2.5 standard deviations above and below the mean, let me be careful, this isn't just 2.5, we have to add another digit of precision. This is 2.5, and the next digit of precision is given by this column over here. We have to look all the way up into the second to last column, and we have to add a digit of 8 here. This is 2.58 standard deviations. We have 2.5 over here, and then we get the next digit, 8, from the column. 2.58 standard deviations above and below the standard deviation encompasses a little over 99% of the total probability."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We have to look all the way up into the second to last column, and we have to add a digit of 8 here. This is 2.58 standard deviations. We have 2.5 over here, and then we get the next digit, 8, from the column. 2.58 standard deviations above and below the standard deviation encompasses a little over 99% of the total probability. There's a little over 99% chance that any sample mean that I select from the sampling distribution of the sample mean will fall within this much of the standard deviation. Let me put it this way. There is a 99.2% chance."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "2.58 standard deviations above and below the standard deviation encompasses a little over 99% of the total probability. There's a little over 99% chance that any sample mean that I select from the sampling distribution of the sample mean will fall within this much of the standard deviation. Let me put it this way. There is a 99.2% chance. If you multiply this times 2, you get 0.99. You get 0.9902. Let's say roughly 99% chance that any sample that a random sample mean is within 2.58 standard deviations of the sampling mean of our population mean, of the mean of the sampling distribution of the sampling mean, which is the same thing as our actual population mean, which is the same thing as our population proportion, so of p. We know what this value is right here."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "There is a 99.2% chance. If you multiply this times 2, you get 0.99. You get 0.9902. Let's say roughly 99% chance that any sample that a random sample mean is within 2.58 standard deviations of the sampling mean of our population mean, of the mean of the sampling distribution of the sampling mean, which is the same thing as our actual population mean, which is the same thing as our population proportion, so of p. We know what this value is right here. At least we have a decent estimate for this value. We don't know exactly what this is, but our best estimate for this value is this over here. We could rewrite this, and we could say that we are confident because we are really using an estimator to get this value over here."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let's say roughly 99% chance that any sample that a random sample mean is within 2.58 standard deviations of the sampling mean of our population mean, of the mean of the sampling distribution of the sampling mean, which is the same thing as our actual population mean, which is the same thing as our population proportion, so of p. We know what this value is right here. At least we have a decent estimate for this value. We don't know exactly what this is, but our best estimate for this value is this over here. We could rewrite this, and we could say that we are confident because we are really using an estimator to get this value over here. We are confident that there is a 99% chance that a random X, a random sample mean, is within, and let's figure out this value right here using a calculator. It is 2.58 times our best estimate of the standard deviation of the sampling distribution, so times 0.031 is equal to 0.0, well, let's just round this up because it's so close to 0.08, is within 0.08 of the population proportion, or you could say that you are confident that the population proportion is within 0.08 of your sample mean. That's the exact same statement."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We could rewrite this, and we could say that we are confident because we are really using an estimator to get this value over here. We are confident that there is a 99% chance that a random X, a random sample mean, is within, and let's figure out this value right here using a calculator. It is 2.58 times our best estimate of the standard deviation of the sampling distribution, so times 0.031 is equal to 0.0, well, let's just round this up because it's so close to 0.08, is within 0.08 of the population proportion, or you could say that you are confident that the population proportion is within 0.08 of your sample mean. That's the exact same statement. If we want our confidence interval, our actual number that we got for there, our actual sample mean, we got was 0.568. We could replace this, and actually let me do it. I can delete this right here, let me clear it."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's the exact same statement. If we want our confidence interval, our actual number that we got for there, our actual sample mean, we got was 0.568. We could replace this, and actually let me do it. I can delete this right here, let me clear it. I can replace this, because we actually did take a sample, so I can replace this with 0.568. We could be confident that there is a 99% chance that 0.568 is within 0.08 of the actual sample, of the population proportion, which is the same thing as the population mean, which is the same thing as the mean of the sampling distribution of the sample mean, so forth and so on. Just to make it clear, we can actually swap these two."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I can delete this right here, let me clear it. I can replace this, because we actually did take a sample, so I can replace this with 0.568. We could be confident that there is a 99% chance that 0.568 is within 0.08 of the actual sample, of the population proportion, which is the same thing as the population mean, which is the same thing as the mean of the sampling distribution of the sample mean, so forth and so on. Just to make it clear, we can actually swap these two. It wouldn't change the meaning. If this is within 0.08 of that, then that is within 0.08 of this. Let me switch this up a little bit."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Just to make it clear, we can actually swap these two. It wouldn't change the meaning. If this is within 0.08 of that, then that is within 0.08 of this. Let me switch this up a little bit. We could put a P is within of 0.568. Now, linguistically, it sounds a little bit more like a confidence interval. We are confident that there is a 99% chance that P is within 0.08 of the sample mean of 0.568."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me switch this up a little bit. We could put a P is within of 0.568. Now, linguistically, it sounds a little bit more like a confidence interval. We are confident that there is a 99% chance that P is within 0.08 of the sample mean of 0.568. What would be our confidence interval? It would be 0.568 plus or minus 0.08. What would that be?"}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We are confident that there is a 99% chance that P is within 0.08 of the sample mean of 0.568. What would be our confidence interval? It would be 0.568 plus or minus 0.08. What would that be? If you add 0.08 to this right over here, at the upper end, you're going to have 0.648, and at the lower end of our range, so this is the upper end, the lower end, if we subtract 8 from this, we get 0.488. We are 99% confident that the true population proportion is between these two numbers. Or another way, that the true percentage of teachers who think those computers are good ideas is between."}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What would that be? If you add 0.08 to this right over here, at the upper end, you're going to have 0.648, and at the lower end of our range, so this is the upper end, the lower end, if we subtract 8 from this, we get 0.488. We are 99% confident that the true population proportion is between these two numbers. Or another way, that the true percentage of teachers who think those computers are good ideas is between. We're 99% confident, we're confident that there's a 99% chance that the true percentage of teachers that like the computers is between 48.8% and 64.8%. Now, that we answered the first part of the question. The second part, how could the survey be changed to narrow the confidence interval, but to maintain the 99% confidence interval?"}, {"video_title": "Confidence interval example   Inferential statistics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way, that the true percentage of teachers who think those computers are good ideas is between. We're 99% confident, we're confident that there's a 99% chance that the true percentage of teachers that like the computers is between 48.8% and 64.8%. Now, that we answered the first part of the question. The second part, how could the survey be changed to narrow the confidence interval, but to maintain the 99% confidence interval? Well, you could just take more samples. If you take more samples, then our estimate of the standard deviation of this distribution will go down, because this denominator will be higher. If that denominator is higher, then this whole thing will go down."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Liliana runs a cake decorating business for which 10% of her orders come over the telephone. Let C be the number of cake orders Liliana receives in a month until she first gets an order over the telephone. Assume the method of placing each cake order is independent. So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three?"}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four?"}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that. Let's get a calculator out."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that. Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439."}, {"video_title": "Cumulative geometric probability (less than a value)   AP Statistics   Khan Academy.mp3", "Sentence": "Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439. So this is equal to 0.3439. And we're done. That's the probability that it takes fewer than five orders for her to get her first telephone order of the month."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Three of them are unfair in that they have a 45% chance of coming up tails when flipped. The rest are fair. So for the rest of them, you have a 50% chance of tails or a 50% chance of heads. You randomly choose one coin from the bag and flip it four times. What is the percent probability of getting four heads? So let's think about it. When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "You randomly choose one coin from the bag and flip it four times. What is the percent probability of getting four heads? So let's think about it. When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin. And three of the four coins are unfair. So there's a 3 4 probability that we get an unfair coin. And then there is only one out of the four coins that's fair."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin. And three of the four coins are unfair. So there's a 3 4 probability that we get an unfair coin. And then there is only one out of the four coins that's fair. So there was a 1 4 probability that I get a fair coin. Now, given that I have unfair, let's remind ourselves. An unfair coin has a 45% chance of coming up tails."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then there is only one out of the four coins that's fair. So there was a 1 4 probability that I get a fair coin. Now, given that I have unfair, let's remind ourselves. An unfair coin has a 45% chance of coming up tails. So this means that I have a 45% chance of tails, which also means, and we have to be careful here because they're asking us about heads, if I have a 45% chance of getting tails, that means I have a 55% chance of getting heads. Whatever, I have 100% chance of getting one of these two. If it's 45% for tails, 100 minus 45 is 55 for heads."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "An unfair coin has a 45% chance of coming up tails. So this means that I have a 45% chance of tails, which also means, and we have to be careful here because they're asking us about heads, if I have a 45% chance of getting tails, that means I have a 55% chance of getting heads. Whatever, I have 100% chance of getting one of these two. If it's 45% for tails, 100 minus 45 is 55 for heads. For the fair coin, I have a 50% chance of tails and a 50% chance of heads. Fair enough. Now, I want to know, in either of these scenarios, what is the percent probability of getting four heads?"}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "If it's 45% for tails, 100 minus 45 is 55 for heads. For the fair coin, I have a 50% chance of tails and a 50% chance of heads. Fair enough. Now, I want to know, in either of these scenarios, what is the percent probability of getting four heads? So if given I've got the unfair coin, the probability of getting four heads is going to be 55% for each of those flips. So the probability of getting exactly four heads is going to be 0.55 times 0.55 times 0.55 times 0.55. And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3 4ths times all of this business over here."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, I want to know, in either of these scenarios, what is the percent probability of getting four heads? So if given I've got the unfair coin, the probability of getting four heads is going to be 55% for each of those flips. So the probability of getting exactly four heads is going to be 0.55 times 0.55 times 0.55 times 0.55. And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3 4ths times all of this business over here. So that's 3 4ths times, and this is 0.55 times itself four times. So I could write it as 0.55 to the fourth power. And we'll get the calculator out in a second to actually calculate what this is."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3 4ths times all of this business over here. So that's 3 4ths times, and this is 0.55 times itself four times. So I could write it as 0.55 to the fourth power. And we'll get the calculator out in a second to actually calculate what this is. Now, let's do the same thing for the fair coin. If I did pick a fair coin, the probability of getting heads four times in a row is going to be 0.5 times 0.5 times 0.5 times 0.5. Or the probability of getting the fair coin, which is 1 4th chance, times the probability and getting four heads in a row is going to be 1 4th times all of this."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we'll get the calculator out in a second to actually calculate what this is. Now, let's do the same thing for the fair coin. If I did pick a fair coin, the probability of getting heads four times in a row is going to be 0.5 times 0.5 times 0.5 times 0.5. Or the probability of getting the fair coin, which is 1 4th chance, times the probability and getting four heads in a row is going to be 1 4th times all of this. So it's going to be 1 4th times, this is just 0.5 times itself, four times. So that's 0.5 to the fourth power. So let's get the calculator out to calculate either one of these."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or the probability of getting the fair coin, which is 1 4th chance, times the probability and getting four heads in a row is going to be 1 4th times all of this. So it's going to be 1 4th times, this is just 0.5 times itself, four times. So that's 0.5 to the fourth power. So let's get the calculator out to calculate either one of these. So we get 3 divided by 4 times, and it knows that when I do the multiplication, it's not in the denominator here. So it's 3 4ths times, and I'll just do it in parentheses, which I don't have to do in parentheses because it knows order of operation. So 0.55 to the fourth power is equal to 0."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's get the calculator out to calculate either one of these. So we get 3 divided by 4 times, and it knows that when I do the multiplication, it's not in the denominator here. So it's 3 4ths times, and I'll just do it in parentheses, which I don't have to do in parentheses because it knows order of operation. So 0.55 to the fourth power is equal to 0. So let me write it down. Let me take it off the screen so I can write it down properly. Actually, let me just do both of these calculations."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 0.55 to the fourth power is equal to 0. So let me write it down. Let me take it off the screen so I can write it down properly. Actually, let me just do both of these calculations. So this probability is that one right over there. And then this one down here is 1 divided by 4 times 0.5 to the fourth power. So it's equal to that right over there."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me just do both of these calculations. So this probability is that one right over there. And then this one down here is 1 divided by 4 times 0.5 to the fourth power. So it's equal to that right over there. So let's be clear. The probability of picking the unfair coin and then getting four heads in a row is this top number. It's like roughly 6.9% chance that you get the unfair coin and then get four heads in a row."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's equal to that right over there. So let's be clear. The probability of picking the unfair coin and then getting four heads in a row is this top number. It's like roughly 6.9% chance that you get the unfair coin and then get four heads in a row. The probability that you get the fair coin and then get four heads in a row is even lower. It's only a 1.6% chance. Now, the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's like roughly 6.9% chance that you get the unfair coin and then get four heads in a row. The probability that you get the fair coin and then get four heads in a row is even lower. It's only a 1.6% chance. Now, the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be. Let me keep my calculator out. So it's going to be equal to. I can just take the previous answer."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be. Let me keep my calculator out. So it's going to be equal to. I can just take the previous answer. Let me just retype it so I don't confuse you. So 0.015625 plus 0.0686296875. I'm going to round it anyway, so it won't matter too much."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "I can just take the previous answer. Let me just retype it so I don't confuse you. So 0.015625 plus 0.0686296875. I'm going to round it anyway, so it won't matter too much. So if I take the sum. Let me take this off screen so I can still see it, and then let me write it. So what I got here, this one is 0.068629, and I'll round it 7."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm going to round it anyway, so it won't matter too much. So if I take the sum. Let me take this off screen so I can still see it, and then let me write it. So what I got here, this one is 0.068629, and I'll round it 7. And this down here was 0.015625. And when you add these two up, because we just care about getting four heads either way, there's a probability of getting it this way with the unfair coin. This is the probability of getting it with the fair coin."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what I got here, this one is 0.068629, and I'll round it 7. And this down here was 0.015625. And when you add these two up, because we just care about getting four heads either way, there's a probability of getting it this way with the unfair coin. This is the probability of getting it with the fair coin. We want it either way. So let's add the two, which we already did in our calculator. So if you add that number to that number, you get 0.08425, and it keeps going."}, {"video_title": "Dependent probability example 2   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the probability of getting it with the fair coin. We want it either way. So let's add the two, which we already did in our calculator. So if you add that number to that number, you get 0.08425, and it keeps going. But I'm just going to round it. So this is the same thing as this is equal to 8.425% if I want to round it a little bit more, 8.43% chance of getting four heads in a row. And once again, that's a slightly higher number than if all of the coins were fair, because there's a 3 4th chance that I get a coin that has a better than even chance of getting heads."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "The company gathered the following data about consumers' preference of soda. So they have year by year, percentage of respondents who preferred Yummy Cola, percentage of respondents who preferred Thrill Cola, and then these are people who had no preference. So in 2006, 80% liked Yummy, only 12% liked Thrill, and 8% didn't like either one or didn't have any preference. And so actually just from here you see that many, many more people liked Yummy Cola than Thrill Cola, actually every year over here. So Thrill Cola definitely has something, they have an uphill battle. But then they said the advertising company created the following two graphs to promote Thrill Soda. And so let's see what's happening over here."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And so actually just from here you see that many, many more people liked Yummy Cola than Thrill Cola, actually every year over here. So Thrill Cola definitely has something, they have an uphill battle. But then they said the advertising company created the following two graphs to promote Thrill Soda. And so let's see what's happening over here. So let's think about whether this is misleading or not. So if we look at this graph over here, in 2006, sure enough, 80% liked Yummy Cola, then 2007, 76%, then it keeps going, then 77%, then 73%, then 73, then 68. So this is actually accurate data."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "And so let's see what's happening over here. So let's think about whether this is misleading or not. So if we look at this graph over here, in 2006, sure enough, 80% liked Yummy Cola, then 2007, 76%, then it keeps going, then 77%, then 73%, then 73, then 68. So this is actually accurate data. It actually represents the data that's given right here. I'll do it in the same, it actually represents this data very faithfully. Then right over here, if we look at this chart, percentage of people who prefer Thrill Soda."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So this is actually accurate data. It actually represents the data that's given right here. I'll do it in the same, it actually represents this data very faithfully. Then right over here, if we look at this chart, percentage of people who prefer Thrill Soda. So over here in 2006, 12% preferred Thrill Soda, 2006, 12%, 2007, 19%, 2008, 19%, then we go up to 20, 21, and 25. So the graphs are actually accurate. They're not lying."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "Then right over here, if we look at this chart, percentage of people who prefer Thrill Soda. So over here in 2006, 12% preferred Thrill Soda, 2006, 12%, 2007, 19%, 2008, 19%, then we go up to 20, 21, and 25. So the graphs are actually accurate. They're not lying. These are actually the data points of the percentage who prefer Thrill Soda. Now what's misleading is if someone were to just look at these two graphs without actually looking at the scales over here, they'll see two things. They'll say, oh look, you see a declining trend, and that's what line graphs are good for, for seeing trends."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "They're not lying. These are actually the data points of the percentage who prefer Thrill Soda. Now what's misleading is if someone were to just look at these two graphs without actually looking at the scales over here, they'll see two things. They'll say, oh look, you see a declining trend, and that's what line graphs are good for, for seeing trends. They say, look, I see a declining trend in the number of, in the percentage of people who prefer Yummy Cola, and I see this increasing trend in the percentage of people who prefer Thrill Cola. And that's true. You have a declining trend here, and you have an increasing trend here."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "They'll say, oh look, you see a declining trend, and that's what line graphs are good for, for seeing trends. They say, look, I see a declining trend in the number of, in the percentage of people who prefer Yummy Cola, and I see this increasing trend in the percentage of people who prefer Thrill Cola. And that's true. You have a declining trend here, and you have an increasing trend here. But what's misleading here is the way that they've plotted the scales. These scales are not the same. So when you look at this, you say not only is there an increasing trend of people who prefer Thrill Soda, but the way they set up the scale, it looks like the trend is above."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "You have a declining trend here, and you have an increasing trend here. But what's misleading here is the way that they've plotted the scales. These scales are not the same. So when you look at this, you say not only is there an increasing trend of people who prefer Thrill Soda, but the way they set up the scale, it looks like the trend is above. It looks like if you look, the human brain is tempted to compare these, and they say, look, not only is this an upward trend, but it's above this trend right over here. Even in 2006, this data point looks higher than these data points right over here. But the reality is that it's only because the scale is distorted."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So when you look at this, you say not only is there an increasing trend of people who prefer Thrill Soda, but the way they set up the scale, it looks like the trend is above. It looks like if you look, the human brain is tempted to compare these, and they say, look, not only is this an upward trend, but it's above this trend right over here. Even in 2006, this data point looks higher than these data points right over here. But the reality is that it's only because the scale is distorted. And this is the oldest trick in the book when plotting line graphs. It all depends on the scale. So this just looks good because they used this scale that went from 0 to 30 as opposed to 0, 100."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "But the reality is that it's only because the scale is distorted. And this is the oldest trick in the book when plotting line graphs. It all depends on the scale. So this just looks good because they used this scale that went from 0 to 30 as opposed to 0, 100. The better thing to do, or the more genuine thing to do, or the more honest thing to do would have actually been to plot them on the same graph. Although if they did that, that wouldn't have painted a very good picture for Thrill Soda. So if we plotted it on the same graph, Thrill Soda, let's try that out."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So this just looks good because they used this scale that went from 0 to 30 as opposed to 0, 100. The better thing to do, or the more genuine thing to do, or the more honest thing to do would have actually been to plot them on the same graph. Although if they did that, that wouldn't have painted a very good picture for Thrill Soda. So if we plotted it on the same graph, Thrill Soda, let's try that out. So in 2006, 12, and actually this is even worse. You actually wouldn't even be able to plot Thrill Soda on this graph because they started this graph right over here at 50%. They didn't even start it at 0%."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So if we plotted it on the same graph, Thrill Soda, let's try that out. So in 2006, 12, and actually this is even worse. You actually wouldn't even be able to plot Thrill Soda on this graph because they started this graph right over here at 50%. They didn't even start it at 0%. So you actually would not even be able to plot Thrill Soda on this graph. If you did, you would have to extend this graph all the way down. So you would have to extend this graph all the way down to, you know, so this would have to be 40, this would be 30, this would be 20%, this would be 10%, and then down all the way over here would be 0%."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "They didn't even start it at 0%. So you actually would not even be able to plot Thrill Soda on this graph. If you did, you would have to extend this graph all the way down. So you would have to extend this graph all the way down to, you know, so this would have to be 40, this would be 30, this would be 20%, this would be 10%, and then down all the way over here would be 0%. And then the Thrill Soda graph would be all the way down here. So it was like 12% and it goes all the way up to like 25%. So the Thrill Soda, so it would have looked something like this."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So you would have to extend this graph all the way down to, you know, so this would have to be 40, this would be 30, this would be 20%, this would be 10%, and then down all the way over here would be 0%. And then the Thrill Soda graph would be all the way down here. So it was like 12% and it goes all the way up to like 25%. So the Thrill Soda, so it would have looked something like this. The graph would have looked something like this, which is nowhere near, if you plotted these on the same scale, on the same graph, then it would have still been pretty obvious that a lot more people, even though the trend is downward, a lot more people prefer Yummy Cola to Thrill Cola. So there's two very disingenuous things going on over here. One is the actual scale."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "So the Thrill Soda, so it would have looked something like this. The graph would have looked something like this, which is nowhere near, if you plotted these on the same scale, on the same graph, then it would have still been pretty obvious that a lot more people, even though the trend is downward, a lot more people prefer Yummy Cola to Thrill Cola. So there's two very disingenuous things going on over here. One is the actual scale. For this amount of distance on this scale, they represent 10%. So whatever the gain is, it looks like it's a huge gain. But over here, that same amount, they're actually representing a larger amount."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "One is the actual scale. For this amount of distance on this scale, they represent 10%. So whatever the gain is, it looks like it's a huge gain. But over here, that same amount, they're actually representing a larger amount. They're representing closer to 15% or 16%. And then the main thing is they started the scale here at 50%. So they're not showing how many people really prefer, how large 80% or even 70% really is."}, {"video_title": "Misleading line graphs   Applying mathematical reasoning   Pre-Algebra   Khan Academy.mp3", "Sentence": "But over here, that same amount, they're actually representing a larger amount. They're representing closer to 15% or 16%. And then the main thing is they started the scale here at 50%. So they're not showing how many people really prefer, how large 80% or even 70% really is. And over here they start at 0% and they just have a larger scale. So it makes it look like out the gate a lot of people prefer, or a comparable amount of people prefer Thrill, and that the trend is up. But the reality is still way more people prefer Yummy Cola."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "In this video, I want to give you an example of what it means to fit data to a line. Instead of doing my traditional video using my little pen tablet, I'm going to do it straight on Excel so you could see how to do this for yourself if you have Excel or some other type of spreadsheet program. And we're not going to go into the math of it. I really just want you to get the conceptual understanding of what it means to fit data with line or to do a linear regression. So here, let's just read the problem. The following table shows the median California income. Remember, median is the middle California income from 1995 to 2002, as reported by the US Census Bureau."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I really just want you to get the conceptual understanding of what it means to fit data with line or to do a linear regression. So here, let's just read the problem. The following table shows the median California income. Remember, median is the middle California income from 1995 to 2002, as reported by the US Census Bureau. Draw a scatter plot and find the equation. What would you expect the median annual income of a California family to be in the year 2010? What are the meanings of the slope and the y-intercept of this problem?"}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Remember, median is the middle California income from 1995 to 2002, as reported by the US Census Bureau. Draw a scatter plot and find the equation. What would you expect the median annual income of a California family to be in the year 2010? What are the meanings of the slope and the y-intercept of this problem? So the first thing you'd want to do, I just copied and pasted this image. We have to get the data in a form that the spreadsheet can understand it. So let's make some tables here."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "What are the meanings of the slope and the y-intercept of this problem? So the first thing you'd want to do, I just copied and pasted this image. We have to get the data in a form that the spreadsheet can understand it. So let's make some tables here. Let's say years since 1995. Let's make that one column. Let me make this a little bit wider."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's make some tables here. Let's say years since 1995. Let's make that one column. Let me make this a little bit wider. And then let me put median income. This is the median income in California for a family. So we start off with one year or zero years since 1995."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me make this a little bit wider. And then let me put median income. This is the median income in California for a family. So we start off with one year or zero years since 1995. Zero, one, two, three, four. And actually, if you want, it'll figure out the trend. If you just keep going down, it'll figure out that you're just incrementing by one."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we start off with one year or zero years since 1995. Zero, one, two, three, four. And actually, if you want, it'll figure out the trend. If you just keep going down, it'll figure out that you're just incrementing by one. And then the income, I'll just copy in these numbers right there. So that's $53,807, $55,217, $55,209, $55,415, $63,100, $63,206, $63,761, and then we have $65,766. So I don't need these over here."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you just keep going down, it'll figure out that you're just incrementing by one. And then the income, I'll just copy in these numbers right there. So that's $53,807, $55,217, $55,209, $55,415, $63,100, $63,206, $63,761, and then we have $65,766. So I don't need these over here. So I'm going to get rid of them. I can clear them. So let me make sure I have enough entries."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I don't need these over here. So I'm going to get rid of them. I can clear them. So let me make sure I have enough entries. This is one, two, three, four, five, six, seven, eight. And I have one, two, three, four, five, six, seven, eight entries, and I want to make sure I got my data right. $53,807, $55,217, $55,209, $45,415, $100, $206, $761, $766."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let me make sure I have enough entries. This is one, two, three, four, five, six, seven, eight. And I have one, two, three, four, five, six, seven, eight entries, and I want to make sure I got my data right. $53,807, $55,217, $55,209, $45,415, $100, $206, $761, $766. OK, there we go. Now, you're going to find that in Excel this is incredibly easy, if you know what to click on, to one, plot this data, create a scatter plot, and then even better, create a regression of that data. So all you have to do is you select the data, and then you go to Insert, and I'm going to insert a scatter plot."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "$53,807, $55,217, $55,209, $45,415, $100, $206, $761, $766. OK, there we go. Now, you're going to find that in Excel this is incredibly easy, if you know what to click on, to one, plot this data, create a scatter plot, and then even better, create a regression of that data. So all you have to do is you select the data, and then you go to Insert, and I'm going to insert a scatter plot. And then you can pick the different types of scatter plots. I just want to plot the data. And there you go."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So all you have to do is you select the data, and then you go to Insert, and I'm going to insert a scatter plot. And then you can pick the different types of scatter plots. I just want to plot the data. And there you go. It plotted the data for me. So there you go. If you go by, this is the actual income, and this is by year since 1995."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And there you go. It plotted the data for me. So there you go. If you go by, this is the actual income, and this is by year since 1995. So this is 1995. It was $53,807. In 1996, it's $55,217."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you go by, this is the actual income, and this is by year since 1995. So this is 1995. It was $53,807. In 1996, it's $55,217. So it plotted all the data. Now what I want to do is fit a line. So this isn't exactly a line, but let's see, if we assume that a line can model this data well, I'm going to get Excel to fit a line for me."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "In 1996, it's $55,217. So it plotted all the data. Now what I want to do is fit a line. So this isn't exactly a line, but let's see, if we assume that a line can model this data well, I'm going to get Excel to fit a line for me. So what I can do is I have all of these options up here for different ways to fit a line, all of these different options, and I'm going to pick this one here. You might not be able to see it. It looks like it has a line between dots, and it also has fx, which tells me it's going to tell me the equation of the line."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this isn't exactly a line, but let's see, if we assume that a line can model this data well, I'm going to get Excel to fit a line for me. So what I can do is I have all of these options up here for different ways to fit a line, all of these different options, and I'm going to pick this one here. You might not be able to see it. It looks like it has a line between dots, and it also has fx, which tells me it's going to tell me the equation of the line. So if I click on that, there you go. It not only fit, it re-plotted that same data on a different graph. Let me make it a little bit bigger."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "It looks like it has a line between dots, and it also has fx, which tells me it's going to tell me the equation of the line. So if I click on that, there you go. It not only fit, it re-plotted that same data on a different graph. Let me make it a little bit bigger. We can cover up the data now just because I think we know what's going on. So let me cover it up right like that. So not only did it plot the various data points, it actually fit a line to that data, and it gave me the equation of that line."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me make it a little bit bigger. We can cover up the data now just because I think we know what's going on. So let me cover it up right like that. So not only did it plot the various data points, it actually fit a line to that data, and it gave me the equation of that line. It says the equation of this line is y. Let me see if I can make this a little bit bigger. I'll move it out of the way so you can read it at least."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So not only did it plot the various data points, it actually fit a line to that data, and it gave me the equation of that line. It says the equation of this line is y. Let me see if I can make this a little bit bigger. I'll move it out of the way so you can read it at least. So it tells me right here that the equation for this line is y is equal to 1,882.3x plus 52,847. So if you remember what we know about slope and y-intercept, the y-intercept is 52,847, which is if you use this line as your measure, where this line intersects at year 0, or in 1995. So if you use this line as a model, in 1995 the line would say that you're going to make 52,847."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'll move it out of the way so you can read it at least. So it tells me right here that the equation for this line is y is equal to 1,882.3x plus 52,847. So if you remember what we know about slope and y-intercept, the y-intercept is 52,847, which is if you use this line as your measure, where this line intersects at year 0, or in 1995. So if you use this line as a model, in 1995 the line would say that you're going to make 52,847. The actual data was a little bit off of that. It was a little bit higher, 53,807. So it was a little bit higher, but we're trying to get a line that gets as close as possible to all of this data."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So if you use this line as a model, in 1995 the line would say that you're going to make 52,847. The actual data was a little bit off of that. It was a little bit higher, 53,807. So it was a little bit higher, but we're trying to get a line that gets as close as possible to all of this data. It's actually trying to minimize the distance, the square of the distance, between each of these points in the line. And we won't go into the math there. But it gave us this nice equation."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it was a little bit higher, but we're trying to get a line that gets as close as possible to all of this data. It's actually trying to minimize the distance, the square of the distance, between each of these points in the line. And we won't go into the math there. But it gave us this nice equation. Now we can use this nice equation to predict things. If we say that this is a good model for the data, let me bring this down a little bit, let's try to answer our question. So we drew a scatter plot, really Excel did it for us."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "But it gave us this nice equation. Now we can use this nice equation to predict things. If we say that this is a good model for the data, let me bring this down a little bit, let's try to answer our question. So we drew a scatter plot, really Excel did it for us. We found the equation right there. They say, what would you expect the median annual income of a California family to be in the year 2010? So here we can just use the equation they gave us."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we drew a scatter plot, really Excel did it for us. We found the equation right there. They say, what would you expect the median annual income of a California family to be in the year 2010? So here we can just use the equation they gave us. This right here was 2002. So I could write down the year. This was the year of 2002."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So here we can just use the equation they gave us. This right here was 2002. So I could write down the year. This was the year of 2002. So the year 2010 is 8 more years. And let me make a little column here. So this is the year, 1995, 1996."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "This was the year of 2002. So the year 2010 is 8 more years. And let me make a little column here. So this is the year, 1995, 1996. And then Excel will be able to figure out, if I select those and I go to this little bottom right square, and I scroll down, Excel will actually figure out that I want to increment by 1 year every time. And if I say years since 1995, once again I can just continue this trend right here. So 2010 would be 15 years."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is the year, 1995, 1996. And then Excel will be able to figure out, if I select those and I go to this little bottom right square, and I scroll down, Excel will actually figure out that I want to increment by 1 year every time. And if I say years since 1995, once again I can just continue this trend right here. So 2010 would be 15 years. And so we can just apply this equation. We could say it's going to be equal to, according to this line. I'm just going to type it in."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 2010 would be 15 years. And so we can just apply this equation. We could say it's going to be equal to, according to this line. I'm just going to type it in. Hopefully you can read what I'm saying. 1,882.3 times x. x here is the year since 1995. So times, I could just select this cell."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm just going to type it in. Hopefully you can read what I'm saying. 1,882.3 times x. x here is the year since 1995. So times, I could just select this cell. Or I could type in the number 15. That means times this cell, times 15. And then plus 52,847."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "So times, I could just select this cell. Or I could type in the number 15. That means times this cell, times 15. And then plus 52,847. Plus that right there. Click Enter. And it predicts $81,081.50."}, {"video_title": "Fitting a line to data   Regression   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then plus 52,847. Plus that right there. Click Enter. And it predicts $81,081.50. So if you just continue this line for another 8 or so years, it predicts that the median income in California for a family will be $81,000. Anyway, hopefully you found that interesting. This is, spreadsheets are very useful tools for manipulating data."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So I have some data here in a spreadsheet. You could use Microsoft Excel or you could use Google Spreadsheets. And we're gonna use the spreadsheet to quickly calculate some parameters. Let's say this is the population. Let's say this is, we're looking at a population of students and we wanna calculate some parameters, and this is their ages, and we wanna calculate some parameters on that. And so, first I'm gonna calculate it using the spreadsheet. And then we're gonna think about how those parameters change as we do things to the data."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say this is the population. Let's say this is, we're looking at a population of students and we wanna calculate some parameters, and this is their ages, and we wanna calculate some parameters on that. And so, first I'm gonna calculate it using the spreadsheet. And then we're gonna think about how those parameters change as we do things to the data. If we were to shift the data up or down, or if we were to multiply all the points by some value, what does that do to the actual parameters? So the first parameter I'm gonna calculate is the mean. Then I'm gonna calculate the standard deviation."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "And then we're gonna think about how those parameters change as we do things to the data. If we were to shift the data up or down, or if we were to multiply all the points by some value, what does that do to the actual parameters? So the first parameter I'm gonna calculate is the mean. Then I'm gonna calculate the standard deviation. Then I wanna calculate the median. And then I wanna calculate, let's say the interquartile range. Inter, I'll call it IQR."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Then I'm gonna calculate the standard deviation. Then I wanna calculate the median. And then I wanna calculate, let's say the interquartile range. Inter, I'll call it IQR. So let's do this. Let's first look at the measures of central tendency. So the mean, the function on most spreadsheets is the average function."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Inter, I'll call it IQR. So let's do this. Let's first look at the measures of central tendency. So the mean, the function on most spreadsheets is the average function. And then I could use my mouse and select all of these, or I could press shift with my arrow button and select all of those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So the mean, the function on most spreadsheets is the average function. And then I could use my mouse and select all of these, or I could press shift with my arrow button and select all of those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice, for every data point I had before, I now have five more than that."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice, for every data point I had before, I now have five more than that. So this is my new data set, which I'm calling data plus five. And let's see what the mean of that is. So the mean of that, notice, is exactly five more."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "And notice, for every data point I had before, I now have five more than that. So this is my new data set, which I'm calling data plus five. And let's see what the mean of that is. So the mean of that, notice, is exactly five more. And the same would have been true if I added or subtracted any number. The mean would change by the amount that I add or subtract. And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So the mean of that, notice, is exactly five more. And the same would have been true if I added or subtracted any number. The mean would change by the amount that I add or subtract. And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. And so, if all the numbers are five more, you're gonna add five, in this case, how many numbers are there? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. And so, if all the numbers are five more, you're gonna add five, in this case, how many numbers are there? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. Let's think about how the mean changes if you multiply. So if you take your data and if I were to multiply it times five, what happens? So this equals this times five."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. Let's think about how the mean changes if you multiply. So if you take your data and if I were to multiply it times five, what happens? So this equals this times five. So now, all the data points are five times more. Now what happens to my mean? Notice, my mean is now five times as much."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So this equals this times five. So now, all the data points are five times more. Now what happens to my mean? Notice, my mean is now five times as much. So the measures of central tendency, if I add or subtract, well, I'm gonna add or subtract the mean by that amount. And if I scale it up by five or if I scaled it down by five, well, my mean would scale up or down by that same amount. And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Notice, my mean is now five times as much. So the measures of central tendency, if I add or subtract, well, I'm gonna add or subtract the mean by that amount. And if I scale it up by five or if I scaled it down by five, well, my mean would scale up or down by that same amount. And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. Now let's look at the other measure, the other typical measure of central tendency, and that is the median, to see if that has the same properties. So let's calculate the median here. So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. Now let's look at the other measure, the other typical measure of central tendency, and that is the median, to see if that has the same properties. So let's calculate the median here. So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. So that's the median for that data set. What do you think the median's gonna be if you take all of the data plus five? Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. So that's the median for that data set. What do you think the median's gonna be if you take all of the data plus five? Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. So this should be 10.5. And yes, it is indeed 10.5. And what would happen if you multiply everything by five?"}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. So this should be 10.5. And yes, it is indeed 10.5. And what would happen if you multiply everything by five? Well, once again, you still have the same ordering, and so it should just multiply that by five. Yep, the middle number is now gonna be five times larger. So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "And what would happen if you multiply everything by five? Well, once again, you still have the same ordering, and so it should just multiply that by five. Yep, the middle number is now gonna be five times larger. So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. Now let's think about these measures of spread. See if that's the same with these measures of spread. So standard deviation."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. Now let's think about these measures of spread. See if that's the same with these measures of spread. So standard deviation. So S-T-D-E-V, I'm gonna take the population standard deviation. I'm assuming that this is my entire population. So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So standard deviation. So S-T-D-E-V, I'm gonna take the population standard deviation. I'm assuming that this is my entire population. So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when I shift everything by five. Actually, pause the video. What do you think is going to happen?"}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when I shift everything by five. Actually, pause the video. What do you think is going to happen? This is a measure of spread. So if you shift, I'll tell you what I think. If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "What do you think is going to happen? This is a measure of spread. So if you shift, I'll tell you what I think. If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. So the standard deviation should not change, I don't think, in this example. And indeed, it does not change. So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. So the standard deviation should not change, I don't think, in this example. And indeed, it does not change. So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. But if we scale it, well, I think it should change, because you can imagine a very simple data set, the things that were a certain amount of distance from the mean are now going to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiply this by five."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. But if we scale it, well, I think it should change, because you can imagine a very simple data set, the things that were a certain amount of distance from the mean are now going to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiply this by five. So scaling the data set will scale the standard deviation in a similar way. What about interquartile range? Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "If I multiply this by five. So scaling the data set will scale the standard deviation in a similar way. What about interquartile range? Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. And so let's do that. We can have the quartile function equals quartile, and then we wanna look at our data, and we want the third quartile. So that's gonna calculate the third quartile minus quartile, same data set."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. And so let's do that. We can have the quartile function equals quartile, and then we wanna look at our data, and we want the third quartile. So that's gonna calculate the third quartile minus quartile, same data set. So now we wanna select it again. So same data set, but this is now gonna be the first quartile. So this is gonna give us our interquartile range."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So that's gonna calculate the third quartile minus quartile, same data set. So now we wanna select it again. So same data set, but this is now gonna be the first quartile. So this is gonna give us our interquartile range. This is the, calculates the third quartile on that data set, and this calculates the first quartile on that data set. And we get 2.75. Now let's think about what the inter, whether the interquartile range should change."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So this is gonna give us our interquartile range. This is the, calculates the third quartile on that data set, and this calculates the first quartile on that data set. And we get 2.75. Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there."}, {"video_title": "How parameters change as data is shifted and scaled   AP Statistics   Khan Academy.mp3", "Sentence": "So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there. So the big takeaway here, and I just use the example of shifting up by five and scaling up by five, but you could subtract by any number, and you could divide by a number as well. The typical measures of central tendency, mean and median, they both shift and scale as you shift and scale the data. But your typical measures of spread, standard deviation and interquartile range, they don't change if you shift the data, but they do change and they scale as you scale the data."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, the first thing I want to do in this video is calculate the total sum of squares. So I'll call that SST. S-S, sum of squares, total. And you could view it as really the numerator when you calculate variance. So you're just going to take the distance between each of these data points and the mean of all of these data points squared. I mean, just take that sum. We're not going to divide by the degree of freedom, which you'd normally do if you were calculating sample variance."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And you could view it as really the numerator when you calculate variance. So you're just going to take the distance between each of these data points and the mean of all of these data points squared. I mean, just take that sum. We're not going to divide by the degree of freedom, which you'd normally do if you were calculating sample variance. Now, what is this going to be? Well, the first thing we need to do, we have to figure out the mean of all of this stuff over here. And I'm actually going to call that the grand mean."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're not going to divide by the degree of freedom, which you'd normally do if you were calculating sample variance. Now, what is this going to be? Well, the first thing we need to do, we have to figure out the mean of all of this stuff over here. And I'm actually going to call that the grand mean. I'm going to call that the grand mean. And I'm going to show you in a second that it's the same thing as the mean of the means of each of these data sets. So let's calculate the grand mean."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I'm actually going to call that the grand mean. I'm going to call that the grand mean. And I'm going to show you in a second that it's the same thing as the mean of the means of each of these data sets. So let's calculate the grand mean. So it's going to be 3 plus 2 plus 1, 3 plus 2 plus 1, plus 5 plus 3 plus 4, plus 5 plus 3 plus 4, plus 5 plus 6 plus 7. Plus 5 plus 6 plus 7. And then we have 9 data points here."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's calculate the grand mean. So it's going to be 3 plus 2 plus 1, 3 plus 2 plus 1, plus 5 plus 3 plus 4, plus 5 plus 3 plus 4, plus 5 plus 6 plus 7. Plus 5 plus 6 plus 7. And then we have 9 data points here. We have 9 data points, so we'll divide by 9. And what is this going to be equal to? 3 plus 2 plus 1 is 6."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then we have 9 data points here. We have 9 data points, so we'll divide by 9. And what is this going to be equal to? 3 plus 2 plus 1 is 6. 6 plus, let me just add, so these are 6. 5 plus 3 plus 4 is, that's 12. And then 5 plus 6 plus 7 is 18."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "3 plus 2 plus 1 is 6. 6 plus, let me just add, so these are 6. 5 plus 3 plus 4 is, that's 12. And then 5 plus 6 plus 7 is 18. And then 6 plus 12 is 18, plus another 18 is 36. Divided by 9 is equal to 4. Let me show you that that's the exact same thing as the mean of the means."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then 5 plus 6 plus 7 is 18. And then 6 plus 12 is 18, plus another 18 is 36. Divided by 9 is equal to 4. Let me show you that that's the exact same thing as the mean of the means. So the mean of this group 1 over here, let me do it in that same green. The mean of group 1 over here is 3 plus 2 plus 1, that's that 6 right over here, divided by 3 data points. So that will be equal to 2."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me show you that that's the exact same thing as the mean of the means. So the mean of this group 1 over here, let me do it in that same green. The mean of group 1 over here is 3 plus 2 plus 1, that's that 6 right over here, divided by 3 data points. So that will be equal to 2. The mean of group 2, the sum here is 12, we saw that right over here. 5 plus 3 plus 4 is 12, divided by 3 is 4, because we have 3 data points. And then the mean of group 3, 5 plus 6 plus 7 is 18, divided by 3 is 6."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that will be equal to 2. The mean of group 2, the sum here is 12, we saw that right over here. 5 plus 3 plus 4 is 12, divided by 3 is 4, because we have 3 data points. And then the mean of group 3, 5 plus 6 plus 7 is 18, divided by 3 is 6. So if you were to take the mean of the means, which is another way of viewing this grand mean, you have 2 plus 4 plus 6, which is 12, divided by 3 means here, and once again you would get 4. So you could view this as the mean of all of the data in all of the groups, or the mean of the means of each of these groups. But either way, now that we've calculated it, we can actually figure out the total sum of squares."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then the mean of group 3, 5 plus 6 plus 7 is 18, divided by 3 is 6. So if you were to take the mean of the means, which is another way of viewing this grand mean, you have 2 plus 4 plus 6, which is 12, divided by 3 means here, and once again you would get 4. So you could view this as the mean of all of the data in all of the groups, or the mean of the means of each of these groups. But either way, now that we've calculated it, we can actually figure out the total sum of squares. So let's do that. So it's going to be equal to 3 minus 4, the 4 is this 4 right over here, squared, plus 2 minus 4 squared, plus 1 minus 4 squared, now I'll do these guys over here in purple, plus 5 minus 4 squared, plus 3 minus 4 squared, plus 4 minus 4 squared, let me scroll over a little bit, plus 4 minus 4 squared, now we only have 3 left, plus 5 minus 4 squared, plus 6 minus 4 squared, plus 7 minus 4 squared. And what does this give us?"}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "But either way, now that we've calculated it, we can actually figure out the total sum of squares. So let's do that. So it's going to be equal to 3 minus 4, the 4 is this 4 right over here, squared, plus 2 minus 4 squared, plus 1 minus 4 squared, now I'll do these guys over here in purple, plus 5 minus 4 squared, plus 3 minus 4 squared, plus 4 minus 4 squared, let me scroll over a little bit, plus 4 minus 4 squared, now we only have 3 left, plus 5 minus 4 squared, plus 6 minus 4 squared, plus 7 minus 4 squared. And what does this give us? Well up here, this first, so this is going to be equal to 3 minus 4, difference is 1, you square it, it's actually negative 1, but you square it, you get 1, plus, you get negative 2 squared is 4, plus negative 3 squared, negative 3 squared is 9. And then we have here in the magenta, 5 minus 4 is 1, squared is still 1, 3 minus 4 squared is 1, you square it again, you still get 1, 5 minus 4 is just a 0, well I'll just write the 0 there just to show you that we actually calculated that, and then we have these last 3 data points, 5 minus 4 squared, that's 1, 6 minus 4 squared, that is 4, that's 2 squared, and then plus 7 minus 4 is 3, squared is 9. So what's this going to be equal to?"}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what does this give us? Well up here, this first, so this is going to be equal to 3 minus 4, difference is 1, you square it, it's actually negative 1, but you square it, you get 1, plus, you get negative 2 squared is 4, plus negative 3 squared, negative 3 squared is 9. And then we have here in the magenta, 5 minus 4 is 1, squared is still 1, 3 minus 4 squared is 1, you square it again, you still get 1, 5 minus 4 is just a 0, well I'll just write the 0 there just to show you that we actually calculated that, and then we have these last 3 data points, 5 minus 4 squared, that's 1, 6 minus 4 squared, that is 4, that's 2 squared, and then plus 7 minus 4 is 3, squared is 9. So what's this going to be equal to? So I have 1 plus 4 plus 9 right over here, that's 5 plus 9, this right over here is 14, and then we also have another 14 right over here, because we have a 1 plus 4 plus 9, so that right over there is also 14, and then we have 2 over here, so it's going to be 28, 14 times 2, 14 plus 14 is 28, plus 2 is 30, is equal to 30. So our total sum of squares, and actually if we wanted the variance here, we would divide this by the degrees of freedom, and we've learned multiple times the degrees of freedom here, so let's say that we have, so we know that we have m groups over here, so let me just write it as m, and I'm not going to prove things rigorously here, but I want to show you where some of these strange formulas that show up in statistics books actually come from, without proving it rigorously, more to give you the intuition. So we have m groups here, and each group here has n members."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's this going to be equal to? So I have 1 plus 4 plus 9 right over here, that's 5 plus 9, this right over here is 14, and then we also have another 14 right over here, because we have a 1 plus 4 plus 9, so that right over there is also 14, and then we have 2 over here, so it's going to be 28, 14 times 2, 14 plus 14 is 28, plus 2 is 30, is equal to 30. So our total sum of squares, and actually if we wanted the variance here, we would divide this by the degrees of freedom, and we've learned multiple times the degrees of freedom here, so let's say that we have, so we know that we have m groups over here, so let me just write it as m, and I'm not going to prove things rigorously here, but I want to show you where some of these strange formulas that show up in statistics books actually come from, without proving it rigorously, more to give you the intuition. So we have m groups here, and each group here has n members. So how many total members do we have here? Well we had m times n, or 9, 3 times 3 total members. So our degrees of freedom, and remember, you have this many, however many data points you had, minus 1 degrees of freedom, because if you know, if you knew the mean of means, if you know the mean of means, if you assume you knew that, then you only would, then only 9 minus 1, only 8 of these are going to give you new information, because if you know that, you could calculate the last one, or it really doesn't have to be the last one, if you have the other 8, you could calculate this one, if you have 8 of them, you can always calculate the 9th one, using the mean of means."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we have m groups here, and each group here has n members. So how many total members do we have here? Well we had m times n, or 9, 3 times 3 total members. So our degrees of freedom, and remember, you have this many, however many data points you had, minus 1 degrees of freedom, because if you know, if you knew the mean of means, if you know the mean of means, if you assume you knew that, then you only would, then only 9 minus 1, only 8 of these are going to give you new information, because if you know that, you could calculate the last one, or it really doesn't have to be the last one, if you have the other 8, you could calculate this one, if you have 8 of them, you can always calculate the 9th one, using the mean of means. So one way to think about it is that there's only 8 independent measurements here, or if we want to talk in terms of general, if we want to talk generally, there are m times n, so that tells us the total number of samples, minus 1 degrees of freedom. So if we were actually calculating the variance here, we would just divide 30 by m times n minus 1, or this is another way of saying 8 degrees of freedom, for this exact example. We would take 30 divided by 8, and we would actually have the variance for this entire group, for the group of 9, when you combine them."}, {"video_title": "ANOVA 1 Calculating SST (total sum of squares)   Probability and Statistics   Khan Academy.mp3", "Sentence": "So our degrees of freedom, and remember, you have this many, however many data points you had, minus 1 degrees of freedom, because if you know, if you knew the mean of means, if you know the mean of means, if you assume you knew that, then you only would, then only 9 minus 1, only 8 of these are going to give you new information, because if you know that, you could calculate the last one, or it really doesn't have to be the last one, if you have the other 8, you could calculate this one, if you have 8 of them, you can always calculate the 9th one, using the mean of means. So one way to think about it is that there's only 8 independent measurements here, or if we want to talk in terms of general, if we want to talk generally, there are m times n, so that tells us the total number of samples, minus 1 degrees of freedom. So if we were actually calculating the variance here, we would just divide 30 by m times n minus 1, or this is another way of saying 8 degrees of freedom, for this exact example. We would take 30 divided by 8, and we would actually have the variance for this entire group, for the group of 9, when you combine them. I'll leave you here in this video, in the next video, we're going to try to figure out how much of this total variance, how much of this total sum of squared, the total squared sum, total variation, comes from the variation within each of these groups, versus the variation between the groups. And I think you get a sense of where this whole analysis of variance is coming from. It's the sense that, look, there's a variance of this entire sample of 9, but some of that variance, if these groups are different in some way, might come from the variation from being in different groups, versus the variation from being within a group."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So I've defined some sets here. And just to make things interesting, I haven't only put numbers in these sets. I've even put some colors and some little yellow stars here. And what I want you to figure out is what would this set be, this crazy thing that involves relative complements, intersections, unions, absolute complements. So I encourage you to pause it and try to figure out what this set would be. Well, let's give it a shot. And the key here is to really break it down, work on the parentheses, the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "And what I want you to figure out is what would this set be, this crazy thing that involves relative complements, intersections, unions, absolute complements. So I encourage you to pause it and try to figure out what this set would be. Well, let's give it a shot. And the key here is to really break it down, work on the parentheses, the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. And then it should hopefully make a little bit of sense. So a good place to start might be to try to figure out what is the relative complement of C in B. Or another way of thinking about it is what is B minus C?"}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the key here is to really break it down, work on the parentheses, the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. And then it should hopefully make a little bit of sense. So a good place to start might be to try to figure out what is the relative complement of C in B. Or another way of thinking about it is what is B minus C? What is B if you take out all the stuff with C in it? So let me write this down. The relative complement of C in B, or you could call this B minus C, this is all the stuff in B with all the stuff in C taken out of it."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or another way of thinking about it is what is B minus C? What is B if you take out all the stuff with C in it? So let me write this down. The relative complement of C in B, or you could call this B minus C, this is all the stuff in B with all the stuff in C taken out of it. So let's think about what this would be. B has a 0. Does C have a 0?"}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "The relative complement of C in B, or you could call this B minus C, this is all the stuff in B with all the stuff in C taken out of it. So let's think about what this would be. B has a 0. Does C have a 0? No, so we don't have to take out the 0. B has a 17. Does C have a 17?"}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "Does C have a 0? No, so we don't have to take out the 0. B has a 17. Does C have a 17? Yes, it does. So we take out the 17. B has a 3."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "Does C have a 17? Yes, it does. So we take out the 17. B has a 3. But C has a 3, so we take that out. B has a blue. C does not have a blue, so we leave the blue in."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "B has a 3. But C has a 3, so we take that out. B has a blue. C does not have a blue, so we leave the blue in. So let me write down, we leave the blue in. And then B has a gold star. C also has a gold star, so we take the gold star out."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "C does not have a blue, so we leave the blue in. So let me write down, we leave the blue in. And then B has a gold star. C also has a gold star, so we take the gold star out. So the relative complement of C in B is just the set of 0 and this blue written in blue. So let me write this down. Now it gets interesting."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "C also has a gold star, so we take the gold star out. So the relative complement of C in B is just the set of 0 and this blue written in blue. So let me write this down. Now it gets interesting. We're going to take the absolute complement of that. So let me write this down. So B, the absolute complement of this business is going to be all things, let me write this, the set of all things in our universe that are neither a 0 or a blue."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now it gets interesting. We're going to take the absolute complement of that. So let me write this down. So B, the absolute complement of this business is going to be all things, let me write this, the set of all things in our universe that are neither a 0 or a blue. That's the only way I can describe it right now. I haven't really defined the universe well. We already see that our universe definitely contains some integers."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So B, the absolute complement of this business is going to be all things, let me write this, the set of all things in our universe that are neither a 0 or a blue. That's the only way I can describe it right now. I haven't really defined the universe well. We already see that our universe definitely contains some integers. It contains colors. It contains some stars. So this is all I can really say."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "We already see that our universe definitely contains some integers. It contains colors. It contains some stars. So this is all I can really say. This is a set of all things in the universe that are neither a 0 or a blue. So fair enough. So far we've figured out all of this stuff."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is all I can really say. This is a set of all things in the universe that are neither a 0 or a blue. So fair enough. So far we've figured out all of this stuff. Let me box this off. So that is that right over there. And now we want to find the intersection."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So far we've figured out all of this stuff. Let me box this off. So that is that right over there. And now we want to find the intersection. We need to find the intersection of A and this business. Let me write that down. So it's going to be A intersected with the relative complement of C in B and the absolute complement of that."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "And now we want to find the intersection. We need to find the intersection of A and this business. Let me write that down. So it's going to be A intersected with the relative complement of C in B and the absolute complement of that. So this is going to be the intersection of the set A and the set of all things in the universe that are neither a 0 or a blue. So it's essentially the things that satisfy both of these, that it has to be in set A and it has to be in the set of all things in the universe that are neither a 0 or a blue. So let's think about what this is."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's going to be A intersected with the relative complement of C in B and the absolute complement of that. So this is going to be the intersection of the set A and the set of all things in the universe that are neither a 0 or a blue. So it's essentially the things that satisfy both of these, that it has to be in set A and it has to be in the set of all things in the universe that are neither a 0 or a blue. So let's think about what this is. So the number 3 is in set A and it's in the set of all things in the universe that are neither a 0 or blue. So let's throw a 3 in there. The number 7, it's an A and it's in the set of all things in the universe that are neither a 0 or a blue."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's think about what this is. So the number 3 is in set A and it's in the set of all things in the universe that are neither a 0 or blue. So let's throw a 3 in there. The number 7, it's an A and it's in the set of all things in the universe that are neither a 0 or a blue. So let's put a 7 there. Negative 5 also meets that constraint. A 0 does not meet that constraint."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "The number 7, it's an A and it's in the set of all things in the universe that are neither a 0 or a blue. So let's put a 7 there. Negative 5 also meets that constraint. A 0 does not meet that constraint. A 0 is an A, but it's not in the set of all things in the universe that are neither a 0 or a blue, because it is a 0. So we're not going to throw a 0 in there. And then a 13 is an A, and it's in the set of all things in the universe that are neither a 0 or a blue."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "A 0 does not meet that constraint. A 0 is an A, but it's not in the set of all things in the universe that are neither a 0 or a blue, because it is a 0. So we're not going to throw a 0 in there. And then a 13 is an A, and it's in the set of all things in the universe that are neither a 0 or a blue. So we could throw a 13 in there. So we've simplified things a good bit. This whole crazy business, all of this crazy business, has simplified to this set right over here."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then a 13 is an A, and it's in the set of all things in the universe that are neither a 0 or a blue. So we could throw a 13 in there. So we've simplified things a good bit. This whole crazy business, all of this crazy business, has simplified to this set right over here. Now, we want to find the relative complement of this business in A. So let me pick another color here. So we want to find the relative complement of this business in A."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "This whole crazy business, all of this crazy business, has simplified to this set right over here. Now, we want to find the relative complement of this business in A. So let me pick another color here. So we want to find the relative complement of this business in A. So now I'll just actually write out the set. 3, 7, negative 5, 13. Actually, let me write out both of them, just to make it, just so that we can really just visualize them both right over here."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we want to find the relative complement of this business in A. So now I'll just actually write out the set. 3, 7, negative 5, 13. Actually, let me write out both of them, just to make it, just so that we can really just visualize them both right over here. So A is this. It is 3, 7, negative 5, 0, and 13. And I could write the relative complement sign."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, let me write out both of them, just to make it, just so that we can really just visualize them both right over here. So A is this. It is 3, 7, negative 5, 0, and 13. And I could write the relative complement sign. Or actually, let me just write relative complement. I was going to write minus. And so in all of this business, we already figured out is a 3, a 7, a negative 5, and a 13."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "And I could write the relative complement sign. Or actually, let me just write relative complement. I was going to write minus. And so in all of this business, we already figured out is a 3, a 7, a negative 5, and a 13. So it's essentially, start with this set and take out all the stuff that are in this set. So this is going to be equal to, so you see, we're going to have to take out a 3 out of this set. We're going to take out a 7."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so in all of this business, we already figured out is a 3, a 7, a negative 5, and a 13. So it's essentially, start with this set and take out all the stuff that are in this set. So this is going to be equal to, so you see, we're going to have to take out a 3 out of this set. We're going to take out a 7. We're going to take out a negative 5. And we're going to take out a 13. So we're just left with the set that contains a 0."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "We're going to take out a 7. We're going to take out a negative 5. And we're going to take out a 13. So we're just left with the set that contains a 0. So all of this business right over here has simplified to a set that only contains 0. Now let's think about what B intersect C is. These are all of the things that are in both B and C. So this is going to be B intersect C. Let's see."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we're just left with the set that contains a 0. So all of this business right over here has simplified to a set that only contains 0. Now let's think about what B intersect C is. These are all of the things that are in both B and C. So this is going to be B intersect C. Let's see. 0 is not in both of them. 17 is in both of them. So we'll throw a 17 there."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are all of the things that are in both B and C. So this is going to be B intersect C. Let's see. 0 is not in both of them. 17 is in both of them. So we'll throw a 17 there. The number 3 is in both of them. Blue is not in both of them. The star is in both of them."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "So we'll throw a 17 there. The number 3 is in both of them. Blue is not in both of them. The star is in both of them. So I'll put the little gold star right over there. And so that's B intersect C. And so we're essentially going to take the union of this crazy thing, which ended up just being a set with a 0 in it. We're taking the union of that and B intersect C. And we deserve a drum roll now."}, {"video_title": "Bringing the set operations together   Probability and Statistics   Khan Academy.mp3", "Sentence": "The star is in both of them. So I'll put the little gold star right over there. And so that's B intersect C. And so we're essentially going to take the union of this crazy thing, which ended up just being a set with a 0 in it. We're taking the union of that and B intersect C. And we deserve a drum roll now. This is all going to be equal to, we're just going to combine these two sets. It's going to be the set with a 0, a 17, a 3, and our gold star. I should make the brackets in a different color."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let A represent the event that he eats a bagel for breakfast and let B represent the event that he eats pizza for lunch. Fair enough. On a randomly selected day, the probability that Rahul will eat a bagel for breakfast, probability of A, is.6. Let me write that down. The probability that he eats a bagel for breakfast is 0.6. The probability that he will eat a pizza for lunch, probability of event B, so the probability of, let me do that in that red color, the probability of event B, that he eats a pizza for lunch, is 0.5. And the conditional probability that he eats a bagel for breakfast, given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch, is equal to 0.7, which is interesting."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me write that down. The probability that he eats a bagel for breakfast is 0.6. The probability that he will eat a pizza for lunch, probability of event B, so the probability of, let me do that in that red color, the probability of event B, that he eats a pizza for lunch, is 0.5. And the conditional probability that he eats a bagel for breakfast, given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch, is equal to 0.7, which is interesting. Let me write this down. The probability of A, given that B is true, given B, is not 0.6. It's equal to 0.7."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And the conditional probability that he eats a bagel for breakfast, given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch, is equal to 0.7, which is interesting. Let me write this down. The probability of A, given that B is true, given B, is not 0.6. It's equal to 0.7. And because these two things are not the same, because probability of A by itself is different than the probability of A, given that B is true, this tells us that these two events are not independent, that we're dealing with dependent probability. This shows us the fact that B being true has changed the probability of A being true. This tells us that A and B are dependent."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's equal to 0.7. And because these two things are not the same, because probability of A by itself is different than the probability of A, given that B is true, this tells us that these two events are not independent, that we're dealing with dependent probability. This shows us the fact that B being true has changed the probability of A being true. This tells us that A and B are dependent. Before I start going on my little soapbox about dependent events, let's just think about what they actually want us to figure out. The probability of A given B is equal to 0.7. That's what we wrote right over here."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This tells us that A and B are dependent. Before I start going on my little soapbox about dependent events, let's just think about what they actually want us to figure out. The probability of A given B is equal to 0.7. That's what we wrote right over here. Based on this information, what is the probability of B given A? They want us to figure out the probability of B given A. The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "That's what we wrote right over here. Based on this information, what is the probability of B given A? They want us to figure out the probability of B given A. The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth. How would we think about this? I encourage you to pause this video before I work through it. I'm assuming you've given a go at it."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth. How would we think about this? I encourage you to pause this video before I work through it. I'm assuming you've given a go at it. The best way to tackle this is to just think about what's the probability that both A and B are going to happen? The probability of A and B happening, let me do this in a new color, the probability of A and B happening, I want to stay true to the colors, is equal to, there's a couple of ways you could write this out. This is equivalent to the probability of A given B times the probability of B. Hopefully that makes sense."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "I'm assuming you've given a go at it. The best way to tackle this is to just think about what's the probability that both A and B are going to happen? The probability of A and B happening, let me do this in a new color, the probability of A and B happening, I want to stay true to the colors, is equal to, there's a couple of ways you could write this out. This is equivalent to the probability of A given B times the probability of B. Hopefully that makes sense. The probability that B happens, and that given that B has happened, the probability that A happens. That would also be equal to, obviously this is A and B is happening, or you could do it the other way around. You could do it as the probability that B, the probability that B given A happens, times the probability of A."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is equivalent to the probability of A given B times the probability of B. Hopefully that makes sense. The probability that B happens, and that given that B has happened, the probability that A happens. That would also be equal to, obviously this is A and B is happening, or you could do it the other way around. You could do it as the probability that B, the probability that B given A happens, times the probability of A. This also makes sense. What's the probability that A happened? And that given A happened, what's the probability of B?"}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "You could do it as the probability that B, the probability that B given A happens, times the probability of A. This also makes sense. What's the probability that A happened? And that given A happened, what's the probability of B? You multiply those together, you get the probability that both happened. Why is this helpful for us? We know a lot of this information."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that given A happened, what's the probability of B? You multiply those together, you get the probability that both happened. Why is this helpful for us? We know a lot of this information. We know the probability of A given B is 0.7. Let me write that, I'll scroll down a little bit. This is 0.7."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know a lot of this information. We know the probability of A given B is 0.7. Let me write that, I'll scroll down a little bit. This is 0.7. We know that the probability of B is 0.5. This is 0.5. We know that the probability of A and B is the product of these two things."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is 0.7. We know that the probability of B is 0.5. This is 0.5. We know that the probability of A and B is the product of these two things. That's going to be 0.35. Seven times five is 35, or I guess you could say half of 0.7 is 0.35, 0.5 of 0.7. That is going to be equal to what we need to figure out, the probability of B given A times the probability of A."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that the probability of A and B is the product of these two things. That's going to be 0.35. Seven times five is 35, or I guess you could say half of 0.7 is 0.35, 0.5 of 0.7. That is going to be equal to what we need to figure out, the probability of B given A times the probability of A. But we know the probability of A. We know that that is 0.6. We know that this is 0.6."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "That is going to be equal to what we need to figure out, the probability of B given A times the probability of A. But we know the probability of A. We know that that is 0.6. We know that this is 0.6. Just like that, we've set up a situation, an equation where we can solve for the probability of B given A. The probability of B given A, notice, let me just rewrite it right over here. Actually, I'll write this part first."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "We know that this is 0.6. Just like that, we've set up a situation, an equation where we can solve for the probability of B given A. The probability of B given A, notice, let me just rewrite it right over here. Actually, I'll write this part first. 0.6 times the probability of B given A, times that right over there, and I'll just copy and paste it so I don't have to keep changing colors. That over there is equal to 0.35. To solve for the probability of B given A, we can just divide both sides by 0.6."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Actually, I'll write this part first. 0.6 times the probability of B given A, times that right over there, and I'll just copy and paste it so I don't have to keep changing colors. That over there is equal to 0.35. To solve for the probability of B given A, we can just divide both sides by 0.6. We get the probability of B given A is equal to, let me get our calculator out, 0.35 divided by 0.6, and we deserve a little bit of a drum roll here, is 0.5833, keeps going. They tell us to round to the nearest hundredth. It's 0.58."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "To solve for the probability of B given A, we can just divide both sides by 0.6. We get the probability of B given A is equal to, let me get our calculator out, 0.35 divided by 0.6, and we deserve a little bit of a drum roll here, is 0.5833, keeps going. They tell us to round to the nearest hundredth. It's 0.58. It is approximately 0.58. Notice, this is approximately equal to 0.58. Once again, verifying that these are dependent."}, {"video_title": "Calculating conditional probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's 0.58. It is approximately 0.58. Notice, this is approximately equal to 0.58. Once again, verifying that these are dependent. The probability that B happens given A is true is higher than just the probability that B by itself, or without knowing anything else. Just the probability of B is lower than the probability of B given that you know A has happened, or event A is true. And we're done."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Let's say I go to 16 students, and I ask them to measure how many glasses of water they drink per day for the last 30 days, and then to average it. And so this data point right over here tells us one student drank an average of 0.5 glasses of water per day. That person is probably very dehydrated. This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1."}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width?"}, {"video_title": "Density Curves   Modeling data distributions   AP Statistics   Khan Academy.mp3", "Sentence": "So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width? The width here, if we're going from 2.9 to 3.1, the width is going to be 0.2 wide. And so we could approximate this area by approximating this rectangle, the area of the rectangle. 0.2 times 0.2, that would give us an area of 0.04."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And this density curve doesn't look like the ones we typically see that are a little bit curvier, but this is a little easier for us to work with and figure out areas. And so they ask us to find the percent of the area under the density curve where x is more than two. So what area represents when x is more than two? So this is when x is equal to two. So they're talking about this area right over here. And so we need to figure out the percent of the total area under the curve that this blue area actually represents. So first let's find the total area under the density curve."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "So this is when x is equal to two. So they're talking about this area right over here. And so we need to figure out the percent of the total area under the curve that this blue area actually represents. So first let's find the total area under the density curve. And the density only has area, the density curve only has area from x equals one to x equals three. So it does amount to finding the area of this larger trapezoid. So let me highlight this trapezoid in red."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "So first let's find the total area under the density curve. And the density only has area, the density curve only has area from x equals one to x equals three. So it does amount to finding the area of this larger trapezoid. So let me highlight this trapezoid in red. So we wanna find the area of this trapezoid right over here. And then that should be equal to one because all density curves have an area of one under the total curve. So let's first verify that."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "So let me highlight this trapezoid in red. So we wanna find the area of this trapezoid right over here. And then that should be equal to one because all density curves have an area of one under the total curve. So let's first verify that. So there's a couple of ways to think about it. We could split it up into two shapes or you could just use the formula for an area of a trapezoid. In fact, let's use the formula for an area of a trapezoid."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "So let's first verify that. So there's a couple of ways to think about it. We could split it up into two shapes or you could just use the formula for an area of a trapezoid. In fact, let's use the formula for an area of a trapezoid. The formula for an area of a trapezoid is you would take the average of, you would take the average of this length, let me do that in another color, this length and this length, and then multiply that times the base. So the average of this length and this length, let's see, this is 0.25, 0.25 plus this height, 0.75 divided by two, so that's the average of those two sides, times the base, times this right over here, which is two. And so this is going to give us, as it should have, 0.25 plus 0.75, which is equal to one."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "In fact, let's use the formula for an area of a trapezoid. The formula for an area of a trapezoid is you would take the average of, you would take the average of this length, let me do that in another color, this length and this length, and then multiply that times the base. So the average of this length and this length, let's see, this is 0.25, 0.25 plus this height, 0.75 divided by two, so that's the average of those two sides, times the base, times this right over here, which is two. And so this is going to give us, as it should have, 0.25 plus 0.75, which is equal to one. So the area under the entire density curve is one, which we need to be true for this to be a density curve. Now let's think about what percentage of that area is represented in blue right over here. Well, we could do the same thing."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And so this is going to give us, as it should have, 0.25 plus 0.75, which is equal to one. So the area under the entire density curve is one, which we need to be true for this to be a density curve. Now let's think about what percentage of that area is represented in blue right over here. Well, we could do the same thing. We could say, all right, this is a trapezoid. We wanna take the average of this side and this side and multiply it times the base. So this side is 0.5 high, 0.5, plus 0.75, 0.75 high, and we're gonna take the average of that, divided by two, times the base."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Well, we could do the same thing. We could say, all right, this is a trapezoid. We wanna take the average of this side and this side and multiply it times the base. So this side is 0.5 high, 0.5, plus 0.75, 0.75 high, and we're gonna take the average of that, divided by two, times the base. Well, the base going from two to three is only equal to, is equal to one, so times one. And so this is going to give us 1.25, 1.25 over two. And what is that going to be equal to?"}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "So this side is 0.5 high, 0.5, plus 0.75, 0.75 high, and we're gonna take the average of that, divided by two, times the base. Well, the base going from two to three is only equal to, is equal to one, so times one. And so this is going to give us 1.25, 1.25 over two. And what is that going to be equal to? Well, that would be the same thing as 0. what? Let's see, 0.625. Did I do that right?"}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "And what is that going to be equal to? Well, that would be the same thing as 0. what? Let's see, 0.625. Did I do that right? Yep, if I multiply two times this, I would get 1.25. So the percent of the area under the density curve where x is more than two, this is the decimal expression of it, but if we wanted to write it as a percent, it would be 62.5%. Let's do another example."}, {"video_title": "Worked example finding area under density curves   AP Statistics   Khan Academy.mp3", "Sentence": "Did I do that right? Yep, if I multiply two times this, I would get 1.25. So the percent of the area under the density curve where x is more than two, this is the decimal expression of it, but if we wanted to write it as a percent, it would be 62.5%. Let's do another example. Consider the density curve below, all right, we have another one of these somewhat angular density curves. Find the percent of the area under the density curve where x is more than three. So we're talking about, see, this is where x is equal to three, x is more than three, we're talking about this entire area right over here."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So we're told that Amanda Young wants to win some prizes. A cereal company is giving away a prize in each box of cereal, and they advertise, collect all six prizes. Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes?"}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through?"}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three."}, {"video_title": "Random number list to run experiment   Probability   AP Statistics   Khan Academy (2).mp3", "Sentence": "One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get? No, we don't know that."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "He may choose the same number both times. If his ticket matches the two numbers in one letter drawn in order, he wins the grand prize and receives $10,405. If just his letter matches, but one or both of his numbers do not match, he wins the small prize of $100. Under any other outcome, he loses and receives nothing. The game costs him $5 to play. So under any other outcome, he loses and receives nothing. He has chosen the ticket 0 for R. So we're assuming he's paying the $5 to play and he picks the ticket 0 for R. So let's say we define a random variable X."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Under any other outcome, he loses and receives nothing. The game costs him $5 to play. So under any other outcome, he loses and receives nothing. He has chosen the ticket 0 for R. So we're assuming he's paying the $5 to play and he picks the ticket 0 for R. So let's say we define a random variable X. Let's say that this random variable is the net profit from playing this lottery game. What is the expected from playing 0 for R? So M is particular ticket right over here."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "He has chosen the ticket 0 for R. So we're assuming he's paying the $5 to play and he picks the ticket 0 for R. So let's say we define a random variable X. Let's say that this random variable is the net profit from playing this lottery game. What is the expected from playing 0 for R? So M is particular ticket right over here. So let's just say X is the random variable, it's the net profit from playing this ticket. What I want to think about in this video is what is the expected value of that? What is the expected net profit from playing 0 for R?"}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So M is particular ticket right over here. So let's just say X is the random variable, it's the net profit from playing this ticket. What I want to think about in this video is what is the expected value of that? What is the expected net profit from playing 0 for R? I encourage you to pause the video and think through it on your own. So let's think about what expected value is. It's the probability of each of those outcomes times the net profit from those outcomes."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the expected net profit from playing 0 for R? I encourage you to pause the video and think through it on your own. So let's think about what expected value is. It's the probability of each of those outcomes times the net profit from those outcomes. So there's the probability of the grand prize. Let me do that in that red color. So there is the probability of getting the grand prize."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "It's the probability of each of those outcomes times the net profit from those outcomes. So there's the probability of the grand prize. Let me do that in that red color. So there is the probability of getting the grand prize. And now what would times his net payoff from the grand prize? What would that be? Well, he gets $10,405, but that's not his net payoff or his net profit, I should say."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there is the probability of getting the grand prize. And now what would times his net payoff from the grand prize? What would that be? Well, he gets $10,405, but that's not his net payoff or his net profit, I should say. His net profit is what he gets minus what he paid to play. So he paid $5 to play. So that's that."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, he gets $10,405, but that's not his net payoff or his net profit, I should say. His net profit is what he gets minus what he paid to play. So he paid $5 to play. So that's that. So plus the probability of getting the small prize times the payoff of the small prize, which is going to be $100 or times the net profit, I guess, if you get the small prize. So you get a payoff of $100 minus you have to pay $5 to play. And then finally you have the probability of neither."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So that's that. So plus the probability of getting the small prize times the payoff of the small prize, which is going to be $100 or times the net profit, I guess, if you get the small prize. So you get a payoff of $100 minus you have to pay $5 to play. And then finally you have the probability of neither. So you're essentially not winning. And there in that situation, what is the net profit? Well, in that situation, your net profit is negative 5."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "And then finally you have the probability of neither. So you're essentially not winning. And there in that situation, what is the net profit? Well, in that situation, your net profit is negative 5. You paid $5 and you got nothing in return. So to figure out the expected value, you just have to figure out these probabilities. So what's the probability of the grand prize?"}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, in that situation, your net profit is negative 5. You paid $5 and you got nothing in return. So to figure out the expected value, you just have to figure out these probabilities. So what's the probability of the grand prize? Do that over here. Probability of grand prize. Well, the probability that he gets the first letter right is 1 in 10."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what's the probability of the grand prize? Do that over here. Probability of grand prize. Well, the probability that he gets the first letter right is 1 in 10. There's 10 digits there. Probability that he gets the second letter right is 1 in 10. These are all independent."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, the probability that he gets the first letter right is 1 in 10. There's 10 digits there. Probability that he gets the second letter right is 1 in 10. These are all independent. And probability he gets the letter right, there's 26 equally likely letters that might be in the actual one. So he has a 1 in 26 chance of that one as well. So the probability of the grand prize is 1 in 2,600."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "These are all independent. And probability he gets the letter right, there's 26 equally likely letters that might be in the actual one. So he has a 1 in 26 chance of that one as well. So the probability of the grand prize is 1 in 2,600. So this is 1 in 2,600. Now what's the probability of getting the small prize? Well, let's see."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of the grand prize is 1 in 2,600. So this is 1 in 2,600. Now what's the probability of getting the small prize? Well, let's see. He has a 1 in 26 chance. The small prize is getting the letter right, but not getting both of the numbers right. So he has a 1 in 26 chance of getting the letter right."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's see. He has a 1 in 26 chance. The small prize is getting the letter right, but not getting both of the numbers right. So he has a 1 in 26 chance of getting the letter right. But we're not done here just with the 1 in 26. Because this 1 in 26, this includes all the scenarios where he gets the letter right, including the scenarios where he wins the grand prize, where he gets the letter and he gets the two numbers right. So what we need to do is we need to subtract out the situation, the probability of getting the letter and the two numbers right."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So he has a 1 in 26 chance of getting the letter right. But we're not done here just with the 1 in 26. Because this 1 in 26, this includes all the scenarios where he gets the letter right, including the scenarios where he wins the grand prize, where he gets the letter and he gets the two numbers right. So what we need to do is we need to subtract out the situation, the probability of getting the letter and the two numbers right. And we already know what that is. It's 1 in 2,600. So it's 1 in 26 minus 1 in 2,600."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So what we need to do is we need to subtract out the situation, the probability of getting the letter and the two numbers right. And we already know what that is. It's 1 in 2,600. So it's 1 in 26 minus 1 in 2,600. The reason why I have to subtract out this 2,600 is he has a 1 in 26 chance of getting this letter right. So that includes the scenario where he gets everything right. But the small prize is only where you get the letter and one or none of these."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it's 1 in 26 minus 1 in 2,600. The reason why I have to subtract out this 2,600 is he has a 1 in 26 chance of getting this letter right. So that includes the scenario where he gets everything right. But the small prize is only where you get the letter and one or none of these. If you get both of these, then you're in the grand prize case. So you essentially have to subtract out the probability that you won the grand prize, that you got all three of them, to figure out the probability of the small prize. Now, what's the probability of essentially losing?"}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "But the small prize is only where you get the letter and one or none of these. If you get both of these, then you're in the grand prize case. So you essentially have to subtract out the probability that you won the grand prize, that you got all three of them, to figure out the probability of the small prize. Now, what's the probability of essentially losing? The probability of neither. Well, it's just kind of, you know, that's everything else. So it would be 1 minus these probabilities right over here."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, what's the probability of essentially losing? The probability of neither. Well, it's just kind of, you know, that's everything else. So it would be 1 minus these probabilities right over here. So it would be 1 minus the probability of the small prize minus the probability of the grand. These are the possible outcomes, so they have to add up to 1 or 100%. So this is 1 minus probability of small minus probability of large."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So it would be 1 minus these probabilities right over here. So it would be 1 minus the probability of the small prize minus the probability of the grand. These are the possible outcomes, so they have to add up to 1 or 100%. So this is 1 minus probability of small minus probability of large. Or I should say, not grand prize. Grand prize. So let's fill this in."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is 1 minus probability of small minus probability of large. Or I should say, not grand prize. Grand prize. So let's fill this in. So the probability of the small one, this right over here, I'm using that red too much, this right over here is 1 in 26 minus 1 in 2,600. And then this right over here is 1 minus the small, which is 1 in 26 minus 1 in 2,600. Minus 1 in 2,600."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's fill this in. So the probability of the small one, this right over here, I'm using that red too much, this right over here is 1 in 26 minus 1 in 2,600. And then this right over here is 1 minus the small, which is 1 in 26 minus 1 in 2,600. Minus 1 in 2,600. And this simplifies to, let's see, this is 1 minus 1 over 26 plus 1 in 2,600, plus or minus 1 in 2,600. These cancel and you're left with 1 in 1 in 2,600. Now, why does this make sense?"}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Minus 1 in 2,600. And this simplifies to, let's see, this is 1 minus 1 over 26 plus 1 in 2,600, plus or minus 1 in 2,600. These cancel and you're left with 1 in 1 in 2,600. Now, why does this make sense? Well, the way you lose, the way you get nothing, is if you get the letter wrong. So you have a 1 in 26 chance of getting the letter right, and then you're going to be in one of these two categories. Or you have a 1 minus 1 in 26, which is equal to 25 out of 26."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, why does this make sense? Well, the way you lose, the way you get nothing, is if you get the letter wrong. So you have a 1 in 26 chance of getting the letter right, and then you're going to be in one of these two categories. Or you have a 1 minus 1 in 26, which is equal to 25 out of 26. You have a 25 in 26 chance of getting the letter wrong, in which case you get nothing, in which case you completely lose. So let's just take our calculator out and calculate this. And we'll round to the nearest penny here."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Or you have a 1 minus 1 in 26, which is equal to 25 out of 26. You have a 25 in 26 chance of getting the letter wrong, in which case you get nothing, in which case you completely lose. So let's just take our calculator out and calculate this. And we'll round to the nearest penny here. So let's see, it is going to be 1,2,600. So 1 divided by 2,600 times, let's see, 10,004 minus 5 is going to be 10,400. Times 10,400, that's your net profit when you win the grand prize."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we'll round to the nearest penny here. So let's see, it is going to be 1,2,600. So 1 divided by 2,600 times, let's see, 10,004 minus 5 is going to be 10,400. Times 10,400, that's your net profit when you win the grand prize. And then you're going to have plus 1 divided by 26 minus 1 divided by 2,600 times your net profit for the small prize, 100 minus 5, which is 95. And then finally, plus 25,26. So 25 divided by 26, actually I'll put parentheses around here just to make it consistent."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "Times 10,400, that's your net profit when you win the grand prize. And then you're going to have plus 1 divided by 26 minus 1 divided by 2,600 times your net profit for the small prize, 100 minus 5, which is 95. And then finally, plus 25,26. So 25 divided by 26, actually I'll put parentheses around here just to make it consistent. So 25 divided by 26 times that net payoff, when you get nothing, well you have to pay out $5 and you've got nothing in return, times negative 5. Actually I don't know if it's going to recognize that as times, so I'll just write times negative 5, and then we delete that. And we deserve a drum roll now."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "So 25 divided by 26, actually I'll put parentheses around here just to make it consistent. So 25 divided by 26 times that net payoff, when you get nothing, well you have to pay out $5 and you've got nothing in return, times negative 5. Actually I don't know if it's going to recognize that as times, so I'll just write times negative 5, and then we delete that. And we deserve a drum roll now. We get a expected net profit of playing as $2.81, if we round up to the nearest penny. So this is all going to be equal to $2.81. And so this is actually a very unusual lottery game where you have a positive expected net profit as a player."}, {"video_title": "Expected profit from lottery ticket   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we deserve a drum roll now. We get a expected net profit of playing as $2.81, if we round up to the nearest penny. So this is all going to be equal to $2.81. And so this is actually a very unusual lottery game where you have a positive expected net profit as a player. Usually the person operating the lottery, the state, who are the casino, whoever it is, they're the ones who have the expected net profit and then the player has the expected net loss. But this actually would make rational sense to play, which is not the case with most lottery games. That if by playing you actually expect a $2.81 net profit."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "And these are going to be in some ways similar to the conditions for inference that we thought about when we were doing hypothesis testing and confidence intervals for means and for proportions, but there's also going to be a few new conditions. So to help us remember these conditions, you might want to think about the LINER acronym, L-I-N-E-R, and if it isn't obvious to you, this almost is linear. Liner, with an A, it would be linear, and this is valuable because remember, we're thinking about linear regression. So the L right over here actually does stand for linear, and here, the condition is is that the actual relationship in the population between your X and Y variables actually is a linear relationship. So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "So the L right over here actually does stand for linear, and here, the condition is is that the actual relationship in the population between your X and Y variables actually is a linear relationship. So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you could be thinking about your 10% rule that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you could be thinking about your 10% rule that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks like this. And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks like this. And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far."}, {"video_title": "Conditions for inference on slope   More on regression   AP Statistics   Khan Academy.mp3", "Sentence": "So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. So I'll leave you there. It's good to know it will show up on some exams, but many times when it comes to problem solving, in an introductory statistics class, they will tell you, hey, just assume the conditions for inference have been met, or what are the conditions for inference? But they're not going to actually make you prove, for example, the normal or the equal variance condition."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Each employee receives an annual rating, the best of which is exceeds expectations. Management claimed that 10% of employees earned this rating but Jules suspected it was actually less common. She obtained an anonymous random sample of 10 ratings for employees on her team. She wants to use the sample data to test her null hypothesis that the true proportion is 10% versus her alternative hypothesis that the true proportion is less than 10% where P is the proportion of all employees on her team who earned exceeds expectations. Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "She wants to use the sample data to test her null hypothesis that the true proportion is 10% versus her alternative hypothesis that the true proportion is less than 10% where P is the proportion of all employees on her team who earned exceeds expectations. Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then we want to calculate a p value."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then we want to calculate a p value. And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less than or equal to the one that you just calculated for a sample size of n equals 10 given that your null hypothesis is true. And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And then we want to calculate a p value. And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less than or equal to the one that you just calculated for a sample size of n equals 10 given that your null hypothesis is true. And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition?"}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition?"}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition? The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10. So they need to be greater than or equal to 10. Now, what are they for this particular scenario?"}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Now, what about the normal condition? The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10. So they need to be greater than or equal to 10. Now, what are they for this particular scenario? Well, n is equal to 10, n is equal to 10, and our true proportion, remember, we're going to assume, when we do the significance test, we assume the null hypothesis is true, and the null hypothesis tells us that our true proportion is 0.1. So this is 0.1. This is one minus 0.1, which is 0.9."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "Now, what are they for this particular scenario? Well, n is equal to 10, n is equal to 10, and our true proportion, remember, we're going to assume, when we do the significance test, we assume the null hypothesis is true, and the null hypothesis tells us that our true proportion is 0.1. So this is 0.1. This is one minus 0.1, which is 0.9. Well, 10 times 0.1 is one, so that's not greater than or equal to 10, so just off of that, we don't meet the normal condition, but even the second one, 10 times 0.9 is nine. That's also not greater than or equal to 10, so we don't meet this normal condition. We can't feel good that the sampling distribution is roughly normal, which we normally assume when we're trying to make this type of calculation."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "This is one minus 0.1, which is 0.9. Well, 10 times 0.1 is one, so that's not greater than or equal to 10, so just off of that, we don't meet the normal condition, but even the second one, 10 times 0.9 is nine. That's also not greater than or equal to 10, so we don't meet this normal condition. We can't feel good that the sampling distribution is roughly normal, which we normally assume when we're trying to make this type of calculation. And then last but not least, independence. Independence is to feel good that each of the data points in your sample are independent. The results of whether they are a success or a failure is independent of each other."}, {"video_title": "Conditions for a z test about a proportion   AP Statistics   Khan Academy.mp3", "Sentence": "We can't feel good that the sampling distribution is roughly normal, which we normally assume when we're trying to make this type of calculation. And then last but not least, independence. Independence is to feel good that each of the data points in your sample are independent. The results of whether they are a success or a failure is independent of each other. Now, if she was surveying these people with replacement, if each data point was with replacement, you would definitely meet this independence condition, but she didn't do it with replacement, but there's another way to go about it. You could use your 10% rule. If your sample size is less than 10% of the population size, then it's okay."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Sunil and his friends have been using a group messaging app for over a year to chat with each other. He suspects that on average, they send each other more than 100 messages per day. Sunil takes a random sample of seven days from their chat history and records how many messages were sent on those days. The sample data are strongly skewed to the right with a mean of 125 messages and a standard deviation of 44 messages. He wants to use these sample data to conduct a t-test about the mean. Which conditions for performing this type of significance test have been met? So let's just think about what's going on here."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "The sample data are strongly skewed to the right with a mean of 125 messages and a standard deviation of 44 messages. He wants to use these sample data to conduct a t-test about the mean. Which conditions for performing this type of significance test have been met? So let's just think about what's going on here. Sunil might have some type of a null hypothesis. Maybe he got this 100, maybe he read a magazine article that says that on average, the average teenager sends 100 text messages per day. And so maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which was signified by mu, maybe the null is 100, that they're no different than all other teenagers, and maybe he suspects, and actually they say it right over here, his alternative hypothesis would be what he suspects, that they'd send more than 100 text messages per day."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "So let's just think about what's going on here. Sunil might have some type of a null hypothesis. Maybe he got this 100, maybe he read a magazine article that says that on average, the average teenager sends 100 text messages per day. And so maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which was signified by mu, maybe the null is 100, that they're no different than all other teenagers, and maybe he suspects, and actually they say it right over here, his alternative hypothesis would be what he suspects, that they'd send more than 100 text messages per day. And so what he does is he takes a sample from the population of days, and there's over 365. They say they've been using the group messaging app for over a year. And he takes seven of those days."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And so maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which was signified by mu, maybe the null is 100, that they're no different than all other teenagers, and maybe he suspects, and actually they say it right over here, his alternative hypothesis would be what he suspects, that they'd send more than 100 text messages per day. And so what he does is he takes a sample from the population of days, and there's over 365. They say they've been using the group messaging app for over a year. And he takes seven of those days. So n is equal to seven. And from that, he calculates sample statistics. He calculates the sample mean, which is trying to estimate the true population mean right over here."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And he takes seven of those days. So n is equal to seven. And from that, he calculates sample statistics. He calculates the sample mean, which is trying to estimate the true population mean right over here. And he also is able to calculate a sample standard deviation. And what you do in a significance test is you say, well, what is the probability of getting this sample mean or something even more extreme, assuming the null hypothesis? And if that probability is below a preset threshold, then you would reject the null hypothesis, and it would suggest the alternative."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "He calculates the sample mean, which is trying to estimate the true population mean right over here. And he also is able to calculate a sample standard deviation. And what you do in a significance test is you say, well, what is the probability of getting this sample mean or something even more extreme, assuming the null hypothesis? And if that probability is below a preset threshold, then you would reject the null hypothesis, and it would suggest the alternative. But in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test. The first is is that this is truly a random sample, and that's known as the random condition. And you have seen this before when we did significance tests with proportions."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And if that probability is below a preset threshold, then you would reject the null hypothesis, and it would suggest the alternative. But in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test. The first is is that this is truly a random sample, and that's known as the random condition. And you have seen this before when we did significance tests with proportions. Here we're doing it with means, population mean, sample mean. In the past, we did it with population proportion and sample proportion. Well, the random condition, it says it right here, Sunil takes a random sample of seven days from their chat history."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And you have seen this before when we did significance tests with proportions. Here we're doing it with means, population mean, sample mean. In the past, we did it with population proportion and sample proportion. Well, the random condition, it says it right here, Sunil takes a random sample of seven days from their chat history. They don't say how he did it, but we'll just take their word for it that it was a random sample. The next condition is sometimes known as the independence, independence condition. And that's that the individual observations in our sample are roughly independent."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Well, the random condition, it says it right here, Sunil takes a random sample of seven days from their chat history. They don't say how he did it, but we'll just take their word for it that it was a random sample. The next condition is sometimes known as the independence, independence condition. And that's that the individual observations in our sample are roughly independent. One way that they would be independent for sure is if Sunil is sampling with replacement. They don't say that, but another condition, so you either could have replacement, sampling with replacement, or another way where you could feel that it's roughly independent is if your sample size is less than or equal to 10% of the population. Now in this situation, he took seven, he took a sample size of seven, and then the population of days, it says that they've been using the group messaging app for over a year, so they've been using it for over 365 days."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "And that's that the individual observations in our sample are roughly independent. One way that they would be independent for sure is if Sunil is sampling with replacement. They don't say that, but another condition, so you either could have replacement, sampling with replacement, or another way where you could feel that it's roughly independent is if your sample size is less than or equal to 10% of the population. Now in this situation, he took seven, he took a sample size of seven, and then the population of days, it says that they've been using the group messaging app for over a year, so they've been using it for over 365 days. So seven is for sure less than or equal to 10% of 365, which would be 36.5. So we meet this condition, which allows us to meet the independence condition. Now the last condition is often known as the normal condition, and this is to feel good that the sampling distribution of the sample means right over here is approximately normal."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Now in this situation, he took seven, he took a sample size of seven, and then the population of days, it says that they've been using the group messaging app for over a year, so they've been using it for over 365 days. So seven is for sure less than or equal to 10% of 365, which would be 36.5. So we meet this condition, which allows us to meet the independence condition. Now the last condition is often known as the normal condition, and this is to feel good that the sampling distribution of the sample means right over here is approximately normal. And this is going to be a little bit different than what we saw with significance tests when we dealt with proportions. There's a few ways to feel good that the sampling distribution of the sample means is normal. One is is if the underlying parent population normal."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "Now the last condition is often known as the normal condition, and this is to feel good that the sampling distribution of the sample means right over here is approximately normal. And this is going to be a little bit different than what we saw with significance tests when we dealt with proportions. There's a few ways to feel good that the sampling distribution of the sample means is normal. One is is if the underlying parent population normal. So parent, parent population normal. Now they don't tell us anything that there's actually a normal distribution for the amount of time that they spend on a given day. So we don't know this one for sure, but sometimes you might."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "One is is if the underlying parent population normal. So parent, parent population normal. Now they don't tell us anything that there's actually a normal distribution for the amount of time that they spend on a given day. So we don't know this one for sure, but sometimes you might. Another way is to feel good that our sample size is greater than or equal to 30. And this comes from the central limit theorem that then our sampling distribution is going to be roughly normal. But we see very clearly our sample size is not greater than or equal to 30, so we don't meet that constraint either."}, {"video_title": "Conditions for a t test about a mean   AP Statistics   Khan Academy.mp3", "Sentence": "So we don't know this one for sure, but sometimes you might. Another way is to feel good that our sample size is greater than or equal to 30. And this comes from the central limit theorem that then our sampling distribution is going to be roughly normal. But we see very clearly our sample size is not greater than or equal to 30, so we don't meet that constraint either. Now the third way that we could feel good that our sampling distribution of our sample mean is roughly normal is if our sample, is if our sample is symmetric, symmetric, and there are no outliers, or maybe even you could say no significant outliers. Now is this the case? Well it says right over here, the sample data are strongly skewed to the right with a mean of 125 messages and a standard deviation of 44 messages."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Suppose that Erica simultaneously rolls a six-sided die and a four-sided die. Let A be the event that she rolls doubles. Let me write this. A be the event that she rolls doubles and B be the event that the four-sided die is a four. Use the sample space of possible outcomes below to answer each of the following questions. Fair enough. What is probability of A, the probability that Erica rolls doubles?"}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "A be the event that she rolls doubles and B be the event that the four-sided die is a four. Use the sample space of possible outcomes below to answer each of the following questions. Fair enough. What is probability of A, the probability that Erica rolls doubles? Well, over here we have our sample space of possible outcomes. Each of these are equally likely. And so let's see how many of them there are."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is probability of A, the probability that Erica rolls doubles? Well, over here we have our sample space of possible outcomes. Each of these are equally likely. And so let's see how many of them there are. There are one, two, three, four by one, two, three, four, five, six. So there are 24 possible outcomes, which makes sense. There's four possible outcomes for the four-sided die and six possible outcomes for the six-sided die."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And so let's see how many of them there are. There are one, two, three, four by one, two, three, four, five, six. So there are 24 possible outcomes, which makes sense. There's four possible outcomes for the four-sided die and six possible outcomes for the six-sided die. So you have a total of 24 equally likely outcomes. So probability of, let me write it here. So probability of A, probability of A is going to be the fraction of the 24 equally likely outcomes that involve event A, that she rolls doubles."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "There's four possible outcomes for the four-sided die and six possible outcomes for the six-sided die. So you have a total of 24 equally likely outcomes. So probability of, let me write it here. So probability of A, probability of A is going to be the fraction of the 24 equally likely outcomes that involve event A, that she rolls doubles. So let's think about that. This is, she has rolled doubles, one and a one. They don't look the same, but they're both ones."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So probability of A, probability of A is going to be the fraction of the 24 equally likely outcomes that involve event A, that she rolls doubles. So let's think about that. This is, she has rolled doubles, one and a one. They don't look the same, but they're both ones. Let's see, we have a two and a two. We have a three and a three. And we have a four and a four."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "They don't look the same, but they're both ones. Let's see, we have a two and a two. We have a three and a three. And we have a four and a four. And it's impossible to have a five and a five because the four-sided die only goes up to four. So there's four possibilities, four of the 24 equally likely possibilities involve rolling doubles. So there's a 424th probability, or if we divide the numerator and denominator by four, it is a 1 6th probability that Erica, a 1 6th probability that Erica rolls doubles."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we have a four and a four. And it's impossible to have a five and a five because the four-sided die only goes up to four. So there's four possibilities, four of the 24 equally likely possibilities involve rolling doubles. So there's a 424th probability, or if we divide the numerator and denominator by four, it is a 1 6th probability that Erica, a 1 6th probability that Erica rolls doubles. What is probability of B? The probability that the four-sided die is a four. So the probability of B, well, once again, there's 24 equally likely possibilities."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there's a 424th probability, or if we divide the numerator and denominator by four, it is a 1 6th probability that Erica, a 1 6th probability that Erica rolls doubles. What is probability of B? The probability that the four-sided die is a four. So the probability of B, well, once again, there's 24 equally likely possibilities. And how many of them involve the four-sided die being a four? Well, you have all of these right over here involve a four-sided die being a four. So this is one, two, three, four, five, six of the 24 equally likely possibilities."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of B, well, once again, there's 24 equally likely possibilities. And how many of them involve the four-sided die being a four? Well, you have all of these right over here involve a four-sided die being a four. So this is one, two, three, four, five, six of the 24 equally likely possibilities. Or you could say 1 4th of the equally likely possibilities, or the probability is 1 4th, which makes sense because probability of B, it kind of ignores the six-sided die. And it just says, well, what's the probability that the four-sided die is four? Well, that's one out of the four possible outcomes for that four-sided die."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is one, two, three, four, five, six of the 24 equally likely possibilities. Or you could say 1 4th of the equally likely possibilities, or the probability is 1 4th, which makes sense because probability of B, it kind of ignores the six-sided die. And it just says, well, what's the probability that the four-sided die is four? Well, that's one out of the four possible outcomes for that four-sided die. What is the probability of A given B? The probability that Erica rolls doubles given that the four-sided die is a four. So let's just think about this a little bit."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, that's one out of the four possible outcomes for that four-sided die. What is the probability of A given B? The probability that Erica rolls doubles given that the four-sided die is a four. So let's just think about this a little bit. Probability of A given that B has happened. Given that B has happened. So essentially, we are restricting our equally likely possibilities now to the situation where B has happened."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's just think about this a little bit. Probability of A given that B has happened. Given that B has happened. So essentially, we are restricting our equally likely possibilities now to the situation where B has happened. Given B means we're assuming that B has happened. So now we're restricting our sample space of possible outcomes where B has happened to this right over here. So now there are one, two, three, four, five, six equally likely outcomes."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So essentially, we are restricting our equally likely possibilities now to the situation where B has happened. Given B means we're assuming that B has happened. So now we're restricting our sample space of possible outcomes where B has happened to this right over here. So now there are one, two, three, four, five, six equally likely outcomes. And how many of them involve A happening? Well, this one right over here that we had already circled. This is the one out of the six equally likely outcomes that involve doubles."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So now there are one, two, three, four, five, six equally likely outcomes. And how many of them involve A happening? Well, this one right over here that we had already circled. This is the one out of the six equally likely outcomes that involve doubles. So there's a 1 6th probability. Now that makes sense. Let me just write this down."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "This is the one out of the six equally likely outcomes that involve doubles. So there's a 1 6th probability. Now that makes sense. Let me just write this down. This is one over six. Why does this make sense? Because with a four-sided die, we're assuming is a four."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Let me just write this down. This is one over six. Why does this make sense? Because with a four-sided die, we're assuming is a four. So it's essentially, this is analogous to saying when you roll a six-sided die, what's the probability that you get a four as well? Because that's the only way you're going to get doubles given that the four-sided die is four. And we see that right over here."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Because with a four-sided die, we're assuming is a four. So it's essentially, this is analogous to saying when you roll a six-sided die, what's the probability that you get a four as well? Because that's the only way you're going to get doubles given that the four-sided die is four. And we see that right over here. The six-sided die has to be a four as well in order for this to be doubles because we're assuming where it's given that B, we're given of event B. We're restricting our sample space with event B. What is the probability of B given A?"}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we see that right over here. The six-sided die has to be a four as well in order for this to be doubles because we're assuming where it's given that B, we're given of event B. We're restricting our sample space with event B. What is the probability of B given A? The probability that the four-sided die is four given that Erica rolls doubles. So let's just think about that a little bit. So the probability of B given A."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "What is the probability of B given A? The probability that the four-sided die is four given that Erica rolls doubles. So let's just think about that a little bit. So the probability of B given A. B given that A is true. So what's this going to be? Well, we've already, so this means we're going to restrict our sample space to the essentially four equally likely outcomes that A has happened."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of B given A. B given that A is true. So what's this going to be? Well, we've already, so this means we're going to restrict our sample space to the essentially four equally likely outcomes that A has happened. So where A is true, I guess I could say. So there's one, two, three, four. And how many of them involve event B being true?"}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, we've already, so this means we're going to restrict our sample space to the essentially four equally likely outcomes that A has happened. So where A is true, I guess I could say. So there's one, two, three, four. And how many of them involve event B being true? Well, the only one of these four that involve event B being true is this one right over here where we've got our doubles. So there is a 1 1 4th probability that if we assume, given that we've gotten doubles, the probability that the four-sided die is a four. So this is a 1 1 4th probability."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "And how many of them involve event B being true? Well, the only one of these four that involve event B being true is this one right over here where we've got our doubles. So there is a 1 1 4th probability that if we assume, given that we've gotten doubles, the probability that the four-sided die is a four. So this is a 1 1 4th probability. And that makes sense. If we've got doubles, and one of them's a four-sided die, we either have doubles at one, doubles at two, doubles at three, or doubles at four. You see that here, doubles one, doubles two, doubles three, doubles four."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is a 1 1 4th probability. And that makes sense. If we've got doubles, and one of them's a four-sided die, we either have doubles at one, doubles at two, doubles at three, or doubles at four. You see that here, doubles one, doubles two, doubles three, doubles four. Well, given that, what's the probability that the four-sided die is a four? Well, that means that's one of these four outcomes where it's a double fours right over here. All right, what is the probability of A and B?"}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "You see that here, doubles one, doubles two, doubles three, doubles four. Well, given that, what's the probability that the four-sided die is a four? Well, that means that's one of these four outcomes where it's a double fours right over here. All right, what is the probability of A and B? The probability that Erica rolls doubles and the second die is four. So this means both A and B happened. Well, let's look at this."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "All right, what is the probability of A and B? The probability that Erica rolls doubles and the second die is four. So this means both A and B happened. Well, let's look at this. Actually, let me write it here. Let me do it in this, let me do it in a new color. So the probability of, I'll write and here in a neutral color, probability of A and B, probability of A and B is equal to, well, now we're looking at, once again, we have 24 equally likely outcomes."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, let's look at this. Actually, let me write it here. Let me do it in this, let me do it in a new color. So the probability of, I'll write and here in a neutral color, probability of A and B, probability of A and B is equal to, well, now we're looking at, once again, we have 24 equally likely outcomes. We have 24 equally likely outcomes. And how many of them involve A and B? Well, to get A and B, you have to have doubles and the four-sided die needs to be a four."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So the probability of, I'll write and here in a neutral color, probability of A and B, probability of A and B is equal to, well, now we're looking at, once again, we have 24 equally likely outcomes. We have 24 equally likely outcomes. And how many of them involve A and B? Well, to get A and B, you have to have doubles and the four-sided die needs to be a four. Essentially, you have to have doubles four. Well, there's only one outcome out of the 24 equally likely outcomes that meets that situation, this one right over here. So there is a 1 4th, sorry, 1 24th probability."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, to get A and B, you have to have doubles and the four-sided die needs to be a four. Essentially, you have to have doubles four. Well, there's only one outcome out of the 24 equally likely outcomes that meets that situation, this one right over here. So there is a 1 4th, sorry, 1 24th probability. So one over 24. What is probability of A times probability of B given A? Well, here we could just go back to our numbers right over here."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "So there is a 1 4th, sorry, 1 24th probability. So one over 24. What is probability of A times probability of B given A? Well, here we could just go back to our numbers right over here. Probability of A, that's going to be one over six. Let me do that in a magenta color. I like to keep my colors, be careful about my colors."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, here we could just go back to our numbers right over here. Probability of A, that's going to be one over six. Let me do that in a magenta color. I like to keep my colors, be careful about my colors. That's 1 6th times probability of B given A. So probability of B given A is 1 4th right over here, times 1 4th, which is, curious enough, 1 24th. 1 24th."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "I like to keep my colors, be careful about my colors. That's 1 6th times probability of B given A. So probability of B given A is 1 4th right over here, times 1 4th, which is, curious enough, 1 24th. 1 24th. What is probability of B times probability of A given B? Well, probability of B, we figured out, is 1 4th. 1 4th."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 24th. What is probability of B times probability of A given B? Well, probability of B, we figured out, is 1 4th. 1 4th. And the probability of A given B is 1 6th, times 1 6th, times 1 6th, which is equal to 1 24th. Now, does it make sense that the probability of A and B is 1 24th, that the probability of A times probability of B given A is 1 24th, and the probability of B times probability of A given B, they're all 1 24th. Is this always going to be the case?"}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "1 4th. And the probability of A given B is 1 6th, times 1 6th, times 1 6th, which is equal to 1 24th. Now, does it make sense that the probability of A and B is 1 24th, that the probability of A times probability of B given A is 1 24th, and the probability of B times probability of A given B, they're all 1 24th. Is this always going to be the case? Well, sure. Think about what probability of A and B means. Well, it means that they both happened."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Is this always going to be the case? Well, sure. Think about what probability of A and B means. Well, it means that they both happened. But that's the same way as saying, well, what's the probability of, let's just say A is happening. Well, now for B and A to happen, it's just going to be that times the probability that B is true given that A is true, because you could kind of say, well, we're already kind of constraining it, we're already multiplying by the probability of A being true, and now we're multiplying by the probability that B is true given A is true. I actually often like to swap these around, just it gets a little bit clearer in my head."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, it means that they both happened. But that's the same way as saying, well, what's the probability of, let's just say A is happening. Well, now for B and A to happen, it's just going to be that times the probability that B is true given that A is true, because you could kind of say, well, we're already kind of constraining it, we're already multiplying by the probability of A being true, and now we're multiplying by the probability that B is true given A is true. I actually often like to swap these around, just it gets a little bit clearer in my head. So this one, let's just write it like this. The probability of B given A times the probability of A. So this is the probability that event A is true, and this is the probability that B, event B is true, given that we know that A is true."}, {"video_title": "Analyzing dependent probability   Probability and Statistics   Khan Academy.mp3", "Sentence": "I actually often like to swap these around, just it gets a little bit clearer in my head. So this one, let's just write it like this. The probability of B given A times the probability of A. So this is the probability that event A is true, and this is the probability that B, event B is true, given that we know that A is true. And it completely makes sense that this is going to be the same thing as the probability of A and B. Clearly, this is the probability of both of these, both A and B happening, and you can go the other way around. The probability of A given B times the probability of B, that would also be, so we're saying B needs to be true, and that given that B is true, that A needs to be true as well."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So for each of those scenarios, four different people could sit in chair two. Now, for each of these scenarios now, so we have 20 scenarios, five times four, we have 20 scenarios where we've seated seat one and seat two. How many people could we now seat in seat three for each of those 20 scenarios? Well, three people haven't sat down yet, so there's three possibilities there. So now there's five times four times three scenarios for seating the first three people. How many people are left for seat four? Well, two people haven't sat down yet, so there's two possibilities."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, three people haven't sat down yet, so there's three possibilities there. So now there's five times four times three scenarios for seating the first three people. How many people are left for seat four? Well, two people haven't sat down yet, so there's two possibilities. So now there's five times four times three times two scenarios of seating the first four seats. And for each of those, how many possibilities are there for the fifth seat? Well, for each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, two people haven't sat down yet, so there's two possibilities. So now there's five times four times three times two scenarios of seating the first four seats. And for each of those, how many possibilities are there for the fifth seat? Well, for each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. And so the number of permutations, the number of, let me write this down, the number of permutations, permutations of seating these five people in five chairs is five factorial. Five factorial, which is equal to five times four times three times two times one, which of course is equal to, let's see, 20 times six, which is equal to 120. And we've already covered this in a previous video."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, for each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. And so the number of permutations, the number of, let me write this down, the number of permutations, permutations of seating these five people in five chairs is five factorial. Five factorial, which is equal to five times four times three times two times one, which of course is equal to, let's see, 20 times six, which is equal to 120. And we've already covered this in a previous video. But now let's do something maybe more interesting, or maybe you might find it less interesting. Let's say that we still have, let's still say we have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. So let's say that we only have three chairs."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we've already covered this in a previous video. But now let's do something maybe more interesting, or maybe you might find it less interesting. Let's say that we still have, let's still say we have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. So let's say that we only have three chairs. So we have chair one, we have chair two, and we have chair three. So how many ways can you have five people where only three of them are going to sit down in these three chairs? And we care which chair they sit in, and I encourage you to pause the video and think about it."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So let's say that we only have three chairs. So we have chair one, we have chair two, and we have chair three. So how many ways can you have five people where only three of them are going to sit down in these three chairs? And we care which chair they sit in, and I encourage you to pause the video and think about it. So I am assuming you have had your go at it. So let's use the same logic. So how many, if we seat them in order, we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one?"}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And we care which chair they sit in, and I encourage you to pause the video and think about it. So I am assuming you have had your go at it. So let's use the same logic. So how many, if we seat them in order, we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we have five different people. Well, five different people could sit in seat one. Well, for each of these scenarios where one person has already sat in seat one, how many people could sit in seat two?"}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So how many, if we seat them in order, we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we have five different people. Well, five different people could sit in seat one. Well, for each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? Well, in each of these scenarios, if one person has sat down, there's four people left who haven't been seated, so four people could sit in seat two. So we have five times four scenarios where we have seated seats one and seat two. Now, for each of those 20 scenarios, how many people could sit in seat three?"}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, for each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? Well, in each of these scenarios, if one person has sat down, there's four people left who haven't been seated, so four people could sit in seat two. So we have five times four scenarios where we have seated seats one and seat two. Now, for each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't, we haven't seaten or sat three of the people yet, so for each of these 20, we could put three different people in seat three. So that gives us five times four times three scenarios. So this is equal to five times four times three scenarios, which is equal to, this is equal to 60."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, for each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't, we haven't seaten or sat three of the people yet, so for each of these 20, we could put three different people in seat three. So that gives us five times four times three scenarios. So this is equal to five times four times three scenarios, which is equal to, this is equal to 60. So there's 60 permutations of sitting five people in three chairs. Now, this, and this is my brain, you know, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because especially, you know, I don't like formulas."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So this is equal to five times four times three scenarios, which is equal to, this is equal to 60. So there's 60 permutations of sitting five people in three chairs. Now, this, and this is my brain, you know, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because especially, you know, I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, you know, when we just did five different people in five different chairs and we cared which seat they sit in, we had this five factorial. Well, you know, factorial is kind of a neat little operation."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I just literally draw it out because especially, you know, I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, you know, when we just did five different people in five different chairs and we cared which seat they sit in, we had this five factorial. Well, you know, factorial is kind of a neat little operation. How can I relate factorial to what we did just now? Well, it looks like we kind of did factorial, but then we stopped. We stopped at, we didn't go times two times one."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Well, you know, factorial is kind of a neat little operation. How can I relate factorial to what we did just now? Well, it looks like we kind of did factorial, but then we stopped. We stopped at, we didn't go times two times one. So one way to think about what we just did is we just did five times four times three times two times one. But of course, we actually didn't do the two times one. So you could take that and you could divide by two times one."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "We stopped at, we didn't go times two times one. So one way to think about what we just did is we just did five times four times three times two times one. But of course, we actually didn't do the two times one. So you could take that and you could divide by two times one. And if you did that, then this two times one would cancel with that two times one and you'd be left with five times four times three. And the whole reason I'm writing this way is that now I could write it in terms of factorial. I could write this as five factorial over two factorial."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So you could take that and you could divide by two times one. And if you did that, then this two times one would cancel with that two times one and you'd be left with five times four times three. And the whole reason I'm writing this way is that now I could write it in terms of factorial. I could write this as five factorial over two factorial. But then you might have the question, well, where did this two come from? You know, I have three seats. Where did this two come from?"}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could write this as five factorial over two factorial. But then you might have the question, well, where did this two come from? You know, I have three seats. Where did this two come from? Well, think about it. I multiplied five times four times three. I kept going until I had that many seats and then I didn't do the remainder."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Where did this two come from? Well, think about it. I multiplied five times four times three. I kept going until I had that many seats and then I didn't do the remainder. So the number of, so the things that I left out, the things that I left out, that was essentially the number of people minus the number of chairs. So I was trying to put five things in three places. So five minus three, that gave me two left over."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I kept going until I had that many seats and then I didn't do the remainder. So the number of, so the things that I left out, the things that I left out, that was essentially the number of people minus the number of chairs. So I was trying to put five things in three places. So five minus three, that gave me two left over. So I could write it like this. I could write it as five, let me use those same colors. I could write it as five factorial over, over five minus three, which of course is two."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "So five minus three, that gave me two left over. So I could write it like this. I could write it as five, let me use those same colors. I could write it as five factorial over, over five minus three, which of course is two. Five minus three factorial. And so another way of thinking about it, if we wanted to generalize, is if you're trying to put, if you're trying to figure out the number of permutations, and there's a bunch of notations for writing this. If you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations, you could put n people in r seats, and there's other notations as well."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "I could write it as five factorial over, over five minus three, which of course is two. Five minus three factorial. And so another way of thinking about it, if we wanted to generalize, is if you're trying to put, if you're trying to figure out the number of permutations, and there's a bunch of notations for writing this. If you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations, you could put n people in r seats, and there's other notations as well. Well this is just going to be n factorial over n minus r factorial. Here n was five, r was three, and five minus three is two. Now, you'll see this in a probability or statistics class, and people might memorize this thing, it seems like this kind of daunting thing."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "If you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations, you could put n people in r seats, and there's other notations as well. Well this is just going to be n factorial over n minus r factorial. Here n was five, r was three, and five minus three is two. Now, you'll see this in a probability or statistics class, and people might memorize this thing, it seems like this kind of daunting thing. I'll just tell you right now, the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook, or what you might see in a class, or when you see this type of formula, you see that it's not coming out of, it's not some type of voodoo magic, but I will tell you that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going to wait, does this formula apply there? What's n, what's r?"}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "Now, you'll see this in a probability or statistics class, and people might memorize this thing, it seems like this kind of daunting thing. I'll just tell you right now, the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook, or what you might see in a class, or when you see this type of formula, you see that it's not coming out of, it's not some type of voodoo magic, but I will tell you that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going to wait, does this formula apply there? What's n, what's r? But if you reason it through, it comes out of straight logic. You don't have to memorize anything, you don't feel like you're just memorizing without understanding, you're just using your deductive reasoning, your logic. And that's especially valuable, because as we'll see, not every scenario is going to fit so cleanly into what we did."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "What's n, what's r? But if you reason it through, it comes out of straight logic. You don't have to memorize anything, you don't feel like you're just memorizing without understanding, you're just using your deductive reasoning, your logic. And that's especially valuable, because as we'll see, not every scenario is going to fit so cleanly into what we did. There might be some tweaks on this, where maybe only person B likes sitting in one of the chairs, or who knows what it might be, and then your formula is going to be useless. So I like reasoning through it like this, but I just showed you this so that you could connect it to a formula that you might see in a lecture, or in a class."}, {"video_title": "Permutation formula   Probability and combinatorics   Probability and Statistics   Khan Academy.mp3", "Sentence": "And that's especially valuable, because as we'll see, not every scenario is going to fit so cleanly into what we did. There might be some tweaks on this, where maybe only person B likes sitting in one of the chairs, or who knows what it might be, and then your formula is going to be useless. So I like reasoning through it like this, but I just showed you this so that you could connect it to a formula that you might see in a lecture, or in a class."}]