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Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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%. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Statistical significance of experiment Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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? |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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? |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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? |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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? |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Hypothesis testing and p-values Inferential statistics Probability and Statistics Khan Academy.mp3 | 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. |
Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3 | 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. |
Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3 | 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. |
Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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? |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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? |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Introduction to experiment design Study design AP Statistics Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Reading pictographs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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? |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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? |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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? |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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? |
Conditional probability and combinations Probability and Statistics Khan Academy.mp3 | 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. |
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