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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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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. | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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? | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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. | Example estimating from regression line.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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? | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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%. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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? | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | ck12.org More empirical rule and z-score practice Probability and Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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%. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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? | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
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