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This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days. And so based on this data, because the experimental probability of being delayed given snowy is so much higher than the experimental probability of just being delayed, I would make the statement that these are not independent. So for these days, are the events delayed and snowy independent? No. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
Here are some summary statistics for each exam. So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable. He did roughly two standard deviations if we were around to the nearest standard deviation, but if you wanted to get precise, he did a little bit better, relatively speaking, on the LSAT. He did 2.1 standard deviations here, while over here he did 1.86 or 1.9 standard deviations. | Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3 |
In this video, we are going to introduce ourselves to the idea of permutations, which is a fancy word for a pretty straightforward concept, which is what are the number of ways that we can arrange things? How many different possibilities are there? And to make that a little bit tangible, let's have an example with, say, a sofa. My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one. And we said, all right, how many different possibilities are, how many different people could sit in seat number one, assuming no one has sat down before? Well, three different people could sit in seat number one. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one. And we said, all right, how many different possibilities are, how many different people could sit in seat number one, assuming no one has sat down before? Well, three different people could sit in seat number one. You can see it right over here. This is where A is sitting in seat number one. This is where B is sitting in seat number one. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Well, three different people could sit in seat number one. You can see it right over here. This is where A is sitting in seat number one. This is where B is sitting in seat number one. And this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
This is where B is sitting in seat number one. And this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is tongue-twister, there's two different possibilities for seat number two. And so you're gonna have two different possibilities here. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is tongue-twister, there's two different possibilities for seat number two. And so you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways of getting that, and so there's two people left who could sit in the second seat. And we saw that right over here where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two? | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
And so you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways of getting that, and so there's two people left who could sit in the second seat. And we saw that right over here where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two? Well, you would multiply. For each of these three, you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three? | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
And so how many different permutations are there for seat number one and seat number two? Well, you would multiply. For each of these three, you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial which is going to be equal to six which is exactly what we've got here. And to appreciate the power of this, let's extend our example. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial which is going to be equal to six which is exactly what we've got here. And to appreciate the power of this, let's extend our example. Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
And to appreciate the power of this, let's extend our example. Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. Five times four is 20. 20 times three is 60. And then 60 times two is 120. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. Five times four is 20. 20 times three is 60. And then 60 times two is 120. And then 120 times one is equal to 120. And once again, that makes a lot of sense. There's five different, if no one sat down, there's five different possibilities for seat number one. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
And then 60 times two is 120. And then 120 times one is equal to 120. And once again, that makes a lot of sense. There's five different, if no one sat down, there's five different possibilities for seat number one. And then for each of those possibilities, there's four people who could sit in seat number two. And then for each of those 20 possibilities in seat numbers one and two, well, there's going to be three people who could sit in seat number three. And for each of these 60 possibilities, there's two people who can sit in seat number four. | Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3 |
Near the end of the last video, I wasn't as articulate as I would like to be, mainly because I think 15 minutes into a video my brain starts to really warm up too much. But what I want to do is restate what I was trying to say. We got this confidence interval. We were confident, I'll rewrite it here, we're confident that there's a 95% chance, I'll just restate the confidence interval. So there's the 95% confidence interval for the mean of this distribution, confidence interval for the mean of that distribution, we got as being 1.91 plus or minus 1.21. And near the end of the video I tried to explain why that is neat. Because here we have, it's a confidence interval for this weird mean of the difference between the sampling mean, it seems kind of confusing, but I just want to restate what we saw in previous videos. | Clarification of confidence interval of difference of means Khan Academy.mp3 |
We were confident, I'll rewrite it here, we're confident that there's a 95% chance, I'll just restate the confidence interval. So there's the 95% confidence interval for the mean of this distribution, confidence interval for the mean of that distribution, we got as being 1.91 plus or minus 1.21. And near the end of the video I tried to explain why that is neat. Because here we have, it's a confidence interval for this weird mean of the difference between the sampling mean, it seems kind of confusing, but I just want to restate what we saw in previous videos. This thing right over here, the mean of the difference of the sampling means, we saw two or three videos ago, is the same thing as the mean of the difference of the means of the sampling distributions. And we know that the mean of each of the sampling distributions is actually the same as the mean of the population distributions. So this is the same thing as the mean of population 1 minus the mean of population 2. | Clarification of confidence interval of difference of means Khan Academy.mp3 |
Because here we have, it's a confidence interval for this weird mean of the difference between the sampling mean, it seems kind of confusing, but I just want to restate what we saw in previous videos. This thing right over here, the mean of the difference of the sampling means, we saw two or three videos ago, is the same thing as the mean of the difference of the means of the sampling distributions. And we know that the mean of each of the sampling distributions is actually the same as the mean of the population distributions. So this is the same thing as the mean of population 1 minus the mean of population 2. And this was the neat result about the last video. This isn't just a 95% confidence interval for this parameter right here, it's actually a 95% confidence interval for this parameter right here. And this is the parameter that we really care about. | Clarification of confidence interval of difference of means Khan Academy.mp3 |
So this is the same thing as the mean of population 1 minus the mean of population 2. And this was the neat result about the last video. This isn't just a 95% confidence interval for this parameter right here, it's actually a 95% confidence interval for this parameter right here. And this is the parameter that we really care about. The true difference in weight loss between going on the diet, going on the low-fat diet, and not going on the low-fat diet. And we have a 95% confidence interval that that difference is between 0.7 and 3.12 pounds, which tells us that we have a 95% confidence interval that you're definitely going to lose some weight. We're not 100% sure, but we're confident that there's a 95% probability of that. | Clarification of confidence interval of difference of means Khan Academy.mp3 |
The first time you're exposed to permutations and combinations, it takes a little bit to get your brain around it. So I think it never hurts to do as many examples. But each incremental example, I'm gonna go, I'm gonna review what we've done before, but hopefully go a little bit further. So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that? | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots? | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot? | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations. There's 360 permutations for putting six people into four chairs, but there's only 15 combinations, because we're no longer counting all of the different arrangements for the same four people in the four chairs. We're saying, hey, if it's the same four people, that is now one combination, and you can see how many ways are there to arrange four people into four chairs? Well, that's the four factorial part right over here, the four factorial part right over here, which is four times three times two times one, which is 24, so we essentially just took the 360 divided by 24 to get 15, but once again, I don't want to, I don't think I can stress this enough. | Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Let's say there are a couple of herbs that people believe help prevent the flu. So to test this, what we do is we wait for flu season, and we randomly assign people to three different groups. And over the course of flu season, we have them either in one group taking herb 1, in the second group taking herb 2, and in the third group they take a placebo. And if you don't know what a placebo is, it's something that to the patient or to the person participating, it feels like they're taking something that you've told them might help them, but it does nothing. It could be just a sugar pill, just so it feels like medicine. The reason why you even go through the effort of giving them something is because oftentimes there's something called a placebo effect, where people get better just because they're being told that they're being given something that will make them better. So this could right here just be a sugar pill. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And if you don't know what a placebo is, it's something that to the patient or to the person participating, it feels like they're taking something that you've told them might help them, but it does nothing. It could be just a sugar pill, just so it feels like medicine. The reason why you even go through the effort of giving them something is because oftentimes there's something called a placebo effect, where people get better just because they're being told that they're being given something that will make them better. So this could right here just be a sugar pill. And a very small amount of sugar, so it really can't affect their actual likelihood of getting the flu. So over here we have a table, and this is actually called a contingency table. And it has on it, in each group, the number that got sick, the number that didn't get sick. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So this could right here just be a sugar pill. And a very small amount of sugar, so it really can't affect their actual likelihood of getting the flu. So over here we have a table, and this is actually called a contingency table. And it has on it, in each group, the number that got sick, the number that didn't get sick. And so we also can, from this, calculate the total number. So in group 1, we had a total of 120 people. In group 2, we had a total of 30 plus 110 is 140 people. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And it has on it, in each group, the number that got sick, the number that didn't get sick. And so we also can, from this, calculate the total number. So in group 1, we had a total of 120 people. In group 2, we had a total of 30 plus 110 is 140 people. And in the placebo group, the group that just got the sugar pill, we had a total of 120 people. And then we could also tabulate the total number of people that got sick. So that's 20 plus 30 is 50, plus 30 is 80. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
In group 2, we had a total of 30 plus 110 is 140 people. And in the placebo group, the group that just got the sugar pill, we had a total of 120 people. And then we could also tabulate the total number of people that got sick. So that's 20 plus 30 is 50, plus 30 is 80. This is the total column right over here. And then the total people that didn't get sick over here is 100 plus 110 is 210, plus 90 is 300. And then the total people here are 380. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So that's 20 plus 30 is 50, plus 30 is 80. This is the total column right over here. And then the total people that didn't get sick over here is 100 plus 110 is 210, plus 90 is 300. And then the total people here are 380. Both this column and this row should add up to 380. So with that out of the way, let's think about how we can use this information in the contingency table and our knowledge of the chi-squared distribution to come up with some conclusion. So let's just make a null hypothesis. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And then the total people here are 380. Both this column and this row should add up to 380. So with that out of the way, let's think about how we can use this information in the contingency table and our knowledge of the chi-squared distribution to come up with some conclusion. So let's just make a null hypothesis. Our null hypothesis is that the herbs do nothing. Let me get some space here. So let's assume the null hypothesis that the herbs do nothing. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So let's just make a null hypothesis. Our null hypothesis is that the herbs do nothing. Let me get some space here. So let's assume the null hypothesis that the herbs do nothing. And then we have our alternative hypothesis, our alternate hypothesis, that the herbs do something. Notice, I don't even care whether they actually improve. I'm just saying they do something. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So let's assume the null hypothesis that the herbs do nothing. And then we have our alternative hypothesis, our alternate hypothesis, that the herbs do something. Notice, I don't even care whether they actually improve. I'm just saying they do something. They might even increase your likelihood of getting the flu. We're not testing whether they're actually good. We're just saying, are they different than just doing nothing? | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
I'm just saying they do something. They might even increase your likelihood of getting the flu. We're not testing whether they're actually good. We're just saying, are they different than just doing nothing? So like we did do with all of our hypothesis tests, let's just assume the null. We're going to assume the null. And given that assumption, figure out if the likelihood of getting data like this or more extreme is really low. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
We're just saying, are they different than just doing nothing? So like we did do with all of our hypothesis tests, let's just assume the null. We're going to assume the null. And given that assumption, figure out if the likelihood of getting data like this or more extreme is really low. And if it is really low, then we will reject the null hypothesis. And in this test, like every hypothesis test, we need a significance level. And let's say our significance level we care about, for whatever reason, is 10% or 0.10. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And given that assumption, figure out if the likelihood of getting data like this or more extreme is really low. And if it is really low, then we will reject the null hypothesis. And in this test, like every hypothesis test, we need a significance level. And let's say our significance level we care about, for whatever reason, is 10% or 0.10. That's the significance level that we care about. Now, to do this, we have to calculate a chi-squared statistic for this contingency table. And to do that, we do it very similar to what we did with the restaurant situation. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And let's say our significance level we care about, for whatever reason, is 10% or 0.10. That's the significance level that we care about. Now, to do this, we have to calculate a chi-squared statistic for this contingency table. And to do that, we do it very similar to what we did with the restaurant situation. We figure out, assuming the null hypothesis, the expected results you would have gotten in each of these cells. You can view each of these entries as a cell. That's what we do with it. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And to do that, we do it very similar to what we did with the restaurant situation. We figure out, assuming the null hypothesis, the expected results you would have gotten in each of these cells. You can view each of these entries as a cell. That's what we do with it. You call each of those entries in Excel also a cell, each of the entries in a table. What we do is we figure out what the expected value would have been if you do assume the null hypothesis. Then we find the squared distance from that expected value, and we, I guess you could call it, normalize it by the expected value, take the sum of all of those differences. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
That's what we do with it. You call each of those entries in Excel also a cell, each of the entries in a table. What we do is we figure out what the expected value would have been if you do assume the null hypothesis. Then we find the squared distance from that expected value, and we, I guess you could call it, normalize it by the expected value, take the sum of all of those differences. And if those differences, those squared differences, are really big, the probability of getting it would be really small, and maybe we'll reject the null hypothesis. So let's just figure out how we can get the expected number. So we're assuming the herbs do nothing. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
Then we find the squared distance from that expected value, and we, I guess you could call it, normalize it by the expected value, take the sum of all of those differences. And if those differences, those squared differences, are really big, the probability of getting it would be really small, and maybe we'll reject the null hypothesis. So let's just figure out how we can get the expected number. So we're assuming the herbs do nothing. So if the herbs do nothing, then we can just figure out that this whole population, they just had nothing happen to them. These herbs were useless. And so we can use this population sample, or I shouldn't call it the population. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So we're assuming the herbs do nothing. So if the herbs do nothing, then we can just figure out that this whole population, they just had nothing happen to them. These herbs were useless. And so we can use this population sample, or I shouldn't call it the population. We should use this sample right here to figure out the expected number of people who would get sick or not sick. And so over here we have 80 out of 380 did not get sick. And I want to be careful. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And so we can use this population sample, or I shouldn't call it the population. We should use this sample right here to figure out the expected number of people who would get sick or not sick. And so over here we have 80 out of 380 did not get sick. And I want to be careful. I just said the word population, but we haven't sampled the whole universe of all people taking this herb. This is a sample. I don't want to confuse you. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And I want to be careful. I just said the word population, but we haven't sampled the whole universe of all people taking this herb. This is a sample. I don't want to confuse you. I was using the population in more of the conversational sense than the statistical sense. But anyway, of our sample, and we're using all of the data, because we're assuming there's no difference. We might as well just use the total data to figure out the expected frequency of getting sick and not getting sick. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
I don't want to confuse you. I was using the population in more of the conversational sense than the statistical sense. But anyway, of our sample, and we're using all of the data, because we're assuming there's no difference. We might as well just use the total data to figure out the expected frequency of getting sick and not getting sick. So 80 divided by 380 did not get sick, and that's 21%. So let me write that over here. So that's 21% of the total. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
We might as well just use the total data to figure out the expected frequency of getting sick and not getting sick. So 80 divided by 380 did not get sick, and that's 21%. So let me write that over here. So that's 21% of the total. And then this would be 79% if we just subtract 1 minus 21. We could divide 300 by 380. We should get 79% as well. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So that's 21% of the total. And then this would be 79% if we just subtract 1 minus 21. We could divide 300 by 380. We should get 79% as well. So one would expect that 21% of each of your total, based on the total sample right over here, our best guess is that 21% should be getting sick and 79% should not be getting sick. So let's look at it for each of these groups. If we assume that 21% of these 120 people should have gotten sick, what would have been the expected value right over here? | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
We should get 79% as well. So one would expect that 21% of each of your total, based on the total sample right over here, our best guess is that 21% should be getting sick and 79% should not be getting sick. So let's look at it for each of these groups. If we assume that 21% of these 120 people should have gotten sick, what would have been the expected value right over here? So let's just multiply this 21% times 120. That gets us to 25.3 people should have gotten sick. So the expected, so let me write it over here. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
If we assume that 21% of these 120 people should have gotten sick, what would have been the expected value right over here? So let's just multiply this 21% times 120. That gets us to 25.3 people should have gotten sick. So the expected, so let me write it over here. I'll do expected in yellow. So the expected right over here, if you assume that 21% of each group should have gotten sick, is that you would have expected 25.3 people to get sick in group 1, in herb 1 group, and then the remainder will not get sick. So let's just subtract, or I could actually multiply 79% times 120. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So the expected, so let me write it over here. I'll do expected in yellow. So the expected right over here, if you assume that 21% of each group should have gotten sick, is that you would have expected 25.3 people to get sick in group 1, in herb 1 group, and then the remainder will not get sick. So let's just subtract, or I could actually multiply 79% times 120. Either one of those will be good. But let me just take 120 minus 25.3, and then I get 94.7. So you would have expected 94.7 to not get sick. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So let's just subtract, or I could actually multiply 79% times 120. Either one of those will be good. But let me just take 120 minus 25.3, and then I get 94.7. So you would have expected 94.7 to not get sick. So this is expected again. 94.7 to not get sick. And now let's do that for each of these groups. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So you would have expected 94.7 to not get sick. So this is expected again. 94.7 to not get sick. And now let's do that for each of these groups. So once again, group 2, you would have expected 21% to get sick, 21% of the total people in that group, so that's 140. So that's 29.4. And then the remainder, 140 minus 29.4, should not have gotten sick. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And now let's do that for each of these groups. So once again, group 2, you would have expected 21% to get sick, 21% of the total people in that group, so that's 140. So that's 29.4. And then the remainder, 140 minus 29.4, should not have gotten sick. So that gets us this right here. We have 20, 29.4 should have gotten sick if the herbs did nothing. And then over here, we would have 110.6 should not have gotten sick. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And then the remainder, 140 minus 29.4, should not have gotten sick. So that gets us this right here. We have 20, 29.4 should have gotten sick if the herbs did nothing. And then over here, we would have 110.6 should not have gotten sick. And these are pretty close. So just looking at the numbers, it looks like this herb doesn't do too much relative to all of the groups combined. And then in the placebo group, let's see what happens. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And then over here, we would have 110.6 should not have gotten sick. And these are pretty close. So just looking at the numbers, it looks like this herb doesn't do too much relative to all of the groups combined. And then in the placebo group, let's see what happens. You have 30, sorry, we expect 21% to get sick, 21% of our group of 120, so it's 25.2. So this right over here. And actually, I should make this, this should be a 25 point, since we're rounding. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And then in the placebo group, let's see what happens. You have 30, sorry, we expect 21% to get sick, 21% of our group of 120, so it's 25.2. So this right over here. And actually, I should make this, this should be a 25 point, since we're rounding. Actually, these will be the same number over here. So I said 21%, but it's 21 point something, something, something. The group sizes are the same. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And actually, I should make this, this should be a 25 point, since we're rounding. Actually, these will be the same number over here. So I said 21%, but it's 21 point something, something, something. The group sizes are the same. So we should expect the same proportion to get sick. So I'll say 25.3, just to make it consistent. The reason why I got 25.2 just now is because I lost some of the trailing decimals over here. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
The group sizes are the same. So we should expect the same proportion to get sick. So I'll say 25.3, just to make it consistent. The reason why I got 25.2 just now is because I lost some of the trailing decimals over here. But since I had them over here, I'm going to use them over here as well. And then over here, in this group, you would expect 94.7 to get sick. So if you just actually relied on this data, it looks like ERB2 is actually, to some degree, even worse than the, oh, no, no, I take that back. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
The reason why I got 25.2 just now is because I lost some of the trailing decimals over here. But since I had them over here, I'm going to use them over here as well. And then over here, in this group, you would expect 94.7 to get sick. So if you just actually relied on this data, it looks like ERB2 is actually, to some degree, even worse than the, oh, no, no, I take that back. It's not worse, because you would have expected a small number, and a lot of people got sick here. So this is the placebo. Well, anyway, we don't want to make judgments just staring at the numbers. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So if you just actually relied on this data, it looks like ERB2 is actually, to some degree, even worse than the, oh, no, no, I take that back. It's not worse, because you would have expected a small number, and a lot of people got sick here. So this is the placebo. Well, anyway, we don't want to make judgments just staring at the numbers. Let's figure out our chi-squared statistic. And to do that, let's get our statistic, our chi-squared statistic. I'll write it like this, maybe for fun. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
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