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And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance, so from i equals 1 to n of our unbiased sample variance, so we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance, so from i equals 1 to n of our unbiased sample variance, so we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy. It's actually dependent on how that population is actually distributed. So in statistics, we just define the sample standard deviation. And the one that we typically use is based on the square root of the unbiased sample variance. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
The mean emission of all engines of a new design needs to be below 20 parts per million if the design is to meet new emission requirements. 10 engines are manufactured for testing purposes, and the emission level of each is determined. The emission data is, and they give us 10 data points for the 10 test engines, and I went ahead and calculated the mean of these data points. The sample mean is 17.17, and the standard deviation of these 10 data points right here is 2.98, the sample standard deviation. Does the data supply sufficient evidence to conclude that this type of engine meets the new standard? Assume we are willing to risk a type 1 error with a probability of 0.01, and we'll touch on this in a second. Before we do that, let's just define what our null hypothesis and our alternative hypothesis are going to be. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
The sample mean is 17.17, and the standard deviation of these 10 data points right here is 2.98, the sample standard deviation. Does the data supply sufficient evidence to conclude that this type of engine meets the new standard? Assume we are willing to risk a type 1 error with a probability of 0.01, and we'll touch on this in a second. Before we do that, let's just define what our null hypothesis and our alternative hypothesis are going to be. Our null hypothesis can be that we don't meet the standards, that we just barely don't meet the standards, that the mean of our new engines is exactly 20 parts per million, and you essentially want the best possible value where we still don't meet, or the lowest possible value where we still don't meet the standard. Then our alternative hypothesis is no, we do meet the standard, that the true mean for our new engines is below 20 parts per million. To see if the data that we have is sufficient, what we're going to do is assume that this is true. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
Before we do that, let's just define what our null hypothesis and our alternative hypothesis are going to be. Our null hypothesis can be that we don't meet the standards, that we just barely don't meet the standards, that the mean of our new engines is exactly 20 parts per million, and you essentially want the best possible value where we still don't meet, or the lowest possible value where we still don't meet the standard. Then our alternative hypothesis is no, we do meet the standard, that the true mean for our new engines is below 20 parts per million. To see if the data that we have is sufficient, what we're going to do is assume that this is true. Given that this is true, if we assume this is true, and the probability of this occurring, and the probability of getting a sample mean of that, is less than 1%, then we will reject the null hypothesis. We are going to reject our null hypothesis if the probability of getting a sample mean of 17.17, given the null hypothesis is true, is less than 1%. Notice, if we do it this way, there will be less than a 1% chance that we are making a type 1 error. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
To see if the data that we have is sufficient, what we're going to do is assume that this is true. Given that this is true, if we assume this is true, and the probability of this occurring, and the probability of getting a sample mean of that, is less than 1%, then we will reject the null hypothesis. We are going to reject our null hypothesis if the probability of getting a sample mean of 17.17, given the null hypothesis is true, is less than 1%. Notice, if we do it this way, there will be less than a 1% chance that we are making a type 1 error. A type 1 error is that we're rejecting it even though it's true. Here, there's only a 1% chance, or less than a 1% chance, that we will reject it if it is true. The next thing we have to think about is what type of distribution we should think about. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
Notice, if we do it this way, there will be less than a 1% chance that we are making a type 1 error. A type 1 error is that we're rejecting it even though it's true. Here, there's only a 1% chance, or less than a 1% chance, that we will reject it if it is true. The next thing we have to think about is what type of distribution we should think about. The first thing that rings in my brain is we only have 10 samples here. We have a small sample size right over here. We're going to be dealing with a t distribution and a t statistic. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
The next thing we have to think about is what type of distribution we should think about. The first thing that rings in my brain is we only have 10 samples here. We have a small sample size right over here. We're going to be dealing with a t distribution and a t statistic. With that said, let's think of it this way. We can come up with a t statistic that is based on these statistics right over here. The t statistic is going to be 17.17, our sample mean, minus the assumed population mean, minus 20 parts per million, over our sample standard deviation, 2.98. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
We're going to be dealing with a t distribution and a t statistic. With that said, let's think of it this way. We can come up with a t statistic that is based on these statistics right over here. The t statistic is going to be 17.17, our sample mean, minus the assumed population mean, minus 20 parts per million, over our sample standard deviation, 2.98. This is really the definition of the t statistic. Hopefully, we see now that this really comes from a z score. The t distribution is kind of an engineered version of the normal distribution using t statistics. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
The t statistic is going to be 17.17, our sample mean, minus the assumed population mean, minus 20 parts per million, over our sample standard deviation, 2.98. This is really the definition of the t statistic. Hopefully, we see now that this really comes from a z score. The t distribution is kind of an engineered version of the normal distribution using t statistics. 2.98 divided by the square root of our sample size. We have 10 samples, so it's divided by the square root of 10. This value right here, let me get the calculator out just to get a value in place there. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
The t distribution is kind of an engineered version of the normal distribution using t statistics. 2.98 divided by the square root of our sample size. We have 10 samples, so it's divided by the square root of 10. This value right here, let me get the calculator out just to get a value in place there. This is going to be 17.17 minus 20, close parentheses, divided by 2.98 divided by the square root of 10, and then close parentheses. It is almost exactly negative 3. Our t statistic is almost exactly negative 3. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
This value right here, let me get the calculator out just to get a value in place there. This is going to be 17.17 minus 20, close parentheses, divided by 2.98 divided by the square root of 10, and then close parentheses. It is almost exactly negative 3. Our t statistic is almost exactly negative 3. Negative 3.00. What we need to figure out, because t statistics have a t distribution, what we need to figure out is the probability of getting this t statistic, or a value of t equal to this or less than this, is that less than 1%. The way we can think about it is we have a t distribution. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
Our t statistic is almost exactly negative 3. Negative 3.00. What we need to figure out, because t statistics have a t distribution, what we need to figure out is the probability of getting this t statistic, or a value of t equal to this or less than this, is that less than 1%. The way we can think about it is we have a t distribution. Let's say we have a normalized t distribution. The distribution of all the t statistics would be a normalized t distribution. This is the mean of the t distribution. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
The way we can think about it is we have a t distribution. Let's say we have a normalized t distribution. The distribution of all the t statistics would be a normalized t distribution. This is the mean of the t distribution. There's going to be some threshold t value right here. This is our threshold t value. This is some threshold t value right over here. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is the mean of the t distribution. There's going to be some threshold t value right here. This is our threshold t value. This is some threshold t value right over here. We want a threshold t value such that any t value less than that, or the probability of getting a t value less than that is 1%. That entire area in yellow is 1%. We need to figure out a threshold t value there. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is some threshold t value right over here. We want a threshold t value such that any t value less than that, or the probability of getting a t value less than that is 1%. That entire area in yellow is 1%. We need to figure out a threshold t value there. This is for a t distribution that has n equal to 10, or 10 minus 1 equals 9 degrees of freedom. What is that threshold value over there? Notice that this is a one-sided distribution. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
We need to figure out a threshold t value there. This is for a t distribution that has n equal to 10, or 10 minus 1 equals 9 degrees of freedom. What is that threshold value over there? Notice that this is a one-sided distribution. We care about this is 1%, and then all of this stuff over here is going to be 99%. Just the way most t tables are set up, they don't set up a negative t value that is oriented like this. They'll just give you a positive t value that's oriented the other way. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
Notice that this is a one-sided distribution. We care about this is 1%, and then all of this stuff over here is going to be 99%. Just the way most t tables are set up, they don't set up a negative t value that is oriented like this. They'll just give you a positive t value that's oriented the other way. The way t tables, and I have one that we're going to use in a second right over here, the way t tables are set up is you have your distribution like this, and they will just give a positive t value. They will give a positive t value over here, some threshold value, where the probability of getting a t value above that is going to be 1%, and the probability of getting a t value below that is going to be 99%. You can see that we know t distributions are symmetric around their mean, so whatever value this is, if this number is 2, then this value is just going to be negative 2. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
They'll just give you a positive t value that's oriented the other way. The way t tables, and I have one that we're going to use in a second right over here, the way t tables are set up is you have your distribution like this, and they will just give a positive t value. They will give a positive t value over here, some threshold value, where the probability of getting a t value above that is going to be 1%, and the probability of getting a t value below that is going to be 99%. You can see that we know t distributions are symmetric around their mean, so whatever value this is, if this number is 2, then this value is just going to be negative 2. We just have to keep that in mind, but the t tables actually help us figure out this value. Let's figure out a t value where the probability of getting a t value below that is 99%. Once again, this is going to be a one-sided situation. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
You can see that we know t distributions are symmetric around their mean, so whatever value this is, if this number is 2, then this value is just going to be negative 2. We just have to keep that in mind, but the t tables actually help us figure out this value. Let's figure out a t value where the probability of getting a t value below that is 99%. Once again, this is going to be a one-sided situation. Let's look at that over here. One-sided, this is just straight from Wikipedia, we want the cumulative distribution below that t value to be 99%. We have it right over here, 99%. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
Once again, this is going to be a one-sided situation. Let's look at that over here. One-sided, this is just straight from Wikipedia, we want the cumulative distribution below that t value to be 99%. We have it right over here, 99%. We have 9 degrees of freedom. We have 10 data points, 10 minus 1 is 9. 9 degrees of freedom. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
We have it right over here, 99%. We have 9 degrees of freedom. We have 10 data points, 10 minus 1 is 9. 9 degrees of freedom. Our threshold t value here is 2.821, so our threshold t value in the case that we care about, just flip this over, it's completely symmetric, is negative 2.821. What this tells us is the probability of getting a t value less than negative 2.821 is going to be 1%. Now, we got a value that's a good bit less than that. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
9 degrees of freedom. Our threshold t value here is 2.821, so our threshold t value in the case that we care about, just flip this over, it's completely symmetric, is negative 2.821. What this tells us is the probability of getting a t value less than negative 2.821 is going to be 1%. Now, we got a value that's a good bit less than that. We got a t value of negative 3. We got a t value right here, our t statistic of negative 3 right over here. That definitely goes into our, I guess you could call it our area of rejection. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now, we got a value that's a good bit less than that. We got a t value of negative 3. We got a t value right here, our t statistic of negative 3 right over here. That definitely goes into our, I guess you could call it our area of rejection. This is even less probable than the 1%. We could even figure it out, that the area over here, the probability of getting a t statistic less than negative 3 is even less than, it's a subset of this yellow area right over here. Because the probability of getting the t statistic that we actually got is less than 1%, we can safely reject the null hypothesis and feel pretty good about our alternate hypothesis right over here, that we do meet the emission standards. | Small sample hypothesis test Inferential statistics Probability and Statistics Khan Academy.mp3 |
And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker. You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just write J. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts. So we counted the jack of hearts twice, so we have to subtract one out of that. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts. So we counted the jack of hearts twice, so we have to subtract one out of that. So it's gonna be four plus 13 minus one, or this is going to be 1650 seconds, and both of these things are divisible by four, so this is going to be the same thing as, divide 16 by four, you get four. 52 divided by four is 13. So there's a 413th chance that you get a jack or a hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Which conditions for constructing this confidence interval did Ali's sample meet? So pause this video, and you can select more than one of these. All right, now let's work through this together. So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval. But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval. But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So n is equal to 30. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample? So that would be the random condition. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample? So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that? | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal. And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal. And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150. So his sample size was 30 out of 150, which is the same thing as 1 5th of the population, which is the same thing as 20%. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150. So his sample size was 30 out of 150, which is the same thing as 1 5th of the population, which is the same thing as 20%. And since this is greater than 10%, we are violating the independence condition. We could have met the independence condition if he was sampling with replacement, which it doesn't seem like he is, or if this thing right over here was less than 10%. But we're not meeting that, so we cannot feel good about that constraint. | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
And so I'm randomly sampling a bunch of people, measuring their height, measuring their weight, and then for each person, I'm plotting a point that represents their height and weight combination. So for example, let's say I measure someone who is 60 inches tall, that would be five feet tall, and they weigh 100 pounds. And so I'd go to 60 inches and then 100 pounds right over there. So that point right over there is the point 60 comma 100. One way to think about it, height we could say is being measured on our x-axis or plotted along our x-axis, and then weight along our y-axis. And so this point from this person is the point 60 comma 100 representing 60 inches, 100 pounds. And so so far I've done it for one, two, three, four, five, six, seven, eight, nine people. | Introduction to residuals and least squares regression.mp3 |
So that point right over there is the point 60 comma 100. One way to think about it, height we could say is being measured on our x-axis or plotted along our x-axis, and then weight along our y-axis. And so this point from this person is the point 60 comma 100 representing 60 inches, 100 pounds. And so so far I've done it for one, two, three, four, five, six, seven, eight, nine people. And I could keep going, but even with this, I could say, well look, it looks like there's a roughly linear relationship here. It looks like it's positive, that generally speaking, as height increases, so does weight. Maybe I could try to put a line that can approximate this trend. | Introduction to residuals and least squares regression.mp3 |
And so so far I've done it for one, two, three, four, five, six, seven, eight, nine people. And I could keep going, but even with this, I could say, well look, it looks like there's a roughly linear relationship here. It looks like it's positive, that generally speaking, as height increases, so does weight. Maybe I could try to put a line that can approximate this trend. So let me try to do that. So this is my line tool. I could think about a bunch of lines. | Introduction to residuals and least squares regression.mp3 |
Maybe I could try to put a line that can approximate this trend. So let me try to do that. So this is my line tool. I could think about a bunch of lines. Something like this seems like it would be, you'd be, most of the data is below the line, so that seems like it's not right. I could do something like, I could do something like this, but that doesn't seem like a good fit. Most of the data seems to be above the line. | Introduction to residuals and least squares regression.mp3 |
I could think about a bunch of lines. Something like this seems like it would be, you'd be, most of the data is below the line, so that seems like it's not right. I could do something like, I could do something like this, but that doesn't seem like a good fit. Most of the data seems to be above the line. And so, and once again, I'm just eyeballing it here. In the future you will learn better methods of finding a better fit, but this, something like this, and I'm just eyeballing it, looks about right. So that line, you could view this as a regression line. | Introduction to residuals and least squares regression.mp3 |
Most of the data seems to be above the line. And so, and once again, I'm just eyeballing it here. In the future you will learn better methods of finding a better fit, but this, something like this, and I'm just eyeballing it, looks about right. So that line, you could view this as a regression line. We could view this as y equals mx plus b, where we would have to figure out the slope and the y-intercept, and we could figure it out based on what I just drew, or we could even think of this as weight. Weight is equal to our slope times height, times height plus whatever our y-intercept is, or you could think of it, if you think of the vertical axis as the weight axis, you could think of it as your weight intercept. But either way, this is the model that I'm just, through eyeballing, this is my regression line, something that I'm trying to fit to these points. | Introduction to residuals and least squares regression.mp3 |
So that line, you could view this as a regression line. We could view this as y equals mx plus b, where we would have to figure out the slope and the y-intercept, and we could figure it out based on what I just drew, or we could even think of this as weight. Weight is equal to our slope times height, times height plus whatever our y-intercept is, or you could think of it, if you think of the vertical axis as the weight axis, you could think of it as your weight intercept. But either way, this is the model that I'm just, through eyeballing, this is my regression line, something that I'm trying to fit to these points. But clearly it can't go through, one line won't be able to go through all of these points. There is going to be, for each point, some difference, or not for all of them, but for many of them, some difference between the actual and what would have been predicted by the line. And that idea, the difference between the actual four-point and what would have been predicted given, say, the height, that is called a residual. | Introduction to residuals and least squares regression.mp3 |
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