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We have a pretty big distance right over here. It would be a negative residual. And so this point is definitely bringing down the r, and it's definitely bringing down the slope of the regression line. If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. The only reason why the line isn't doing that is it's trying to get close to this point right over here. So if we remove this outlier, our r would increase. So r would increase. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. The only reason why the line isn't doing that is it's trying to get close to this point right over here. So if we remove this outlier, our r would increase. So r would increase. And also the slope of our line would increase. And slope would increase. We'd have a better fit to this positively correlated data. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
So r would increase. And also the slope of our line would increase. And slope would increase. We'd have a better fit to this positively correlated data. And we would no longer have this point dragging the slope down anymore. So let's see which choices apply. The coefficient of determination, r squared, would increase. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
We'd have a better fit to this positively correlated data. And we would no longer have this point dragging the slope down anymore. So let's see which choices apply. The coefficient of determination, r squared, would increase. Well, if r would increase, then squaring that value would increase as well, so I will circle that. The coefficient, the correlation coefficient r would get close to zero. No, in fact, it would get closer to one because we would have a better fit here. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
The coefficient of determination, r squared, would increase. Well, if r would increase, then squaring that value would increase as well, so I will circle that. The coefficient, the correlation coefficient r would get close to zero. No, in fact, it would get closer to one because we would have a better fit here. And so I will rule that out. The slope of the least squares regression line would increase. Yes, indeed, this point, this outlier is pulling it down. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
No, in fact, it would get closer to one because we would have a better fit here. And so I will rule that out. The slope of the least squares regression line would increase. Yes, indeed, this point, this outlier is pulling it down. If you take it out, it'll allow the slope to increase. So I will circle that as well. Let's do another example. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
Yes, indeed, this point, this outlier is pulling it down. If you take it out, it'll allow the slope to increase. So I will circle that as well. Let's do another example. The scatter plot below displays a set of bivariate data along with its least squares regression line. Same idea. Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
Let's do another example. The scatter plot below displays a set of bivariate data along with its least squares regression line. Same idea. Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. So what would happen this time? So as is, without removing this outlier, we have a negative slope for the regression line. So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. So what would happen this time? So as is, without removing this outlier, we have a negative slope for the regression line. So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. Without even removing the outlier, we know it's not going to be negative one. If r was exactly negative one, then it would be a downward-sloping line that went exactly through all of the points. But if we remove this point, what's going to happen? | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. Without even removing the outlier, we know it's not going to be negative one. If r was exactly negative one, then it would be a downward-sloping line that went exactly through all of the points. But if we remove this point, what's going to happen? Well, this least squares regression is being pulled down here by this outlier. So if you were to remove this point, the least squares regression line could move up on the left-hand side, and so you'll probably have a line that looks more like that. And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
But if we remove this point, what's going to happen? Well, this least squares regression is being pulled down here by this outlier. So if you were to remove this point, the least squares regression line could move up on the left-hand side, and so you'll probably have a line that looks more like that. And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. And so clearly, the new line that I drew after removing the outlier, this has a more negative slope. So removing the outlier would decrease r. r would get closer to negative one. It would be closer to being a perfect negative correlation. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. And so clearly, the new line that I drew after removing the outlier, this has a more negative slope. So removing the outlier would decrease r. r would get closer to negative one. It would be closer to being a perfect negative correlation. And also, it would decrease the slope. Decrease the slope, which choices match that? The coefficient of determination r squared would decrease. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
It would be closer to being a perfect negative correlation. And also, it would decrease the slope. Decrease the slope, which choices match that? The coefficient of determination r squared would decrease. So let's be very careful. r was already negative. If we decrease it, it's going to become more negative. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
The coefficient of determination r squared would decrease. So let's be very careful. r was already negative. If we decrease it, it's going to become more negative. If you square something that is more negative, it's not going to become smaller. Let's say before you remove the data point, r was, I'm just gonna make up a value, let's say it was negative 0.4, and then after removing the outlier, r becomes more negative, and it's going to be equal to negative 0.5. Well, if you square this, this would be positive 0.16, while this would be positive 0.25. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
If we decrease it, it's going to become more negative. If you square something that is more negative, it's not going to become smaller. Let's say before you remove the data point, r was, I'm just gonna make up a value, let's say it was negative 0.4, and then after removing the outlier, r becomes more negative, and it's going to be equal to negative 0.5. Well, if you square this, this would be positive 0.16, while this would be positive 0.25. So if r is already negative, and if you make it more negative, it would not decrease r squared. It actually would increase r squared. So I will rule this one out. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
Well, if you square this, this would be positive 0.16, while this would be positive 0.25. So if r is already negative, and if you make it more negative, it would not decrease r squared. It actually would increase r squared. So I will rule this one out. The slope of the least squares regression line would increase. No, it's going to decrease. It's going to be a stronger negative correlation. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
So I will rule this one out. The slope of the least squares regression line would increase. No, it's going to decrease. It's going to be a stronger negative correlation. Rule that one out. The y-intercept of the least squares regression line would increase. Yes, by getting rid of this outlier, you could think of it as the left side of this line is going to increase, or another way to think about it, the slope of this line is going to decrease. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
It's going to be a stronger negative correlation. Rule that one out. The y-intercept of the least squares regression line would increase. Yes, by getting rid of this outlier, you could think of it as the left side of this line is going to increase, or another way to think about it, the slope of this line is going to decrease. It's going to become more negative. We know that the least squares regression line will always go through the mean of both variables. So we're just gonna pivot around the mean of both variables, which would mean that the y-intercept will go higher. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
Here is computer output from Elise Square's regression analysis on her sample. So just to be clear what's going on, she took a sample of phones, they're not telling us exactly how many, but she took a number of phones and she found a linear relationship between processor speed and prices. So this is price right over here and this is processor speed right over here. And then she plotted her sample for every phone would be a data point. And so you see that. And then she put those data points into her computer and was able to come up with a line, a regression line for her sample. And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And then she plotted her sample for every phone would be a data point. And so you see that. And then she put those data points into her computer and was able to come up with a line, a regression line for her sample. And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there. And for her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there. And for her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe. If she were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe. If she were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate. And just because she sees this positive linear relationship in her sample doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship. And so that's why she's doing this hypothesis test. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
But since this is a sample, it's just an estimate. And just because she sees this positive linear relationship in her sample doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship. And so that's why she's doing this hypothesis test. And in a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population. So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And so that's why she's doing this hypothesis test. And in a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population. So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. So the regression line might look something like that. Where the equation of the regression line for the population, Y hat, would be alpha plus beta times X. And so our null hypothesis is that beta is equal to zero. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. So the regression line might look something like that. Where the equation of the regression line for the population, Y hat, would be alpha plus beta times X. And so our null hypothesis is that beta is equal to zero. And the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero. Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And so our null hypothesis is that beta is equal to zero. And the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero. Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true, assuming this is the actual slope of the population regression line, I guess you could think about it, what is the probability of us getting this result right over here? And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true, assuming this is the actual slope of the population regression line, I guess you could think about it, what is the probability of us getting this result right over here? And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. And for this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this. So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. And for this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this. So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. Maybe they just figured out for us that this probability is 0.004. And we have to be very, very careful here, because here, they're actually giving us, I guess you could call it a two-sided p-value. If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. Maybe they just figured out for us that this probability is 0.004. And we have to be very, very careful here, because here, they're actually giving us, I guess you could call it a two-sided p-value. If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? So if this is t equals zero right here in the middle, and this is 2.999, we care about this region. We care about this right tail. This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? So if this is t equals zero right here in the middle, and this is 2.999, we care about this region. We care about this right tail. This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? So it's giving us both of these areas. So if you want the p-value for this scenario, we would just look at this. And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? So it's giving us both of these areas. So if you want the p-value for this scenario, we would just look at this. And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. So this is going to be equal to 0.002. And what you do in any significance test is then compare your p-value to your level of significance. And so if you look at 0.002 and compare it to 0.01, which of these is greater? | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. So this is going to be equal to 0.002. And what you do in any significance test is then compare your p-value to your level of significance. And so if you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first your eyes might say, hey, two is greater than one, but this is 2,000th versus 1,000th. This is 10,000th right over here. So in this situation, our p-value is less than our level of significance. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
And so if you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first your eyes might say, hey, two is greater than one, but this is 2,000th versus 1,000th. This is 10,000th right over here. So in this situation, our p-value is less than our level of significance. And so we're saying, hey, the probability of getting a result this extreme or more extreme is so low if we assume our null hypothesis that in this situation we will reject, we will decide to reject our null hypothesis, which would suggest the alternative. So is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones? Yes. | Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3 |
Let's imagine ourselves in some type of a strange casino with very strange games. And you walk up to a table, and on that table there is an empty bag. And the guy who runs the table says, look, I've got some marbles here, three green marbles, two orange marbles, and I'm going to stick them in the bag. And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play? | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game? | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game. | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game. Now, one thing I will let you think about is, would you want to play this game if you could replace the green marble, the first pick after the first pick? If you could replace the green marble, would you want to pick? Would you want to play the game in that scenario? | Dependent probability introduction Probability and Statistics Khan Academy.mp3 |
And under which situations does it look skewed right? So does it look something like this? And under which situations does it look skewed left? Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions. So with that in our minds, let's do some examples here. So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions. So with that in our minds, let's do some examples here. So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
All right, so right over here, we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. So the mean is going to be 12%. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. So the mean is going to be 12%. So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So the mean is going to be 12%. So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail. So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
You're gonna have a big, long tail. So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it out. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it out. All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Once again, pause this video and see if you can figure it out. All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88. So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10? | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So P is 0.88. So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
This is actually going to be over 100. So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
I didn't even have to calculate it. I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb. Is it gonna be perfectly normal? No. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb. Is it gonna be perfectly normal? No. In fact, if we didn't have this rule of thumb to kind of draw the line, some might even argue that, well, we still have a longer tail to the left than we do to the right. Maybe it's skewed to the left. But using this threshold, using this rule of thumb, which is the standard in statistics, this would be viewed as approximately normal. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
My idea is to make the background color of my website yellow. But after making that change, how do I feel good about this actually having the intended consequence? Well, that's where significance tests come into play. What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. | P-values and significance tests AP Statistics Khan Academy.mp3 |
What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change. | P-values and significance tests AP Statistics Khan Academy.mp3 |
And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% or 10%. You might see other ones, but we're gonna set a significance level for this particular case. | P-values and significance tests AP Statistics Khan Academy.mp3 |
So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% or 10%. You might see other ones, but we're gonna set a significance level for this particular case. Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what is the probability of getting a sample with the statistics that we get? And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. | P-values and significance tests AP Statistics Khan Academy.mp3 |
You might see other ones, but we're gonna set a significance level for this particular case. Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what is the probability of getting a sample with the statistics that we get? And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. | P-values and significance tests AP Statistics Khan Academy.mp3 |
And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. | P-values and significance tests AP Statistics Khan Academy.mp3 |
In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. | P-values and significance tests AP Statistics Khan Academy.mp3 |
We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going to be. And then we decide whether we can reject the null hypothesis. | P-values and significance tests AP Statistics Khan Academy.mp3 |
It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going to be. And then we decide whether we can reject the null hypothesis. So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. | P-values and significance tests AP Statistics Khan Academy.mp3 |
And then we decide whether we can reject the null hypothesis. So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypothesis. | P-values and significance tests AP Statistics Khan Academy.mp3 |
If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypothesis. And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. | P-values and significance tests AP Statistics Khan Academy.mp3 |
I would just say that we do not, do not reject, reject the null hypothesis. And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which you can interpret as saying that, hey, if we assume the null hypothesis is true, that there's no change due to making the background yellow, I would have a 50% chance of getting this result. And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. | P-values and significance tests AP Statistics Khan Academy.mp3 |
And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which you can interpret as saying that, hey, if we assume the null hypothesis is true, that there's no change due to making the background yellow, I would have a 50% chance of getting this result. And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very, very, very clear. | P-values and significance tests AP Statistics Khan Academy.mp3 |
And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very, very, very clear. Our p-value, the thing that we're using to decide whether or not we reject the null hypothesis, this is the probability of getting your sample statistics given that the null hypothesis is true. Sometimes people confuse this and they say, hey, is this the probability that the null hypothesis is true given the sample, given the sample statistics that we got? And I would say clearly, no, that is not the case. | P-values and significance tests AP Statistics Khan Academy.mp3 |
Now, there's one last point of clarification that I wanna make very, very, very clear. Our p-value, the thing that we're using to decide whether or not we reject the null hypothesis, this is the probability of getting your sample statistics given that the null hypothesis is true. Sometimes people confuse this and they say, hey, is this the probability that the null hypothesis is true given the sample, given the sample statistics that we got? And I would say clearly, no, that is not the case. We are not trying to gauge the probability that the null hypothesis is true or not. What we are trying to do is say, hey, if we assume the null hypothesis were true, what is the probability that we got the result that we did for our sample? And if that probability is low, if it's below some threshold that we set ahead of time, then we decide to reject the null hypothesis and say that we have evidence for the alternative. | P-values and significance tests AP Statistics Khan Academy.mp3 |
Now before I calculate the correlation coefficient, let's just make sure we understand some of these other statistics that they've given us. So we assume that these are samples of the x and the corresponding y from a broader population. And so we have the sample mean for x and the sample standard deviation for x. The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things. All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Until you realize a few things. All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
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