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And let's say the least squares regression line looks something like this. And a least squares regression line comes from trying to minimize the square distance between the line and all of these points. And then this is giving us information on that least squares regression line. And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column. The constant coefficient tells us essentially what is the y-intercept here, so 2.544. And then the coefficient on the caffeine, this is one way of thinking about, well, for every incremental increase in caffeine, how much does the time studying increase? Or you might recognize this as the slope of the least squares regression line. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column. The constant coefficient tells us essentially what is the y-intercept here, so 2.544. And then the coefficient on the caffeine, this is one way of thinking about, well, for every incremental increase in caffeine, how much does the time studying increase? Or you might recognize this as the slope of the least squares regression line. So this is the slope, and this would be equal to 0.164. Now this information right over here, it tells us how well our least squares regression line fits the data. R squared you might already be familiar with. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
Or you might recognize this as the slope of the least squares regression line. So this is the slope, and this would be equal to 0.164. Now this information right over here, it tells us how well our least squares regression line fits the data. R squared you might already be familiar with. It says how much of the variance in the y variable is explainable by the x variable. If it was one or 100%, that means all of it could be explained, and it's a very good fit. If it was zero, that means none of it can be explained. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
R squared you might already be familiar with. It says how much of the variance in the y variable is explainable by the x variable. If it was one or 100%, that means all of it could be explained, and it's a very good fit. If it was zero, that means none of it can be explained. It would be a very bad fit. Capital S, this is the standard deviation of the residuals, and it's another measure of how much these data points vary from this regression line. Now this column right over here is going to prove to be useful for answering the question at hand. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
If it was zero, that means none of it can be explained. It would be a very bad fit. Capital S, this is the standard deviation of the residuals, and it's another measure of how much these data points vary from this regression line. Now this column right over here is going to prove to be useful for answering the question at hand. This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line. You could view this as the estimate of the standard deviation of the sampling distribution of the slope of the regression line. Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
Now this column right over here is going to prove to be useful for answering the question at hand. This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line. You could view this as the estimate of the standard deviation of the sampling distribution of the slope of the regression line. Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population. This would sometimes also be called the standard error of the slope of the least squares regression line. Now these last two columns you don't have to worry about in the context of this video. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population. This would sometimes also be called the standard error of the slope of the least squares regression line. Now these last two columns you don't have to worry about in the context of this video. This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low. It's about a 1% chance that you would have gotten these results if there truly was not a relationship between caffeine intake and time studying. But with all of that out of the way, let's actually answer the question. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
Now these last two columns you don't have to worry about in the context of this video. This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low. It's about a 1% chance that you would have gotten these results if there truly was not a relationship between caffeine intake and time studying. But with all of that out of the way, let's actually answer the question. Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate. We don't actually know the standard deviation of the sampling distribution. So the last thing we have to do is figure out what is this critical t value? | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
But with all of that out of the way, let's actually answer the question. Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate. We don't actually know the standard deviation of the sampling distribution. So the last thing we have to do is figure out what is this critical t value? You can figure it out using either a calculator or using a table. I'll do it using a table, and to do that, we need to know what the degrees of freedom. Well, when you're doing this with a regression slope like we're doing right now, your degrees of freedom are going to be the number of data points you have minus two so our degrees of freedom are going to be 20 minus two which is equal to 18. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
So the last thing we have to do is figure out what is this critical t value? You can figure it out using either a calculator or using a table. I'll do it using a table, and to do that, we need to know what the degrees of freedom. Well, when you're doing this with a regression slope like we're doing right now, your degrees of freedom are going to be the number of data points you have minus two so our degrees of freedom are going to be 20 minus two which is equal to 18. I'm not gonna go into a bunch of depth right now. It actually is beyond the scope of this video for sure as to why you subtract two here, but just so that we can look it up on a table, this is our degrees of freedom. So we care about a 95% confidence level. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
The amount of fuel he uses follows a normal distribution for each part of his commute, but the amount of fuel he uses on the way home varies more. The amounts of fuel he uses for each part of the commute are also independent of each other. Here are summary statistics for the amount of fuel Shinji uses for each part of his commute. So when he goes to work, he uses a mean of 10 liters of fuel with a standard deviation of 1.5 liters. And on the way home, he also has a mean of 10 liters, but there is more variation, there is more spread. He has a standard deviation of two liters. Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So when he goes to work, he uses a mean of 10 liters of fuel with a standard deviation of 1.5 liters. And on the way home, he also has a mean of 10 liters, but there is more variation, there is more spread. He has a standard deviation of two liters. Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. What is the probability that Shinji runs out of fuel? All right, this is really interesting. We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. What is the probability that Shinji runs out of fuel? All right, this is really interesting. We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. They say that right over here, follows a normal distribution. But here we're talking about the total amount of fuel he has to go to work and to go home. So what we wanna do is come up with a total distribution, home and back, I guess you could say. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. They say that right over here, follows a normal distribution. But here we're talking about the total amount of fuel he has to go to work and to go home. So what we wanna do is come up with a total distribution, home and back, I guess you could say. We could call this work plus home, home and back. If you have two random variables that can be described by normal distributions, and you were to define a new random variable as their sum, the distribution of that new random variable will still be a normal distribution, and its mean will be the sum of the means of those other random variables. So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So what we wanna do is come up with a total distribution, home and back, I guess you could say. We could call this work plus home, home and back. If you have two random variables that can be described by normal distributions, and you were to define a new random variable as their sum, the distribution of that new random variable will still be a normal distribution, and its mean will be the sum of the means of those other random variables. So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. He will use a mean of 20 liters in the round trip. Now for the standard deviation from home plus work, you can't just add the standard deviations, going and coming back. But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. He will use a mean of 20 liters in the round trip. Now for the standard deviation from home plus work, you can't just add the standard deviations, going and coming back. But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. And only because they are independent can we add the variances. So what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home. So what's the variance of going to work? | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. And only because they are independent can we add the variances. So what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home. So what's the variance of going to work? Well, 1.5 squared is, so this will be 1.5 squared. And what's the variance coming home? Well, this is going to be two squared, two squared. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So what's the variance of going to work? Well, 1.5 squared is, so this will be 1.5 squared. And what's the variance coming home? Well, this is going to be two squared, two squared. Well, this is 2.25 plus four, which is equal to 6.25. So the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
Well, this is going to be two squared, two squared. Well, this is 2.25 plus four, which is equal to 6.25. So the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. So we have this normal distribution that might look something like this. We know its mean is 20 liters. So this is 20 liters. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. So we have this normal distribution that might look something like this. We know its mean is 20 liters. So this is 20 liters. And we wanna know what is the probability that Shinji runs out of fuel? Well, to run out of fuel, he would need to require more than 25 liters of fuel. So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So this is 20 liters. And we wanna know what is the probability that Shinji runs out of fuel? Well, to run out of fuel, he would need to require more than 25 liters of fuel. So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters. He actually has 25 liters in his tank. So how do we figure out that area right over there? | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters. He actually has 25 liters in his tank. So how do we figure out that area right over there? Well, we could use a z-table. We could say how many standard deviations above the mean is 25 liters? Well, it is five liters above the mean. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So how do we figure out that area right over there? Well, we could use a z-table. We could say how many standard deviations above the mean is 25 liters? Well, it is five liters above the mean. So let me write this down. So the z here, the z is equal to 25 minus the mean, minus 20, divided by the standard deviation for, I guess you could say, this combined normal distribution. This is two standard deviations above the mean, or a z-score of plus two. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
Well, it is five liters above the mean. So let me write this down. So the z here, the z is equal to 25 minus the mean, minus 20, divided by the standard deviation for, I guess you could say, this combined normal distribution. This is two standard deviations above the mean, or a z-score of plus two. So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean. And then if we subtract that from one, we will get the area that we care about. So let's get our z-table out. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
This is two standard deviations above the mean, or a z-score of plus two. So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean. And then if we subtract that from one, we will get the area that we care about. So let's get our z-table out. We care about a z-score of exactly two. So 2.00 is right over here,.9772. So that tells us that this area right over here is 0.9772. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So let's get our z-table out. We care about a z-score of exactly two. So 2.00 is right over here,.9772. So that tells us that this area right over here is 0.9772. And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So that tells us that this area right over here is 0.9772. And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228. Did I do that right? I think I did that right. Yes, 0.0228 is the probability that Shinji runs out of fuel. | Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3 |
So you can have a sample study, and we've already talked about this in several videos, but we'll go over it again in this one. You can have an observational study. Observational study. Or you can have an experiment. Experiment. So let's go through each of these, and always pause this video and see if you can think about what these words likely mean, or you might already know. Well, sample study we have looked at, this is really where you're trying to estimate the value of a parameter for a population. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Or you can have an experiment. Experiment. So let's go through each of these, and always pause this video and see if you can think about what these words likely mean, or you might already know. Well, sample study we have looked at, this is really where you're trying to estimate the value of a parameter for a population. So what's an example of that? So let's say we take the population of people in a city, and so that could be hundreds of thousands of people, and the parameter that you care about is how much time, on average, do they spend on a computer. So the parameter would be for the entire population. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Well, sample study we have looked at, this is really where you're trying to estimate the value of a parameter for a population. So what's an example of that? So let's say we take the population of people in a city, and so that could be hundreds of thousands of people, and the parameter that you care about is how much time, on average, do they spend on a computer. So the parameter would be for the entire population. If it was possible, you would go talk to every, maybe there's a million people in the city, you would talk to all million of those people, and ask them how much time they spend on a computer, and you would get the average, and then that would be the parameter. So population parameter, would be average time on a computer per day. Average daily time on a computer. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
So the parameter would be for the entire population. If it was possible, you would go talk to every, maybe there's a million people in the city, you would talk to all million of those people, and ask them how much time they spend on a computer, and you would get the average, and then that would be the parameter. So population parameter, would be average time on a computer per day. Average daily time on a computer. Now you determine that it's impractical to go talk to everyone, so you're not going to be able to figure out the exact population parameter, average daily time on a computer, so instead you do a sample study. You randomly sample, and there's a lot of thought in thinking about whether your sample is truly random, so you randomly sample, but there's also different techniques of randomly sampling. So you randomly sample people from your population, and then you take the average daily time on a computer for your sample, and that is going to be an estimate for the population parameter. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Average daily time on a computer. Now you determine that it's impractical to go talk to everyone, so you're not going to be able to figure out the exact population parameter, average daily time on a computer, so instead you do a sample study. You randomly sample, and there's a lot of thought in thinking about whether your sample is truly random, so you randomly sample, but there's also different techniques of randomly sampling. So you randomly sample people from your population, and then you take the average daily time on a computer for your sample, and that is going to be an estimate for the population parameter. So that's your classic sample study. Now in an observational study, you're not trying to estimate a parameter, you're trying to understand how two parameters in a population might move together or not. So let's say that you have a population now. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
So you randomly sample people from your population, and then you take the average daily time on a computer for your sample, and that is going to be an estimate for the population parameter. So that's your classic sample study. Now in an observational study, you're not trying to estimate a parameter, you're trying to understand how two parameters in a population might move together or not. So let's say that you have a population now. Let's say you have a population of 1,000 people. And you're curious about whether average daily time on a computer, how it relates to people's blood pressure. So average computer time, actually let me write it this way, instead of average computer time, it should just be computer time. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
So let's say that you have a population now. Let's say you have a population of 1,000 people. And you're curious about whether average daily time on a computer, how it relates to people's blood pressure. So average computer time, actually let me write it this way, instead of average computer time, it should just be computer time. Computer time versus blood pressure. So what you do is you apply a survey to all 1,000 people, and you ask them how much time you spend on a computer, and what is your blood pressure? Or maybe you measure it in some way. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
So average computer time, actually let me write it this way, instead of average computer time, it should just be computer time. Computer time versus blood pressure. So what you do is you apply a survey to all 1,000 people, and you ask them how much time you spend on a computer, and what is your blood pressure? Or maybe you measure it in some way. And then you plot it all, you look at the data, and you see if those two variables move together. So what does that mean? Well, let me draw. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Or maybe you measure it in some way. And then you plot it all, you look at the data, and you see if those two variables move together. So what does that mean? Well, let me draw. So if this axis is, let's say this is computer time, computer time, and this axis is blood pressure. Blood pressure. So let's say that there's one person who doesn't spend a lot of time on a computer, and they have a relatively low blood pressure. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Well, let me draw. So if this axis is, let's say this is computer time, computer time, and this axis is blood pressure. Blood pressure. So let's say that there's one person who doesn't spend a lot of time on a computer, and they have a relatively low blood pressure. There's another person who spends a lot of time, has high blood pressure. There could be someone who doesn't spend much time on a computer, but has a reasonably high blood pressure. But you keep doing this, and you get all these data points for those 1,000 people. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
So let's say that there's one person who doesn't spend a lot of time on a computer, and they have a relatively low blood pressure. There's another person who spends a lot of time, has high blood pressure. There could be someone who doesn't spend much time on a computer, but has a reasonably high blood pressure. But you keep doing this, and you get all these data points for those 1,000 people. And I'm not going to sit here and draw 1,000 points. But you see something like this. And so you see, hey look, it looks like, there's definitely some outliers, but it looks like these two variables move together. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
But you keep doing this, and you get all these data points for those 1,000 people. And I'm not going to sit here and draw 1,000 points. But you see something like this. And so you see, hey look, it looks like, there's definitely some outliers, but it looks like these two variables move together. It looks like in general, the more computer time, the higher the blood pressure, or the higher the blood pressure, the more computer time. And so you can make a conclusion here about these two variables correlating, that they're positively correlated. There is a positive, a reasonable conclusion, if you did the study appropriately, would be that more computer time correlates with higher blood pressure, or that higher blood pressure correlates with more computer time. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And so you see, hey look, it looks like, there's definitely some outliers, but it looks like these two variables move together. It looks like in general, the more computer time, the higher the blood pressure, or the higher the blood pressure, the more computer time. And so you can make a conclusion here about these two variables correlating, that they're positively correlated. There is a positive, a reasonable conclusion, if you did the study appropriately, would be that more computer time correlates with higher blood pressure, or that higher blood pressure correlates with more computer time. Now when you do these observational studies, or when you interpret these observational studies, when you read about someone else's, it's very important not to say, oh well, this shows me that computer time causes blood pressure. Because this is not showing causality. And you also can't say, maybe you might say somehow blood pressure causes more people to spend time in front of a computer. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
There is a positive, a reasonable conclusion, if you did the study appropriately, would be that more computer time correlates with higher blood pressure, or that higher blood pressure correlates with more computer time. Now when you do these observational studies, or when you interpret these observational studies, when you read about someone else's, it's very important not to say, oh well, this shows me that computer time causes blood pressure. Because this is not showing causality. And you also can't say, maybe you might say somehow blood pressure causes more people to spend time in front of a computer. That seems even a little bit sillier, but they're actually the same. Because all you're saying is that there's a correlation, these two variables move together. You can't make a conclusion about causality, that computer time causes blood pressure, or that blood pressure causes some time, high blood pressure causes more computer time. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And you also can't say, maybe you might say somehow blood pressure causes more people to spend time in front of a computer. That seems even a little bit sillier, but they're actually the same. Because all you're saying is that there's a correlation, these two variables move together. You can't make a conclusion about causality, that computer time causes blood pressure, or that blood pressure causes some time, high blood pressure causes more computer time. Why can't you make that? Well, there could be what's called a confounding variable. Sometimes called a lurking variable. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
You can't make a conclusion about causality, that computer time causes blood pressure, or that blood pressure causes some time, high blood pressure causes more computer time. Why can't you make that? Well, there could be what's called a confounding variable. Sometimes called a lurking variable. Where, let's say that, so this is computer time. Computer time. And this is blood pressure, I'll just write it like that. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Sometimes called a lurking variable. Where, let's say that, so this is computer time. Computer time. And this is blood pressure, I'll just write it like that. Blood, looks like building. So blood, blood pressure. And it looks like these two things move together. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And this is blood pressure, I'll just write it like that. Blood, looks like building. So blood, blood pressure. And it looks like these two things move together. We saw that right over here in our data. But there could be a root variable that drives both of these. A confounding variable. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And it looks like these two things move together. We saw that right over here in our data. But there could be a root variable that drives both of these. A confounding variable. And that could just be the amount of physical activity someone has. So there could just be a lack of physical activity driving both. Lack of activity. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
A confounding variable. And that could just be the amount of physical activity someone has. So there could just be a lack of physical activity driving both. Lack of activity. People who are less active spend more time in front of a computer, and people who are less active have higher, have higher blood pressure. And if you were to control for this, if you were to take a bunch of people who had a similar lack of activity, or had a similar level of activity, you might see that computer time does not correlate with blood pressure. That these are just both driven by the same thing. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Lack of activity. People who are less active spend more time in front of a computer, and people who are less active have higher, have higher blood pressure. And if you were to control for this, if you were to take a bunch of people who had a similar lack of activity, or had a similar level of activity, you might see that computer time does not correlate with blood pressure. That these are just both driven by the same thing. And what you're really seeing here is like, okay, people with high lack of activity who aren't active, well it drives both of these variables. So once again, when you do this observational study, and if you do it well, you can draw correlations, and that might give you decent hypotheses for causality, but this does not show causality. Because you could have these confounding variables. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
That these are just both driven by the same thing. And what you're really seeing here is like, okay, people with high lack of activity who aren't active, well it drives both of these variables. So once again, when you do this observational study, and if you do it well, you can draw correlations, and that might give you decent hypotheses for causality, but this does not show causality. Because you could have these confounding variables. Now experiments, and experiments are the basis of the scientific method. Experiments are all about trying to establish causality. And so what you would do is, if you wanted to do an experiment, you would take, and you probably wouldn't be able to do it with a thousand people, experiments in some way are the hardest to do of all of these. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Because you could have these confounding variables. Now experiments, and experiments are the basis of the scientific method. Experiments are all about trying to establish causality. And so what you would do is, if you wanted to do an experiment, you would take, and you probably wouldn't be able to do it with a thousand people, experiments in some way are the hardest to do of all of these. Maybe you take a hundred people. A hundred people. And to avoid having this confounding variable introduce error into your experiment, you randomly assign these hundred people into two groups. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And so what you would do is, if you wanted to do an experiment, you would take, and you probably wouldn't be able to do it with a thousand people, experiments in some way are the hardest to do of all of these. Maybe you take a hundred people. A hundred people. And to avoid having this confounding variable introduce error into your experiment, you randomly assign these hundred people into two groups. So random assign. So random. It's very important that they're randomly assigned. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And to avoid having this confounding variable introduce error into your experiment, you randomly assign these hundred people into two groups. So random assign. So random. It's very important that they're randomly assigned. And that's nice, you might not know all of the confounding variables there, but it makes it likely that each group will have a same amount of people with lack of activity, or that the lack of activity, or the activity levels on average in each of the groups, when they're randomly assigned, it gives you a better chance that one group doesn't have a significantly different activity level than the other. And then what you do is, you have a control group, and you have a treatment group. Once again, you've randomly assigned them. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
It's very important that they're randomly assigned. And that's nice, you might not know all of the confounding variables there, but it makes it likely that each group will have a same amount of people with lack of activity, or that the lack of activity, or the activity levels on average in each of the groups, when they're randomly assigned, it gives you a better chance that one group doesn't have a significantly different activity level than the other. And then what you do is, you have a control group, and you have a treatment group. Once again, you've randomly assigned them. So control, and then treatment. And what you might say is, okay, for some amount of time, all of you in the control group can only spend a max of 30 minutes in front of a computer. And on the, or maybe if you really wanted to do it, you say you have to spend exactly 30 minutes on a computer, and that's maybe a little unrealistic. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
Once again, you've randomly assigned them. So control, and then treatment. And what you might say is, okay, for some amount of time, all of you in the control group can only spend a max of 30 minutes in front of a computer. And on the, or maybe if you really wanted to do it, you say you have to spend exactly 30 minutes on a computer, and that's maybe a little unrealistic. And then the treatment group, you have to say, you have to spend exactly two hours in front of a computer. And I'm making up these numbers at random. And it would be nice to see, okay, what was everyone's blood pressure before the experiment? | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And on the, or maybe if you really wanted to do it, you say you have to spend exactly 30 minutes on a computer, and that's maybe a little unrealistic. And then the treatment group, you have to say, you have to spend exactly two hours in front of a computer. And I'm making up these numbers at random. And it would be nice to see, okay, what was everyone's blood pressure before the experiment? And you can say, okay, well, the averages are similar going into the experiment. And then you go some amount of time, and you measure blood pressure. And if you see that, wow, this group definitely has a higher blood pressure. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And it would be nice to see, okay, what was everyone's blood pressure before the experiment? And you can say, okay, well, the averages are similar going into the experiment. And then you go some amount of time, and you measure blood pressure. And if you see that, wow, this group definitely has a higher blood pressure. This group has a higher blood pressure. So the blood pressure, the blood pressure is higher here. And once again, some of this might have just happened randomly. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And if you see that, wow, this group definitely has a higher blood pressure. This group has a higher blood pressure. So the blood pressure, the blood pressure is higher here. And once again, some of this might have just happened randomly. It might have been, you know, the people you happened to put in there, et cetera, et cetera. But depending, if this was a large enough experiment, and you conducted it well, this says, hey, look, there's, I'm feeling like there's a causality here. That by making these people spend more time in front of a computer, that that actually raised their blood pressure. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
And once again, some of this might have just happened randomly. It might have been, you know, the people you happened to put in there, et cetera, et cetera. But depending, if this was a large enough experiment, and you conducted it well, this says, hey, look, there's, I'm feeling like there's a causality here. That by making these people spend more time in front of a computer, that that actually raised their blood pressure. So once again, sample study, you're trying to estimate a population parameter. Observation study, you are seeing if there is a correlation between two things. And you have to be careful not to say, hey, one is causing the other, because you could have confounding variables. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
That by making these people spend more time in front of a computer, that that actually raised their blood pressure. So once again, sample study, you're trying to estimate a population parameter. Observation study, you are seeing if there is a correlation between two things. And you have to be careful not to say, hey, one is causing the other, because you could have confounding variables. Experiment, you're trying to establish or show causality. And you do that by taking your group, randomly assigning to a controller treatment that should evenly, or hopefully evenly distribute, not always, there's some chance it doesn't, but distribute the confounding variables. And then you, on each group, you change how much of one of these variables they get, and you see if it drives the other variable. | Types of statistical studies Study design AP Statistics Khan Academy.mp3 |
The table below displays the probability distribution of X, the number of shots that Anoush makes in a set of two attempts, along with some summary statistics. So here's the random variable X. It's a discrete random variable. It only takes on a finite number of values. Sometimes people say it takes on a countable number of values, but we see he can either make zero free throws, one, or two of the two. And the probability that he makes zero is here, one is here, two is here. And then they also give us the mean of X and the standard deviation of X. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
It only takes on a finite number of values. Sometimes people say it takes on a countable number of values, but we see he can either make zero free throws, one, or two of the two. And the probability that he makes zero is here, one is here, two is here. And then they also give us the mean of X and the standard deviation of X. Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N? All right, so let's define a new random variable N, which is equal to his net gain, net gain. We can define N in terms of X. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
And then they also give us the mean of X and the standard deviation of X. Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N? All right, so let's define a new random variable N, which is equal to his net gain, net gain. We can define N in terms of X. What is his net gain going to be? Well, let's see, N, it's going to be equal to 10 times however many shots he makes. So it's gonna be 10 times X. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
We can define N in terms of X. What is his net gain going to be? Well, let's see, N, it's going to be equal to 10 times however many shots he makes. So it's gonna be 10 times X. And then no matter what, he has to pay $15 to play, minus 15. In fact, we could set up a little table here for the probability distribution of N. So let me make it right over here. So I'll make it look just like this one. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
So it's gonna be 10 times X. And then no matter what, he has to pay $15 to play, minus 15. In fact, we could set up a little table here for the probability distribution of N. So let me make it right over here. So I'll make it look just like this one. N is equal to net gain. And here we'll have the probability of N. And there's three outcomes here. The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
So I'll make it look just like this one. N is equal to net gain. And here we'll have the probability of N. And there's three outcomes here. The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. And it would have the same probability, 0.16. When he makes one shot, the net gain is gonna be 10 times one minus 15, which is negative five. But it's gonna have the same probability. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. And it would have the same probability, 0.16. When he makes one shot, the net gain is gonna be 10 times one minus 15, which is negative five. But it's gonna have the same probability. He has a 48% chance of making one shot, and so it's a 48% chance of losing $5. And then last but not least, when X is two, his net gain is going to be positive five, plus five. And so this is a 0.36 chance. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
But it's gonna have the same probability. He has a 48% chance of making one shot, and so it's a 48% chance of losing $5. And then last but not least, when X is two, his net gain is going to be positive five, plus five. And so this is a 0.36 chance. So what they want us to figure out are what are the mean and standard deviation of his net gain? So let's first figure out the mean of N. Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
And so this is a 0.36 chance. So what they want us to figure out are what are the mean and standard deviation of his net gain? So let's first figure out the mean of N. Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15. This is 1.2. So it is 12 minus 15, which is equal to negative three. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15. This is 1.2. So it is 12 minus 15, which is equal to negative three. Now the standard deviation of N is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
So it is 12 minus 15, which is equal to negative three. Now the standard deviation of N is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. So this is going to be equal to 10 times the standard deviation of X. Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. So this is going to be equal to 10 times the standard deviation of X. Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, well, your spread should grow by the amount that you're scaling it. But by shifting it, it doesn't affect how much you disperse from the mean. So standard deviation is only affected by the scaling, but not by the shifting here. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, well, your spread should grow by the amount that you're scaling it. But by shifting it, it doesn't affect how much you disperse from the mean. So standard deviation is only affected by the scaling, but not by the shifting here. So this is going to be 10 times 0.69, which is going to, this was an approximation, so I'll say this is approximately equal to 6.9. So this is our new distribution for our net gain. This is the mean of our net gain, and this is roughly the standard deviation of our net gain. | Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3 |
It's the number of successes from n trials, so it's a finite number of trials, where the probability of success is equal to p, so the probability is constant across the trials, for each of these independent trials. So the probability of success in one trial is not dependent on what happened in the other trials. And we also talked in that previous video where we talked about the expected value of this binomial variable, is we said, hey, it could be viewed, this binomial variable can be viewed as the sum of n of what you could really consider to be a Bernoulli variable here. So this variable, this random variable y, the probability that it's equal to one, you could view that as a success, is equal to p. The probability that it's a failure, that y is equal to zero, is one minus p. So you could view y, the outcome of y is really the, or whether y is one or zero is really whether we had a success or not in each of these trials. So if you add up n y's, then you are going to get x. And we use that information to figure out what the expected value of x is going to be. Because the expected value of y is pretty straightforward to directly compute. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So this variable, this random variable y, the probability that it's equal to one, you could view that as a success, is equal to p. The probability that it's a failure, that y is equal to zero, is one minus p. So you could view y, the outcome of y is really the, or whether y is one or zero is really whether we had a success or not in each of these trials. So if you add up n y's, then you are going to get x. And we use that information to figure out what the expected value of x is going to be. Because the expected value of y is pretty straightforward to directly compute. Expected value of y is just probability-weighted outcomes. So it's p times one plus one minus p, one minus p, times zero, times zero. This whole term's gonna be zero. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Because the expected value of y is pretty straightforward to directly compute. Expected value of y is just probability-weighted outcomes. So it's p times one plus one minus p, one minus p, times zero, times zero. This whole term's gonna be zero. And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review. We could say that the expected value of x is just going to be equal to, we know from our expected value properties, that's going to be equal to the sum of the expected values of these n y's. Or you could say it is n times the expected value, times the expected value of y. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
This whole term's gonna be zero. And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review. We could say that the expected value of x is just going to be equal to, we know from our expected value properties, that's going to be equal to the sum of the expected values of these n y's. Or you could say it is n times the expected value, times the expected value of y. The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to. Because we could see, we know from our variance properties, you can't do this with standard deviation, but you could do it with variance. And then once you figure out the variance, you just take the square root for the standard deviation. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Or you could say it is n times the expected value, times the expected value of y. The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to. Because we could see, we know from our variance properties, you can't do this with standard deviation, but you could do it with variance. And then once you figure out the variance, you just take the square root for the standard deviation. The variance of x is similarly going to be the sum of the variances of these n y's. So it's going to be similarly n times the variance, n times the variance of y. So this all boils down to what is the variance of y going to be equal to? | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
And then once you figure out the variance, you just take the square root for the standard deviation. The variance of x is similarly going to be the sum of the variances of these n y's. So it's going to be similarly n times the variance, n times the variance of y. So this all boils down to what is the variance of y going to be equal to? So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here. All right, so we want to figure out the variance of y. So variance of y is going to be equal to what? | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So this all boils down to what is the variance of y going to be equal to? So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here. All right, so we want to figure out the variance of y. So variance of y is going to be equal to what? Well here it's going to be the probability squared distances from the expected value. So we have a probability of p, where what is going to be our squared distance from the expected value? Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So variance of y is going to be equal to what? Well here it's going to be the probability squared distances from the expected value. So we have a probability of p, where what is going to be our squared distance from the expected value? Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight. And then we have, actually let me scroll over a little bit, well I'll just do it right over here. Plus we have a probability of one minus p, one minus p for the other possible outcome. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight. And then we have, actually let me scroll over a little bit, well I'll just do it right over here. Plus we have a probability of one minus p, one minus p for the other possible outcome. So in that outcome, we are at zero. And the difference between zero and our expected value, well that's just going to be zero minus p. And once again, we are going to square that distance. And so this is the expression, or square that quantity. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Plus we have a probability of one minus p, one minus p for the other possible outcome. So in that outcome, we are at zero. And the difference between zero and our expected value, well that's just going to be zero minus p. And once again, we are going to square that distance. And so this is the expression, or square that quantity. And so this is the expression for the variance of y, and we can simplify it a little bit. So this is all going to be equal to, so let me just, p times one minus p squared. And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
And so this is the expression, or square that quantity. And so this is the expression for the variance of y, and we can simplify it a little bit. So this is all going to be equal to, so let me just, p times one minus p squared. And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out. This is just, this whole thing is just a one. So you're left with p times one minus p, which is indeed the variance for a binomial variable. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out. This is just, this whole thing is just a one. So you're left with p times one minus p, which is indeed the variance for a binomial variable. We actually proved that in other videos. I guess it doesn't hurt to see it again. But there you have it. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So you're left with p times one minus p, which is indeed the variance for a binomial variable. We actually proved that in other videos. I guess it doesn't hurt to see it again. But there you have it. We know what the variance of y is. It is p times one minus p. And the variance of x is just n times the variance of y. So there we go. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
But there you have it. We know what the variance of y is. It is p times one minus p. And the variance of x is just n times the variance of y. So there we go. We deserve a little bit of a drum roll. The variance of x is equal to n times p times one minus p. So if we were to take the concrete example of the last video, where if I were to take 10 free throws, so each trial is a shot, is a free throw. So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So there we go. We deserve a little bit of a drum roll. The variance of x is equal to n times p times one minus p. So if we were to take the concrete example of the last video, where if I were to take 10 free throws, so each trial is a shot, is a free throw. So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. And so that would be what? This right over here, so this would be equal to 10 times 0.3 times 0.7 times 0.21. So my variance in this situation is going to be equal to 2.1. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. And so that would be what? This right over here, so this would be equal to 10 times 0.3 times 0.7 times 0.21. So my variance in this situation is going to be equal to 2.1. It is equal to 2.1. And if I wanted to figure out the standard deviation of this right over here, I would just take the square root of this. So if you want the standard deviation, just take the square root of this expression right over here. | Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
The rest are fair coins. So if three are unfair, the rest are eight coins. And when I say that, or when this problem says that they are fair coins, it means that they have a 50-50 chance of coming up either heads or tails. You randomly choose one coin from the bag and flip it two times. What is the percent probability of getting two heads? So this is an interesting question, but if we break it down, essentially with a decision tree, it'll help break it down a little bit better. So let's say that, so we have a bag. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
You randomly choose one coin from the bag and flip it two times. What is the percent probability of getting two heads? So this is an interesting question, but if we break it down, essentially with a decision tree, it'll help break it down a little bit better. So let's say that, so we have a bag. Three of them are unfair. So we can even visualize the bag. You don't have to do this all the time. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So let's say that, so we have a bag. Three of them are unfair. So we can even visualize the bag. You don't have to do this all the time. So we have, out of the fair coins in white, one, two, three, four, four, five fair coins. And we have three unfair coins. One, two, three, and this whole thing is my bag right over here. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
You don't have to do this all the time. So we have, out of the fair coins in white, one, two, three, four, four, five fair coins. And we have three unfair coins. One, two, three, and this whole thing is my bag right over here. That is my bag of coins. If I were to, when I take my hand in, if I were to take any of these white coins, there's a 50% chance that it gets heads on any flip. The odds of getting two heads in a row would be 50% times 50% for these white coins. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
One, two, three, and this whole thing is my bag right over here. That is my bag of coins. If I were to, when I take my hand in, if I were to take any of these white coins, there's a 50% chance that it gets heads on any flip. The odds of getting two heads in a row would be 50% times 50% for these white coins. But I don't know I'm going to get a white coin. If I get one of these orange coins, I have a 60% chance of coming up heads. And the odds of coming up 60%, if I have picked one of these orange coins, the probability of getting heads twice is going to be 60% times 60%. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
The odds of getting two heads in a row would be 50% times 50% for these white coins. But I don't know I'm going to get a white coin. If I get one of these orange coins, I have a 60% chance of coming up heads. And the odds of coming up 60%, if I have picked one of these orange coins, the probability of getting heads twice is going to be 60% times 60%. So how do I factor in this idea that I don't know if I've picked a white fair coin or an orange unfair coin? We'll assume that the coins actually aren't white and orange. They all look like regular coins. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
And the odds of coming up 60%, if I have picked one of these orange coins, the probability of getting heads twice is going to be 60% times 60%. So how do I factor in this idea that I don't know if I've picked a white fair coin or an orange unfair coin? We'll assume that the coins actually aren't white and orange. They all look like regular coins. So what I'll do is I'll draw a little bit of a decision tree here. I guess maybe I could call it a probability tree. So there's some probability that I pick a fair coin. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
They all look like regular coins. So what I'll do is I'll draw a little bit of a decision tree here. I guess maybe I could call it a probability tree. So there's some probability that I pick a fair coin. And there's some probability that I pick an unfair coin. And so what is the probability that I pick a fair coin? Well, 1, 2, 3, 4, 5 out of the total 8 coins are fair. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So there's some probability that I pick a fair coin. And there's some probability that I pick an unfair coin. And so what is the probability that I pick a fair coin? Well, 1, 2, 3, 4, 5 out of the total 8 coins are fair. So there is a 5 8's probability. I'll write it here on the branch, actually. So there's a 5 8's chance that I pick a fair coin. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
Well, 1, 2, 3, 4, 5 out of the total 8 coins are fair. So there is a 5 8's probability. I'll write it here on the branch, actually. So there's a 5 8's chance that I pick a fair coin. And then there is a 3, 1, 2, 3 out of 8 chance that I pick an unfair coin. So if I were to just tell you what's the probability of picking a fair coin, you'd say, oh, 5 8's. What's the probability of an unfair coin? | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So there's a 5 8's chance that I pick a fair coin. And then there is a 3, 1, 2, 3 out of 8 chance that I pick an unfair coin. So if I were to just tell you what's the probability of picking a fair coin, you'd say, oh, 5 8's. What's the probability of an unfair coin? 3 8's. And you could convert that to a decimal or a percentage or whatever you'd like. Now, given that I have picked a fair coin, what is the probability that I will get heads twice? | Dependent probability example Probability and Statistics Khan Academy.mp3 |
What's the probability of an unfair coin? 3 8's. And you could convert that to a decimal or a percentage or whatever you'd like. Now, given that I have picked a fair coin, what is the probability that I will get heads twice? So let me write it this way. And this is just notation right here. So the probability of, I'll call it heads, heads, so I get two heads in a row, given that I have a fair coin. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
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