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And so my average is going to be 65 plus 55 divided by 2. Which is 60. So then my average went up a little bit. Then I had a 45, which will bring my average down a little bit. I won't plot a 45 here. Now I have to average all of these out. What's 45 plus 65? | Law of large numbers Probability and Statistics Khan Academy.mp3 |
Then I had a 45, which will bring my average down a little bit. I won't plot a 45 here. Now I have to average all of these out. What's 45 plus 65? Well, let me actually just get the numbers, just so you get the point. So it's 55 plus 65 is 120. Plus 45 is 165. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
What's 45 plus 65? Well, let me actually just get the numbers, just so you get the point. So it's 55 plus 65 is 120. Plus 45 is 165. Divided by 3 is 53. Right? No, no, no. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
Plus 45 is 165. Divided by 3 is 53. Right? No, no, no. 15. 55. So the average goes down back down to 55. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
No, no, no. 15. 55. So the average goes down back down to 55. And we could keep doing these trials, right? So you might say that the law of large numbers tells us, OK, after we've done three trials and our average is there. So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So the average goes down back down to 55. And we could keep doing these trials, right? So you might say that the law of large numbers tells us, OK, after we've done three trials and our average is there. So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future. That somehow the next couple of trials are going to have to be down here in order to bring our average down. And that's not necessarily the case. Going forward, the probabilities are always the same. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future. That somehow the next couple of trials are going to have to be down here in order to bring our average down. And that's not necessarily the case. Going forward, the probabilities are always the same. The probabilities are always 50% that I'm going to get heads. It's not like if I had a bunch of heads to start off with, or more than I would have expected to start off with, that all of a sudden things would be made up and I'd get more tails. And that would be the gambler's fallacy. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
Going forward, the probabilities are always the same. The probabilities are always 50% that I'm going to get heads. It's not like if I had a bunch of heads to start off with, or more than I would have expected to start off with, that all of a sudden things would be made up and I'd get more tails. And that would be the gambler's fallacy. That if you have a long streak of heads, or you have a disproportionate number of heads, that at some point you're going to have a higher likelihood of having a disproportionate number of tails. And that's not quite true. What the law of large numbers tells us is that it doesn't care, let's say after some finite number of trials, your average actually, it's a low probability of this happening, but let's say your average is actually up here. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And that would be the gambler's fallacy. That if you have a long streak of heads, or you have a disproportionate number of heads, that at some point you're going to have a higher likelihood of having a disproportionate number of tails. And that's not quite true. What the law of large numbers tells us is that it doesn't care, let's say after some finite number of trials, your average actually, it's a low probability of this happening, but let's say your average is actually up here. It's actually at 70, right? Like, wow, we really diverged a good bit from the expected value. But what the law of large numbers says, well, I don't care how many trials this is, we have an infinite number of trials left, right? | Law of large numbers Probability and Statistics Khan Academy.mp3 |
What the law of large numbers tells us is that it doesn't care, let's say after some finite number of trials, your average actually, it's a low probability of this happening, but let's say your average is actually up here. It's actually at 70, right? Like, wow, we really diverged a good bit from the expected value. But what the law of large numbers says, well, I don't care how many trials this is, we have an infinite number of trials left, right? And the expected value for that infinite number of trials, or especially in this type of situation, is going to be this. So when you average a finite number that averages out to some high number, and then an infinite number that's going to converge to this, you're going to, over time, converge back to the expected value. And that was a very informal way of describing it. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
But what the law of large numbers says, well, I don't care how many trials this is, we have an infinite number of trials left, right? And the expected value for that infinite number of trials, or especially in this type of situation, is going to be this. So when you average a finite number that averages out to some high number, and then an infinite number that's going to converge to this, you're going to, over time, converge back to the expected value. And that was a very informal way of describing it. But that's what the law of large numbers tells you. And it's an important thing. It's not telling you that if you get a bunch of heads, that somehow the probability of getting tails is going to increase to kind of make up for the heads. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And that was a very informal way of describing it. But that's what the law of large numbers tells you. And it's an important thing. It's not telling you that if you get a bunch of heads, that somehow the probability of getting tails is going to increase to kind of make up for the heads. What it's telling you is that no matter what happened over a finite number of trials, no matter what the average is over a finite number of trials, you have an infinite number of trials left. And if you do enough of them, it's going to converge back to your expected value. And this is an important thing to think about. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
It's not telling you that if you get a bunch of heads, that somehow the probability of getting tails is going to increase to kind of make up for the heads. What it's telling you is that no matter what happened over a finite number of trials, no matter what the average is over a finite number of trials, you have an infinite number of trials left. And if you do enough of them, it's going to converge back to your expected value. And this is an important thing to think about. But this isn't used in practice every day with the lottery and with casinos. Because they know that if you do large enough samples, and we could even calculate if you do large enough samples, what's the probability that things deviate significantly? But casinos and the lottery every day operate on this principle that if you take enough people, sure, in the short term or with a few samples, a couple people might beat the house. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And this is an important thing to think about. But this isn't used in practice every day with the lottery and with casinos. Because they know that if you do large enough samples, and we could even calculate if you do large enough samples, what's the probability that things deviate significantly? But casinos and the lottery every day operate on this principle that if you take enough people, sure, in the short term or with a few samples, a couple people might beat the house. But over the long term, the house is always going to win because of the parameters of the games that they're making you play. Anyway, this is an important thing in probability. And I think it's fairly intuitive. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
But casinos and the lottery every day operate on this principle that if you take enough people, sure, in the short term or with a few samples, a couple people might beat the house. But over the long term, the house is always going to win because of the parameters of the games that they're making you play. Anyway, this is an important thing in probability. And I think it's fairly intuitive. Although sometimes when you see it formally explained like this with the random variables, then that's a little bit confusing. All it's saying is that as you take more and more samples, the average of that sample is going to approximate the true average, or I should be a little bit more particular, the mean of your sample is going to converge to the true mean of the population, or to the expected value of the random variable. Anyway, see you in the next video. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And that's not quite what random variables are. Random variables are really ways to map outcomes of random processes to numbers. So if you have a random process, like you're flipping a coin, or you are rolling dice, or you are measuring the rain that might fall tomorrow. So random process. You're really just mapping outcomes of that to numbers. You're quantifying the outcomes. So what's an example of a random variable? | Random variables Probability and Statistics Khan Academy.mp3 |
So random process. You're really just mapping outcomes of that to numbers. You're quantifying the outcomes. So what's an example of a random variable? Well, let's define one right over here. So I'm going to define random variable capital X. And they tend to be denoted by capital letters. | Random variables Probability and Statistics Khan Academy.mp3 |
So what's an example of a random variable? Well, let's define one right over here. So I'm going to define random variable capital X. And they tend to be denoted by capital letters. So random variable capital X, I will define it as, is going to be equal to 1 if my fair die rolls heads. And it's going to be equal to 0 if tails. I could have defined this any way I wanted to. | Random variables Probability and Statistics Khan Academy.mp3 |
And they tend to be denoted by capital letters. So random variable capital X, I will define it as, is going to be equal to 1 if my fair die rolls heads. And it's going to be equal to 0 if tails. I could have defined this any way I wanted to. This is actually a fairly typical way of defining a random variable, especially for a coin flip. But I could have defined this as 100. And I could have defined this as 703. | Random variables Probability and Statistics Khan Academy.mp3 |
I could have defined this any way I wanted to. This is actually a fairly typical way of defining a random variable, especially for a coin flip. But I could have defined this as 100. And I could have defined this as 703. And this would still be a legitimate random variable. It might not be as pure a way of thinking about it as defining 1 as heads and 0 as tails. But that would have been a random variable. | Random variables Probability and Statistics Khan Academy.mp3 |
And I could have defined this as 703. And this would still be a legitimate random variable. It might not be as pure a way of thinking about it as defining 1 as heads and 0 as tails. But that would have been a random variable. Notice, we have taken this random process, flipping a coin, and we've mapped the outcomes of that random process. And we've quantified them, 1 if heads, 0 if tails. So we can define another random variable, capital Y, as equal to, let's say, the sum of rolls of, let's say, 7 dice. | Random variables Probability and Statistics Khan Academy.mp3 |
But that would have been a random variable. Notice, we have taken this random process, flipping a coin, and we've mapped the outcomes of that random process. And we've quantified them, 1 if heads, 0 if tails. So we can define another random variable, capital Y, as equal to, let's say, the sum of rolls of, let's say, 7 dice. And when we talk about the sum, we're talking about the sum of the 7. Let me write this. The sum of the upward face after rolling 7 dice. | Random variables Probability and Statistics Khan Academy.mp3 |
So we can define another random variable, capital Y, as equal to, let's say, the sum of rolls of, let's say, 7 dice. And when we talk about the sum, we're talking about the sum of the 7. Let me write this. The sum of the upward face after rolling 7 dice. Once again, we are quantifying an outcome for a random process. We are the random processes rolling these 7 dice and seeing what sides show up on top. And then we are taking those, and we are taking the sum, and we are defining a random variable in that way. | Random variables Probability and Statistics Khan Academy.mp3 |
The sum of the upward face after rolling 7 dice. Once again, we are quantifying an outcome for a random process. We are the random processes rolling these 7 dice and seeing what sides show up on top. And then we are taking those, and we are taking the sum, and we are defining a random variable in that way. So the natural question you might ask is, why are we doing this? What's so useful about defining random variables like this? It will become more apparent as we get a little bit deeper in probability. | Random variables Probability and Statistics Khan Academy.mp3 |
And then we are taking those, and we are taking the sum, and we are defining a random variable in that way. So the natural question you might ask is, why are we doing this? What's so useful about defining random variables like this? It will become more apparent as we get a little bit deeper in probability. But the simple way of thinking about it is, as soon as you quantify outcomes, you can start to do a little bit more math on the outcomes. And you can start to use a little bit more mathematical notation on the outcome. So for example, if you cared about the probability that the sum of the upward faces after rolling 7 dice, if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of, and you would have to write all of what I just wrote here, is less than or equal to 30. | Random variables Probability and Statistics Khan Academy.mp3 |
It will become more apparent as we get a little bit deeper in probability. But the simple way of thinking about it is, as soon as you quantify outcomes, you can start to do a little bit more math on the outcomes. And you can start to use a little bit more mathematical notation on the outcome. So for example, if you cared about the probability that the sum of the upward faces after rolling 7 dice, if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of, and you would have to write all of what I just wrote here, is less than or equal to 30. You would have had to write that big thing. And if you wanted to write, and then you would try to figure it out somehow if you had some information. But now we can just write the probability that capital Y is less than or equal to 30. | Random variables Probability and Statistics Khan Academy.mp3 |
So for example, if you cared about the probability that the sum of the upward faces after rolling 7 dice, if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of, and you would have to write all of what I just wrote here, is less than or equal to 30. You would have had to write that big thing. And if you wanted to write, and then you would try to figure it out somehow if you had some information. But now we can just write the probability that capital Y is less than or equal to 30. It's a little bit cleaner notation. And if someone else cares about the probability that the sum of the upward face after rolling 7 dice, if they say, hey, what's the probability that that's even, instead of having to write all of that over, they can say, well, what's the probability that Y is even? Now, the one thing that I do want to emphasize is how these are different than traditional variables, traditional variables that you see in your algebra class, like x plus 5 is equal to 6, usually denoted by lowercase variables. | Random variables Probability and Statistics Khan Academy.mp3 |
But now we can just write the probability that capital Y is less than or equal to 30. It's a little bit cleaner notation. And if someone else cares about the probability that the sum of the upward face after rolling 7 dice, if they say, hey, what's the probability that that's even, instead of having to write all of that over, they can say, well, what's the probability that Y is even? Now, the one thing that I do want to emphasize is how these are different than traditional variables, traditional variables that you see in your algebra class, like x plus 5 is equal to 6, usually denoted by lowercase variables. Y is equal to x plus 7. These variables, you can essentially assign values. You either can solve for them. | Random variables Probability and Statistics Khan Academy.mp3 |
Now, the one thing that I do want to emphasize is how these are different than traditional variables, traditional variables that you see in your algebra class, like x plus 5 is equal to 6, usually denoted by lowercase variables. Y is equal to x plus 7. These variables, you can essentially assign values. You either can solve for them. So in this case, x is an unknown. You can subtract 5 from both sides and solve for x. Say that x is going to be equal to 1. | Random variables Probability and Statistics Khan Academy.mp3 |
You either can solve for them. So in this case, x is an unknown. You can subtract 5 from both sides and solve for x. Say that x is going to be equal to 1. In this case, you could say, well, x is going to vary. We can assign a value to x and see how y varies as a function of x. You can either assign variable, you can assign values to them, or you can solve for them. | Random variables Probability and Statistics Khan Academy.mp3 |
Say that x is going to be equal to 1. In this case, you could say, well, x is going to vary. We can assign a value to x and see how y varies as a function of x. You can either assign variable, you can assign values to them, or you can solve for them. You could say, hey, x is going to be 1 in this case. That's not going to be the case with a random variable. A random variable can take on many, many, many, many, many different values with different probabilities. | Random variables Probability and Statistics Khan Academy.mp3 |
You can either assign variable, you can assign values to them, or you can solve for them. You could say, hey, x is going to be 1 in this case. That's not going to be the case with a random variable. A random variable can take on many, many, many, many, many different values with different probabilities. And it makes much more sense to talk about the probability of a random variable equaling a value, or the probability that is less than or greater than something, or the probability that it has some property. And you see that in either of these cases. In the next video, we'll continue this discussion. | Random variables Probability and Statistics Khan Academy.mp3 |
And all this error means is that you've rejected, this is the error of rejecting, let me do this in a different color, rejecting the null hypothesis even though it is true. So for example, in actually all of the hypothesis testing examples we've seen, we start assuming that the null hypothesis is true. We always assume that the null hypothesis is true. And given that the null hypothesis is true, we say OK, if the null hypothesis is true, then the mean is usually going to be equal to some value. So we create some distribution assuming that the null hypothesis is true. It normally has some mean value right over there. And then we have some statistic, and we are seeing if the null hypothesis is true, what is the probability of getting that statistic, or getting a result that extreme or more extreme than that statistic? | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
And given that the null hypothesis is true, we say OK, if the null hypothesis is true, then the mean is usually going to be equal to some value. So we create some distribution assuming that the null hypothesis is true. It normally has some mean value right over there. And then we have some statistic, and we are seeing if the null hypothesis is true, what is the probability of getting that statistic, or getting a result that extreme or more extreme than that statistic? So let's say that the statistic gives us some value over here, and we say, gee, you know what? There's only, I don't know, there might be a 1% chance, there's only a 1% probability of getting a result that extreme or greater. And then if that's, I guess, low enough of a threshold for us, we will reject the null hypothesis. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
And then we have some statistic, and we are seeing if the null hypothesis is true, what is the probability of getting that statistic, or getting a result that extreme or more extreme than that statistic? So let's say that the statistic gives us some value over here, and we say, gee, you know what? There's only, I don't know, there might be a 1% chance, there's only a 1% probability of getting a result that extreme or greater. And then if that's, I guess, low enough of a threshold for us, we will reject the null hypothesis. So in this case, we will. So let's say that, actually, let's think of it this way. Let's say that 1% is our threshold. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
And then if that's, I guess, low enough of a threshold for us, we will reject the null hypothesis. So in this case, we will. So let's say that, actually, let's think of it this way. Let's say that 1% is our threshold. We say, look, we're going to assume that the null hypothesis is true. There's some threshold that if we get a value any more extreme than that value, there's less than a 1% chance of that happening. So let's say we're looking at sample means, and we get a sample mean that is way out here. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let's say that 1% is our threshold. We say, look, we're going to assume that the null hypothesis is true. There's some threshold that if we get a value any more extreme than that value, there's less than a 1% chance of that happening. So let's say we're looking at sample means, and we get a sample mean that is way out here. We say, well, there's less than a 1% chance of that happening, given that the null hypothesis is true. So we are going to reject the null hypothesis. So we will reject the null hypothesis. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
So let's say we're looking at sample means, and we get a sample mean that is way out here. We say, well, there's less than a 1% chance of that happening, given that the null hypothesis is true. So we are going to reject the null hypothesis. So we will reject the null hypothesis. Now what does that mean, though? Let's say that this area, the probability of getting a result like that or that much more extreme is just this area right here. So let's say that's half a percent. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
So we will reject the null hypothesis. Now what does that mean, though? Let's say that this area, the probability of getting a result like that or that much more extreme is just this area right here. So let's say that's half a percent. Maybe I can write it this way. Let's say it's half a percent. And because it's so unlikely to get a statistic like that, assuming that the null hypothesis is true, we decide to reject the null hypothesis. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
So let's say that's half a percent. Maybe I can write it this way. Let's say it's half a percent. And because it's so unlikely to get a statistic like that, assuming that the null hypothesis is true, we decide to reject the null hypothesis. Or another way to view it is, there's a 0.5% chance that we have made a type I error in rejecting the null hypothesis. Because if the null hypothesis is true, there's a 0.5% chance that this could still happen. So in rejecting it, we would make a mistake. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
And because it's so unlikely to get a statistic like that, assuming that the null hypothesis is true, we decide to reject the null hypothesis. Or another way to view it is, there's a 0.5% chance that we have made a type I error in rejecting the null hypothesis. Because if the null hypothesis is true, there's a 0.5% chance that this could still happen. So in rejecting it, we would make a mistake. There's a 0.5% chance we've made a type I error. I just want to clear that up. Hopefully that clarified it for you. | Type 1 errors Inferential statistics Probability and Statistics Khan Academy.mp3 |
I have this article right here from WebMD. And the point of this isn't to poke holes at WebMD. I think they have some great articles and they have some great information on their site. But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese? | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying? | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows? | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast. And this also leads to obesity. But once again, if this is the actual reality, doing the advice that that article's saying might actually be a bad thing. If you eat an unhealthy dinner and then force yourself to eat a breakfast when you're not hungry, that might make the obesity even worse. | Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3 |
We are told that a zookeeper took a random sample of 30 days and observed how much food an elephant ate on each of those days. The sample mean was 350 kilograms and the sample standard deviation was 25 kilograms. The resulting 90% confidence interval for the mean amount of food was from 341 kilograms to 359 kilograms. Which of the following statements is a correct interpretation of the 90% confidence level? So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
Which of the following statements is a correct interpretation of the 90% confidence level? So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. So let's think about it this way. Let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. So let's think about it this way. Let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is. So let's say sample mean and this is their first sample. It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is. So let's say sample mean and this is their first sample. It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample and then if I did another sample, let's say this is the mean of that next sample so that's sample mean two and I have an associated confidence interval and that interval, not only the start and end points will change but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample and then if I did another sample, let's say this is the mean of that next sample so that's sample mean two and I have an associated confidence interval and that interval, not only the start and end points will change but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So now with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So now with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval. So it causes a little bit of unease. See choice, so I'm just gonna put a question mark here. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval. So it causes a little bit of unease. See choice, so I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals, yep, that's what it does, every time you sample, you produce an interval, that capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
See choice, so I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals, yep, that's what it does, every time you sample, you produce an interval, that capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about. If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
Yeah, that's exactly what we're talking about. If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out. In repeated sampling, this method produces a sample mean between 341 kilograms and 359 kilograms in about 90% of samples. No, the confidence interval does not put a constrain on that 90% of the time you will have a sample mean between these values. It is not trying to do that. | Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3 |
He randomly selects 20 students at his school and records their caffeine intake in milligrams and the amount of time studying in a given week. Here is a computer output from a least squares regression analysis on his sample. Assume that all conditions for inference have been met. What is a 95% confidence interval for the slope of the least squares regression line? So if you feel inspired, pause the video and see if you can have a go at it. Otherwise, we'll do this together. Okay, so let's first remind ourselves what's even going on. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
What is a 95% confidence interval for the slope of the least squares regression line? So if you feel inspired, pause the video and see if you can have a go at it. Otherwise, we'll do this together. Okay, so let's first remind ourselves what's even going on. So let's visualize the regression. So our horizontal axis, or our x-axis, that would be our caffeine intake in milligrams. And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
Okay, so let's first remind ourselves what's even going on. So let's visualize the regression. So our horizontal axis, or our x-axis, that would be our caffeine intake in milligrams. And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. And Moussa here, he randomly selects 20 students. And so for each of those students, he sees how much caffeine they consumed and how much time they spent studying and plots them here. And so there'll be 20 data points. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. And Moussa here, he randomly selects 20 students. And so for each of those students, he sees how much caffeine they consumed and how much time they spent studying and plots them here. And so there'll be 20 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. He inputs these data points into a computer in order to fit a least squares regression line. And let's say the least squares regression line looks something like this. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
And so there'll be 20 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. He inputs these data points into a computer in order to fit a least squares regression line. 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. | Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3 |
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