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The area under the density curve is one. All right, now let's try to work through it together. If we call this height h, we know how to find the area of a triangle. It's 1 1⁄2 base times height. Area is equal to 1 1⁄2 base times height. We know that the area is one. This is a density curve. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
It's 1 1⁄2 base times height. Area is equal to 1 1⁄2 base times height. We know that the area is one. This is a density curve. So one is going to be equal to, what's the length of the base? We go from one to six. So from one to six, this base, the length of this base is five. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
This is a density curve. So one is going to be equal to, what's the length of the base? We go from one to six. So from one to six, this base, the length of this base is five. 1 1⁄2 times five times height. Or we could say one is equal to 5 1⁄2 times height. Or multiply both sides by 2⁵ to solve for the height. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
So from one to six, this base, the length of this base is five. 1 1⁄2 times five times height. Or we could say one is equal to 5 1⁄2 times height. Or multiply both sides by 2⁵ to solve for the height. And what are we gonna get? We're gonna get the height is equal to 2⁵. So if you have a very clean triangular density curve like this, you can actually figure out the height with even if it was not directly specified. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
So let's say I want to figure out the probability. I'm going to flip a coin eight times, and it's a fair coin. And I want to figure out the probability of getting exactly three out of eight heads. So I say three eight heads. But three of my flips are going to be heads, and the rest are going to be tails. So how do I think about that? Well, let's go back to one of the early definitions we use for probability. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
So I say three eight heads. But three of my flips are going to be heads, and the rest are going to be tails. So how do I think about that? Well, let's go back to one of the early definitions we use for probability. And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true. So the number of events, I guess trials or situations, in which we get three heads. And exactly three heads. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Well, let's go back to one of the early definitions we use for probability. And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true. So the number of events, I guess trials or situations, in which we get three heads. And exactly three heads. We're not saying greater than three heads. So four heads won't count, and two heads won't count, five heads won't. Only three heads. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And exactly three heads. We're not saying greater than three heads. So four heads won't count, and two heads won't count, five heads won't. Only three heads. And then over the total number of equally probable trials. So total number of equally possible outcomes. I should be using the word outcomes. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Only three heads. And then over the total number of equally probable trials. So total number of equally possible outcomes. I should be using the word outcomes. So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes. So let's do the bottom part first. What are the total possible outcomes if I'm flipping a fair coin eight times? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
I should be using the word outcomes. So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes. So let's do the bottom part first. What are the total possible outcomes if I'm flipping a fair coin eight times? Well, the first time I flip it, I either get heads or tails, so I get two outcomes. And then when I flip it again, I get two more outcomes for the second one. And then how many total outcomes? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
What are the total possible outcomes if I'm flipping a fair coin eight times? Well, the first time I flip it, I either get heads or tails, so I get two outcomes. And then when I flip it again, I get two more outcomes for the second one. And then how many total outcomes? Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip. And then essentially, we multiply 2 times the number of flips. So that's 5, 6, 7, 8. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And then how many total outcomes? Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip. And then essentially, we multiply 2 times the number of flips. So that's 5, 6, 7, 8. And that equals 2 to the 8th. So the number of outcomes is just going to be 2 to the total number of flips. And hopefully that makes sense to you. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
So that's 5, 6, 7, 8. And that equals 2 to the 8th. So the number of outcomes is just going to be 2 to the total number of flips. And hopefully that makes sense to you. If not, you might want to re-watch some of the earlier videos. But that's the easy part. So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And hopefully that makes sense to you. If not, you might want to re-watch some of the earlier videos. But that's the easy part. So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. So how many of those outcomes are going to result in exactly three heads? Let's think of it this way. Let's give a name to each of our flips. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. So how many of those outcomes are going to result in exactly three heads? Let's think of it this way. Let's give a name to each of our flips. Let's give a name to them. So let me make a little column. Call these the flips. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Let's give a name to each of our flips. Let's give a name to them. So let me make a little column. Call these the flips. This is my flips column. And I don't know, I could name them anything. I could name them Larry, Curly Moe. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Call these the flips. This is my flips column. And I don't know, I could name them anything. I could name them Larry, Curly Moe. I could name them the, well, I would need five more names for them. But I could name them the seven dwarves, or the eight dwarves, really, because I have eight flips. But I could, you know, I'll just name, I'll number the flips. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
I could name them Larry, Curly Moe. I could name them the, well, I would need five more names for them. But I could name them the seven dwarves, or the eight dwarves, really, because I have eight flips. But I could, you know, I'll just name, I'll number the flips. One, two, three, four, five, six, seven, eight. And I'm the god of probability. And essentially, I need to just pick three of these flips that are going to result in heads. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
But I could, you know, I'll just name, I'll number the flips. One, two, three, four, five, six, seven, eight. And I'm the god of probability. And essentially, I need to just pick three of these flips that are going to result in heads. So another way to think about it is, these could be eight people. And I could pick which of these, you know, how many ways can I pick three of these people to put into the car? Or how many ways can I pick three of these people to sit in chairs? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And essentially, I need to just pick three of these flips that are going to result in heads. So another way to think about it is, these could be eight people. And I could pick which of these, you know, how many ways can I pick three of these people to put into the car? Or how many ways can I pick three of these people to sit in chairs? And it doesn't matter the order that I pick them in, right? It doesn't matter if I say the people that are going to get in the car are going to be people one, two, and three. Or if I say three, two, and one. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Or how many ways can I pick three of these people to sit in chairs? And it doesn't matter the order that I pick them in, right? It doesn't matter if I say the people that are going to get in the car are going to be people one, two, and three. Or if I say three, two, and one. Or if I say two, three, and one. Those are all the same combination, right? So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Or if I say three, two, and one. Or if I say two, three, and one. Those are all the same combination, right? So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. I don't want to confuse you too much. But essentially, I'm just going to choose three things out of the eight. So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. I don't want to confuse you too much. But essentially, I'm just going to choose three things out of the eight. So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three. And that's, you know, how many combinations of three can we pick of eight? And that's, we went over in the last video. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three. And that's, you know, how many combinations of three can we pick of eight? And that's, we went over in the last video. And let's do it with the formula first. So let me write the formula up here, just so you remember it. But I also want to give you the intuition again for the formula. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And that's, we went over in the last video. And let's do it with the formula first. So let me write the formula up here, just so you remember it. But I also want to give you the intuition again for the formula. So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial. So in this situation, it would be, that would equal 8 factorial over 3 factorial times what? 8 minus k times 5 factorial. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
But I also want to give you the intuition again for the formula. So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial. So in this situation, it would be, that would equal 8 factorial over 3 factorial times what? 8 minus k times 5 factorial. Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1. And of course, that and that cancel out. And all you're left with is 8 times 7 times 6 over 3 factorial. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
8 minus k times 5 factorial. Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1. And of course, that and that cancel out. And all you're left with is 8 times 7 times 6 over 3 factorial. And I did this for a reason. Because I want you to re-get the intuition for, at least for this part of the formula. That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And all you're left with is 8 times 7 times 6 over 3 factorial. And I did this for a reason. Because I want you to re-get the intuition for, at least for this part of the formula. That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? And that's essentially saying, well, before I pick anything, I can pick one of eight. Then I have seven left to pick from for the second spot. And then I have six left to pick for the third spot. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? And that's essentially saying, well, before I pick anything, I can pick one of eight. Then I have seven left to pick from for the second spot. And then I have six left to pick for the third spot. And so that's essentially the number of permutations. But since we don't care, you know, if we picked it, what order we pick them in, we need to divide by the number of ways we can rearrange three things. And that's where the 3 factorial comes from. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And then I have six left to pick for the third spot. And so that's essentially the number of permutations. But since we don't care, you know, if we picked it, what order we pick them in, we need to divide by the number of ways we can rearrange three things. And that's where the 3 factorial comes from. And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient. But it's good to have the intuition. And then, well, once we're at this point, we can just calculate this. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And that's where the 3 factorial comes from. And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient. But it's good to have the intuition. And then, well, once we're at this point, we can just calculate this. Well, what's this? This is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1, so that's 6. So 6 cancels out, so it's 8 times 7. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
And then, well, once we're at this point, we can just calculate this. Well, what's this? This is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1, so that's 6. So 6 cancels out, so it's 8 times 7. So there's 8 times 7, or what is that, 56. Right, 56. Yeah, that's equal to 56. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
So 6 cancels out, so it's 8 times 7. So there's 8 times 7, or what is that, 56. Right, 56. Yeah, that's equal to 56. So there's 56 different ways to pick three things out of eight. Or if I have eight people, there's 56 ways of picking three people to sit in the car, or however you want to view it. But if I have eight flips, there's 56 ways of picking three of those flips to be heads. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Yeah, that's equal to 56. So there's 56 different ways to pick three things out of eight. Or if I have eight people, there's 56 ways of picking three people to sit in the car, or however you want to view it. But if I have eight flips, there's 56 ways of picking three of those flips to be heads. So let's go to our original probability problem. What is the probability that I get three out of eight heads? Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
But if I have eight flips, there's 56 ways of picking three of those flips to be heads. So let's go to our original probability problem. What is the probability that I get three out of eight heads? Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? The total number of outcomes is 2 to the 8th. Another way I could write that, 56, let me unseparate, that's 8 times 7 over 2 to the 8th. 8 is 2 to the 3rd, right? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? The total number of outcomes is 2 to the 8th. Another way I could write that, 56, let me unseparate, that's 8 times 7 over 2 to the 8th. 8 is 2 to the 3rd, right? Let me erase some of this. Not with that color. Let me erase all of this so I have space. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
8 is 2 to the 3rd, right? Let me erase some of this. Not with that color. Let me erase all of this so I have space. I will switch colors for variety. OK, so I'm back. All right, so 8 is the same thing as 2 to the 3rd times 7. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Let me erase all of this so I have space. I will switch colors for variety. OK, so I'm back. All right, so 8 is the same thing as 2 to the 3rd times 7. This is all just mathematical simplification, but it's useful, over 2 to the 8th. And so if we just divide both sides, the numerator and the denominator by 2 to the 3rd, this becomes 1, this becomes 2 to the 5th, and so it becomes 7 over 32. Is that right? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
All right, so 8 is the same thing as 2 to the 3rd times 7. This is all just mathematical simplification, but it's useful, over 2 to the 8th. And so if we just divide both sides, the numerator and the denominator by 2 to the 3rd, this becomes 1, this becomes 2 to the 5th, and so it becomes 7 over 32. Is that right? So if I were to pick 3 out of 8, yep, I think that is right. And so what does that turn out to be? Let me get my calculator. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Is that right? So if I were to pick 3 out of 8, yep, I think that is right. And so what does that turn out to be? Let me get my calculator. I often make careless mistakes. Let me see, my calculator seems to have disappeared. Let me get it back. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Let me get my calculator. I often make careless mistakes. Let me see, my calculator seems to have disappeared. Let me get it back. There it is. OK, 7 divided by 32 is equal to 0.21875, which is equal to 21, if I were to round, roughly, 21.9% chance. So there's a little bit better than 1 in 5 chance that I get exactly 3 out of the 8 flips as heads. | Probability using combinations Probability and Statistics Khan Academy.mp3 |
Let me get it back. There it is. OK, 7 divided by 32 is equal to 0.21875, which is equal to 21, if I were to round, roughly, 21.9% chance. So there's a little bit better than 1 in 5 chance that I get exactly 3 out of the 8 flips as heads. Hopefully I didn't confuse you, and now you can apply that to pretty much anything. You could say, well, what is the probability of getting, if I flip a fair coin, of getting exactly 7 out of 8 heads? Or you could say, what's the probability of getting 2 out of 100 heads? | Probability using combinations Probability and Statistics Khan Academy.mp3 |
What we're going to do in this video is dig a little bit deeper into confidence intervals. In other videos, we compute them, we even interpret them, but here we're gonna make sure that we are making the right assumptions so that we can have confidence in our confidence intervals or that we are even calculating them in the right way or in the right context. So just as a bit of review, a lot of what we do in confidence intervals is we are trying to estimate some population parameter. Let's say it's the proportion. Maybe it's the proportion that will vote for a candidate. We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Let's say it's the proportion. Maybe it's the proportion that will vote for a candidate. We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. If you're trying to estimate the proportion of people that are gonna vote for a certain candidate, but you are only surveying people at a senior community, well, that would not be a truly random sample if we were only to survey people on a college campus. So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. If you're trying to estimate the proportion of people that are gonna vote for a certain candidate, but you are only surveying people at a senior community, well, that would not be a truly random sample if we were only to survey people on a college campus. So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume that the distribution of the sample proportions the sampling distribution of the sample proportions has roughly a normal shape like that. But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume that the distribution of the sample proportions the sampling distribution of the sample proportions has roughly a normal shape like that. But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, or 0.5, then you wouldn't meet that normal condition because you would expect five successes and five failures for each sample. Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, or 0.5, then you wouldn't meet that normal condition because you would expect five successes and five failures for each sample. Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem. So you wanna expect, you wanna have at least greater than or equal to 10 successes or failures on each. And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And if we have less than 10 on either one of those, then we are going to have a problem. So you wanna expect, you wanna have at least greater than or equal to 10 successes or failures on each. And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last thing we wanna really make sure is known as the independence condition. Independence condition. And this is the 10% rule. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last thing we wanna really make sure is known as the independence condition. Independence condition. And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's say your population were 100,000 people. And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's say your population were 100,000 people. And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition breaks down. So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition breaks down. So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval that we calculated for that sample. And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval that we calculated for that sample. And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples. This is truly at a 95% confidence level. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples. This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes. And you can see that when I do, once again, I did a bunch of samples here. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes. And you can see that when I do, once again, I did a bunch of samples here. I did over 2,000 samples. Even though I'm trying to set up these confidence intervals that every time I compute it, that over time, that there's kind of a 95% hit rate, so to speak, here there's only a 94% hit rate. And I've done a lot of samples here. | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
100 randomly assigned people are assigned to group 1 and put on the low-fat diet. Another 100 randomly assigned obese people are assigned to group 2 and put on a diet of approximately the same amount of food, but not as low in fat. So group 2 is the control, just the no diet. Group 1 is the low-fat group, so see if it actually works. After 4 months, the mean weight loss was 9.31 pounds for group 1. Let me write this down. The mean weight loss for group 1, let me make it very clear. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
Group 1 is the low-fat group, so see if it actually works. After 4 months, the mean weight loss was 9.31 pounds for group 1. Let me write this down. The mean weight loss for group 1, let me make it very clear. The low-fat group, the mean weight loss was 9.31 pounds. Our sample mean for group 1 is 9.31 pounds, with a sample standard deviation of 4.67. Both of these are obviously very easy to calculate from the actual data. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
The mean weight loss for group 1, let me make it very clear. The low-fat group, the mean weight loss was 9.31 pounds. Our sample mean for group 1 is 9.31 pounds, with a sample standard deviation of 4.67. Both of these are obviously very easy to calculate from the actual data. For our control group, the sample mean is 7.40 pounds for group 2. Our sample mean here for the control is 7.40, with a sample standard deviation of 4.04 pounds. If we look at it superficially, it looks like the low-fat group lost more weight, just based on our samples, than the control group. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
Both of these are obviously very easy to calculate from the actual data. For our control group, the sample mean is 7.40 pounds for group 2. Our sample mean here for the control is 7.40, with a sample standard deviation of 4.04 pounds. If we look at it superficially, it looks like the low-fat group lost more weight, just based on our samples, than the control group. If we take the difference of the low-fat group, or between the low-fat group and the control group, we get 9.31 minus 7.40 is equal to 1.91. The difference of our samples is 1.91. So just based on what we see, it says maybe you lose an incremental 1.91 pounds every 4 months if you are on this diet. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
If we look at it superficially, it looks like the low-fat group lost more weight, just based on our samples, than the control group. If we take the difference of the low-fat group, or between the low-fat group and the control group, we get 9.31 minus 7.40 is equal to 1.91. The difference of our samples is 1.91. So just based on what we see, it says maybe you lose an incremental 1.91 pounds every 4 months if you are on this diet. What we want to do in this video is to get a 95% confidence interval around this number to see that in that 95% confidence interval, maybe do we always lose weight? Or is there a chance that we can actually go the other way with the low-fat diet? So really, just this video, 95% confidence interval. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So just based on what we see, it says maybe you lose an incremental 1.91 pounds every 4 months if you are on this diet. What we want to do in this video is to get a 95% confidence interval around this number to see that in that 95% confidence interval, maybe do we always lose weight? Or is there a chance that we can actually go the other way with the low-fat diet? So really, just this video, 95% confidence interval. In the next video, we'll actually do a hypothesis test using this same data. Now to do a 95% confidence interval, let's think about the distribution that we're thinking about. So let's look at the distribution. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So really, just this video, 95% confidence interval. In the next video, we'll actually do a hypothesis test using this same data. Now to do a 95% confidence interval, let's think about the distribution that we're thinking about. So let's look at the distribution. Of course, we're going to think about the distribution that we're thinking about. We want to think about the distribution of the difference of the means. So it's going to have some true mean here. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So let's look at the distribution. Of course, we're going to think about the distribution that we're thinking about. We want to think about the distribution of the difference of the means. So it's going to have some true mean here. It's going to have some true mean over here, which is the mean of the difference of the sample means. Actually, let me write that. It's not a y. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So it's going to have some true mean here. It's going to have some true mean over here, which is the mean of the difference of the sample means. Actually, let me write that. It's not a y. It's an x1 and x2. So it's the sample mean of x1 minus the sample mean of x2. And then this distribution right here is going to have some standard deviation. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
It's not a y. It's an x1 and x2. So it's the sample mean of x1 minus the sample mean of x2. And then this distribution right here is going to have some standard deviation. It's going to have some standard deviation. So it's the standard deviation of the distribution of x, of the mean of 1 of x1 minus the sample mean of x2. It's going to have some standard deviation here. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
And then this distribution right here is going to have some standard deviation. It's going to have some standard deviation. So it's the standard deviation of the distribution of x, of the mean of 1 of x1 minus the sample mean of x2. It's going to have some standard deviation here. And we want to make an inference about this. I guess this is the best way to think about it. We want to get a 95% confidence interval based on our sample. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
It's going to have some standard deviation here. And we want to make an inference about this. I guess this is the best way to think about it. We want to get a 95% confidence interval based on our sample. We want to create an interval around this where we are confident that there's a 95% chance that this true mean, the true mean of the differences lies within that interval. And to do that, let's just think of it the other way. How can we create a 95% interval around this, around the mean, where we are 95% sure, or construct an interval around this, where we're 95% sure that any sample from this distribution, and this is one of those samples, that there's a 95% chance that we will select from this region right over here. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
We want to get a 95% confidence interval based on our sample. We want to create an interval around this where we are confident that there's a 95% chance that this true mean, the true mean of the differences lies within that interval. And to do that, let's just think of it the other way. How can we create a 95% interval around this, around the mean, where we are 95% sure, or construct an interval around this, where we're 95% sure that any sample from this distribution, and this is one of those samples, that there's a 95% chance that we will select from this region right over here. So we care about a 95% region right over here. So how many standard deviations do we have to go in each direction? And to do that, we just have to look at a z-table. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
How can we create a 95% interval around this, around the mean, where we are 95% sure, or construct an interval around this, where we're 95% sure that any sample from this distribution, and this is one of those samples, that there's a 95% chance that we will select from this region right over here. So we care about a 95% region right over here. So how many standard deviations do we have to go in each direction? And to do that, we just have to look at a z-table. And just remember, if we have 95% in the middle right over here, we're going to have 2.5% over here, and we're going to have 2.5% over here. We have to have 5% split between these two symmetric tails. So when we look at a z-table, we want the critical z-value that they give right over here. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
And to do that, we just have to look at a z-table. And just remember, if we have 95% in the middle right over here, we're going to have 2.5% over here, and we're going to have 2.5% over here. We have to have 5% split between these two symmetric tails. So when we look at a z-table, we want the critical z-value that they give right over here. And we have to be careful here. We're not going to look up 95%, because the z-table gives us a cumulative probability up to that critical z-value. So the z-table is going to be interpreted like this. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So when we look at a z-table, we want the critical z-value that they give right over here. And we have to be careful here. We're not going to look up 95%, because the z-table gives us a cumulative probability up to that critical z-value. So the z-table is going to be interpreted like this. So there's going to be some z-value right over here, where we have 2.5% above it. The probability of getting a more extreme result, or a z-score above that, is 2.5%. And the probability of getting one below that is going to be 97.5%. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So the z-table is going to be interpreted like this. So there's going to be some z-value right over here, where we have 2.5% above it. The probability of getting a more extreme result, or a z-score above that, is 2.5%. And the probability of getting one below that is going to be 97.5%. But if we can find whatever z-value this is right over here, it's going to be the same z-value as that. Instead of thinking about it in terms of a one-tailed scenario, we're going to think of it in a two-tailed scenario. So let's look it up for 97.5% on our z-table. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
And the probability of getting one below that is going to be 97.5%. But if we can find whatever z-value this is right over here, it's going to be the same z-value as that. Instead of thinking about it in terms of a one-tailed scenario, we're going to think of it in a two-tailed scenario. So let's look it up for 97.5% on our z-table. Let's see, we have 97 right here. This is 0.975, or 97.5. And this gives us a z-value of 1.96. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So let's look it up for 97.5% on our z-table. Let's see, we have 97 right here. This is 0.975, or 97.5. And this gives us a z-value of 1.96. So this is z is equal to 1.96. Or, only 2.5% of the results, or of the samples from this population, are going to be more than 1.96 standard deviations away from the mean. So this critical z-value right here is 1.96 standard deviations. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
And this gives us a z-value of 1.96. So this is z is equal to 1.96. Or, only 2.5% of the results, or of the samples from this population, are going to be more than 1.96 standard deviations away from the mean. So this critical z-value right here is 1.96 standard deviations. This is 1.96 times the standard deviation of x1 minus x2. And then this right here is going to be negative 1.96 times the same thing. So let me write that. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So this critical z-value right here is 1.96 standard deviations. This is 1.96 times the standard deviation of x1 minus x2. And then this right here is going to be negative 1.96 times the same thing. So let me write that. So this right here, it's symmetric. This distance is going to be the same as that distance. So this is negative 1.96 times the standard deviation of this distribution. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So let me write that. So this right here, it's symmetric. This distance is going to be the same as that distance. So this is negative 1.96 times the standard deviation of this distribution. And if there's a 95% chance, so let's put it this way. There's a 95% chance that our mean, or I guess we could say that our sample that we got from our distribution, this sample is a difference of these other samples. There's a 95% chance that 1.91 lies within, or let me just write, is within this distance is within 1.96 times the standard deviation of that distribution. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So this is negative 1.96 times the standard deviation of this distribution. And if there's a 95% chance, so let's put it this way. There's a 95% chance that our mean, or I guess we could say that our sample that we got from our distribution, this sample is a difference of these other samples. There's a 95% chance that 1.91 lies within, or let me just write, is within this distance is within 1.96 times the standard deviation of that distribution. So you could view it as a standard error of this statistic. So x1 minus x2. Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got, is within 1.96 times the standard deviation of this distribution of the mean of the distribution, of the true mean of the distribution. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
There's a 95% chance that 1.91 lies within, or let me just write, is within this distance is within 1.96 times the standard deviation of that distribution. So you could view it as a standard error of this statistic. So x1 minus x2. Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got, is within 1.96 times the standard deviation of this distribution of the mean of the distribution, of the true mean of the distribution. Or we could say it the other way around. There's a 95% chance that the true mean of the distribution is within 1.96 times the standard deviation of the distribution of 1.91. These are equivalent statements. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got, is within 1.96 times the standard deviation of this distribution of the mean of the distribution, of the true mean of the distribution. Or we could say it the other way around. There's a 95% chance that the true mean of the distribution is within 1.96 times the standard deviation of the distribution of 1.91. These are equivalent statements. If I say I'm within 3 feet of you, that's equivalent to saying you're within 3 feet of me. That's all that's saying. But when we construct it this way, it becomes pretty clear how do we actually construct the confidence interval. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
These are equivalent statements. If I say I'm within 3 feet of you, that's equivalent to saying you're within 3 feet of me. That's all that's saying. But when we construct it this way, it becomes pretty clear how do we actually construct the confidence interval. We just have to figure out what this distance right over here is. And to figure out what that distance is, we're going to have to figure out what the standard deviation of this distribution is. Well, the standard deviation of the differences of the sample means is going to be equal to, and we saw this in the last video. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
But when we construct it this way, it becomes pretty clear how do we actually construct the confidence interval. We just have to figure out what this distance right over here is. And to figure out what that distance is, we're going to have to figure out what the standard deviation of this distribution is. Well, the standard deviation of the differences of the sample means is going to be equal to, and we saw this in the last video. In fact, I think I have it right at the bottom here. It's going to be equal to the square root of the variances of each of those distributions. Or the variance of this distribution is going to be equal to the sum. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
Well, the standard deviation of the differences of the sample means is going to be equal to, and we saw this in the last video. In fact, I think I have it right at the bottom here. It's going to be equal to the square root of the variances of each of those distributions. Or the variance of this distribution is going to be equal to the sum. Let me write it this way, right over here. So the variance, I'll re-kind of prove it. The variance of the means, or the variance of our distribution, is going to be equal to the sum of the variances of each of these sampling distributions. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
Or the variance of this distribution is going to be equal to the sum. Let me write it this way, right over here. So the variance, I'll re-kind of prove it. The variance of the means, or the variance of our distribution, is going to be equal to the sum of the variances of each of these sampling distributions. And we know that the variance of each of these sampling distributions is equal to the variance of this sampling distribution, is equal to the variance of the population distribution, divided by our sample size, and our sample size in this case is 100. And the variance of this sampling distribution, for our control, let me do this in a new color, for our control is going to be equal to the variance of the population distribution for the control, divided by its sample size. And since we don't know what these are, we can approximate them, especially because our n is greater than 30 for both circumstances, we can approximate these with our sample variances for each of these distributions. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
The variance of the means, or the variance of our distribution, is going to be equal to the sum of the variances of each of these sampling distributions. And we know that the variance of each of these sampling distributions is equal to the variance of this sampling distribution, is equal to the variance of the population distribution, divided by our sample size, and our sample size in this case is 100. And the variance of this sampling distribution, for our control, let me do this in a new color, for our control is going to be equal to the variance of the population distribution for the control, divided by its sample size. And since we don't know what these are, we can approximate them, especially because our n is greater than 30 for both circumstances, we can approximate these with our sample variances for each of these distributions. So let me make this clear. Our sample variances for each of these distributions. So this is going to be our sample variance 1, or actually our sample standard deviation 1 squared, which is the sample variance for that distribution, over 100, plus my sample standard deviation for the control squared, which is the sample variance, standard deviation squared is just the variance, divided by 100. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
And since we don't know what these are, we can approximate them, especially because our n is greater than 30 for both circumstances, we can approximate these with our sample variances for each of these distributions. So let me make this clear. Our sample variances for each of these distributions. So this is going to be our sample variance 1, or actually our sample standard deviation 1 squared, which is the sample variance for that distribution, over 100, plus my sample standard deviation for the control squared, which is the sample variance, standard deviation squared is just the variance, divided by 100. And this will give us the variance for this distribution. And if we want the standard deviation, we just take the square roots of both sides. If we want the standard deviation of this distribution right here, this is the variance right now, so we just need to take the square root. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So this is going to be our sample variance 1, or actually our sample standard deviation 1 squared, which is the sample variance for that distribution, over 100, plus my sample standard deviation for the control squared, which is the sample variance, standard deviation squared is just the variance, divided by 100. And this will give us the variance for this distribution. And if we want the standard deviation, we just take the square roots of both sides. If we want the standard deviation of this distribution right here, this is the variance right now, so we just need to take the square root. So let's calculate this. We actually know these values. S1, our sample standard deviation for group 1 is 4.67. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
If we want the standard deviation of this distribution right here, this is the variance right now, so we just need to take the square root. So let's calculate this. We actually know these values. S1, our sample standard deviation for group 1 is 4.67. We wrote it right here as well. It's 4.67 and 4.04. So this is 4.67, and this number right here is 4.04. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
S1, our sample standard deviation for group 1 is 4.67. We wrote it right here as well. It's 4.67 and 4.04. So this is 4.67, and this number right here is 4.04. The s is 4.67, we're going to have to square it, and the s2 is 4.04, we're going to have to square it. So let's calculate that. So we get, we're going to take the square root of 4.67 squared, divided by 100, plus 4.04 squared, divided by 100, and then close the parentheses, and we get.617. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So this is 4.67, and this number right here is 4.04. The s is 4.67, we're going to have to square it, and the s2 is 4.04, we're going to have to square it. So let's calculate that. So we get, we're going to take the square root of 4.67 squared, divided by 100, plus 4.04 squared, divided by 100, and then close the parentheses, and we get.617. So this is equal to, let me write it right here, this is going to be equal to 0.617. So if we go back up over here, we calculated the standard deviation of this distribution to be 0.617. So now we can actually calculate our interval, because this is going to be 0.617. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So we get, we're going to take the square root of 4.67 squared, divided by 100, plus 4.04 squared, divided by 100, and then close the parentheses, and we get.617. So this is equal to, let me write it right here, this is going to be equal to 0.617. So if we go back up over here, we calculated the standard deviation of this distribution to be 0.617. So now we can actually calculate our interval, because this is going to be 0.617. So if you want 1.96 times that, so we get 1.96 times that.617, I'll just write the answer we just got, so we get 1.21. So this is, this number right here, this number right here is 1.21. So the 95% confidence interval is going to be, is going to be the difference of our means, 1.91 plus or minus this number, 1.21. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So now we can actually calculate our interval, because this is going to be 0.617. So if you want 1.96 times that, so we get 1.96 times that.617, I'll just write the answer we just got, so we get 1.21. So this is, this number right here, this number right here is 1.21. So the 95% confidence interval is going to be, is going to be the difference of our means, 1.91 plus or minus this number, 1.21. So what's our confidence interval? If we subtract it, so the low end of our confidence interval, and I'm running out of space, the low end, 1.91 minus 1.21 is just, what is that? That's just.7. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So the 95% confidence interval is going to be, is going to be the difference of our means, 1.91 plus or minus this number, 1.21. So what's our confidence interval? If we subtract it, so the low end of our confidence interval, and I'm running out of space, the low end, 1.91 minus 1.21 is just, what is that? That's just.7. So the low end is.7, and then the high end, 1.91 plus 1.21, what is that? That's 2.12. Let me just make sure that my brain sometimes doesn't work properly when I'm making these videos. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
That's just.7. So the low end is.7, and then the high end, 1.91 plus 1.21, what is that? That's 2.12. Let me just make sure that my brain sometimes doesn't work properly when I'm making these videos. 3.12, good thing I read it, 3.12, of course. Yeah, 3.12. So let me, so it is 3.12. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
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