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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. So, and just be clear, there's not a pure 95% chance that the true difference of the true means lies in this. We are just confident that there's a 95% chance, and we always have to put that little confidence there because remember, we didn't actually know the population standard deviations or the population variances. We estimated them with our sample, and because of that, we don't know that it's an exact probability. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
So let me, so it is 3.12. So, and just be clear, there's not a pure 95% chance that the true difference of the true means lies in this. We are just confident that there's a 95% chance, and we always have to put that little confidence there because remember, we didn't actually know the population standard deviations or the population variances. We estimated them with our sample, and because of that, we don't know that it's an exact probability. We just have to say we're confident that it's a 95% probability, and that's why it's really just, we just say it's a confidence interval. It's a pure probability. But it's a pretty neat result. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
We estimated them with our sample, and because of that, we don't know that it's an exact probability. We just have to say we're confident that it's a 95% probability, and that's why it's really just, we just say it's a confidence interval. It's a pure probability. But it's a pretty neat result. We are now, we have this 95% confidence interval, so we're confident that there's a 95% chance that the true difference of these two samples, and remember, the sample means, the means of the sample, the difference between the, let me make it very clear, the difference between the means of the sample is, or let me put it, the sample means, the expected value of the sample means is actually the same thing as the expected value of the populations. And so, what this is giving us is actually a confidence interval for the true difference between the populations. If you were to give everyone, every possible person, diet one, and every possible person, diet two, this is giving us a confidence interval for the true population means. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
But it's a pretty neat result. We are now, we have this 95% confidence interval, so we're confident that there's a 95% chance that the true difference of these two samples, and remember, the sample means, the means of the sample, the difference between the, let me make it very clear, the difference between the means of the sample is, or let me put it, the sample means, the expected value of the sample means is actually the same thing as the expected value of the populations. And so, what this is giving us is actually a confidence interval for the true difference between the populations. If you were to give everyone, every possible person, diet one, and every possible person, diet two, this is giving us a confidence interval for the true population means. And so when you look at this, it looks like diet one actually does do something because in any case, even at the low end of the confidence interval, you still have a greater weight loss than diet two. Hopefully that doesn't confuse you too much. In the next video, we're actually going to do a hypothesis test with the same data. | Confidence interval of difference of means Probability and Statistics Khan Academy.mp3 |
You could go on an island beach vacation. Island. Island beach vacation. You could go skiing on a ski vacation. Or you could go camping. Now those aren't the only possibilities because for each of those vacations, there's different amount of time that you could go on them. So you could go for one day. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
You could go skiing on a ski vacation. Or you could go camping. Now those aren't the only possibilities because for each of those vacations, there's different amount of time that you could go on them. So you could go for one day. You could go for two days. Two days. Or you could go for three days. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
So you could go for one day. You could go for two days. Two days. Or you could go for three days. Three, it's in a different color. You could go for three days. You could go for three days. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
Or you could go for three days. Three, it's in a different color. You could go for three days. You could go for three days. So the first question I'd want to know is, well what is the, and they're gonna randomly pick either a one day ski vacation or a two day island vacation. But the first question I want to know is what are all of the possible outcomes here? What is the sample space? | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
You could go for three days. So the first question I'd want to know is, well what is the, and they're gonna randomly pick either a one day ski vacation or a two day island vacation. But the first question I want to know is what are all of the possible outcomes here? What is the sample space? What is the space from which we are going to pick your particular vacation package? Well for the sample space, we can construct a grid which you can see that I've essentially been constructing while I wrote down all of the possibilities. So let me draw out the sample space with these uneven looking grid lines. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
What is the sample space? What is the space from which we are going to pick your particular vacation package? Well for the sample space, we can construct a grid which you can see that I've essentially been constructing while I wrote down all of the possibilities. So let me draw out the sample space with these uneven looking grid lines. Alright. I think you get the picture. Alright, so you could go, and I'll just abbreviate it. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
So let me draw out the sample space with these uneven looking grid lines. Alright. I think you get the picture. Alright, so you could go, and I'll just abbreviate it. You could go on a one day island. A one day island trip. This one I, this is a one day island trip. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
Alright, so you could go, and I'll just abbreviate it. You could go on a one day island. A one day island trip. This one I, this is a one day island trip. You could go on a two day. Two day. Actually let me just write it this way. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
This one I, this is a one day island trip. You could go on a two day. Two day. Actually let me just write it this way. All of these are gonna be one day. Right, cause on the one day column. All of these are going to be two days. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
Actually let me just write it this way. All of these are gonna be one day. Right, cause on the one day column. All of these are going to be two days. Two days. And all of these are going to be three days cause it's on the three day column. And all of the ones in this row are gonna be island trips. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
All of these are going to be two days. Two days. And all of these are going to be three days cause it's on the three day column. And all of the ones in this row are gonna be island trips. So it's a one day island trip, two day island trip, three day island trip. This second row, it's all ski trips. One day ski trip, two day ski trip, three day ski trip. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
And all of the ones in this row are gonna be island trips. So it's a one day island trip, two day island trip, three day island trip. This second row, it's all ski trips. One day ski trip, two day ski trip, three day ski trip. And then finally everything in this third row, they're camping trips. One day camping trip, two day camping trip, three day camping trip. So just like that, we have constructed the sample space right over here. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
One day ski trip, two day ski trip, three day ski trip. And then finally everything in this third row, they're camping trips. One day camping trip, two day camping trip, three day camping trip. So just like that, we have constructed the sample space right over here. You see that there's one, two, three, four, five, six, seven, eight, nine outcomes. And let's say that each of these outcomes are a little piece of paper and they put it in a barrel and they roll it up. And for our purposes, we can assume that they are all equally likely outcomes. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
So just like that, we have constructed the sample space right over here. You see that there's one, two, three, four, five, six, seven, eight, nine outcomes. And let's say that each of these outcomes are a little piece of paper and they put it in a barrel and they roll it up. And for our purposes, we can assume that they are all equally likely outcomes. So we're gonna assume equally likely outcomes. So if we do assume equally likely outcomes, we can figure out a probability. Maybe you live in some place that's cold and you're really not in the mood to go skiing. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
And for our purposes, we can assume that they are all equally likely outcomes. So we're gonna assume equally likely outcomes. So if we do assume equally likely outcomes, we can figure out a probability. Maybe you live in some place that's cold and you're really not in the mood to go skiing. In fact, you'd like to spend several days away from the snow. So let's ask ourselves a question. What is the probability that you're going to win something at least two days on a vacation without snow? | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
Maybe you live in some place that's cold and you're really not in the mood to go skiing. In fact, you'd like to spend several days away from the snow. So let's ask ourselves a question. What is the probability that you're going to win something at least two days on a vacation without snow? Two days on vacation without snow. You're going to randomly pick one of these nine outcomes. What's the probability that it's going to be at least, it's going to give you a vacation that gives you at least a vacation with two days without snow? | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
What is the probability that you're going to win something at least two days on a vacation without snow? Two days on vacation without snow. You're going to randomly pick one of these nine outcomes. What's the probability that it's going to be at least, it's going to give you a vacation that gives you at least a vacation with two days without snow? Well, let's just think a little bit about it. We know the sample space and we know each of the outcomes are equally likely. There are nine equal outcomes here. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
What's the probability that it's going to be at least, it's going to give you a vacation that gives you at least a vacation with two days without snow? Well, let's just think a little bit about it. We know the sample space and we know each of the outcomes are equally likely. There are nine equal outcomes here. So let's write that down. We got nine equal outcomes. Now how many of the outcomes satisfy this event, this constraint, at least two days of vacation, let me write this, two days without snow, whether it falls or touching it or whatever. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
There are nine equal outcomes here. So let's write that down. We got nine equal outcomes. Now how many of the outcomes satisfy this event, this constraint, at least two days of vacation, let me write this, two days without snow, whether it falls or touching it or whatever. So this is, you're essentially avoiding skiing. You want at least two days on something other than skiing. We're assuming you're not going to go camping in some type of alpine. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
Now how many of the outcomes satisfy this event, this constraint, at least two days of vacation, let me write this, two days without snow, whether it falls or touching it or whatever. So this is, you're essentially avoiding skiing. You want at least two days on something other than skiing. We're assuming you're not going to go camping in some type of alpine. You're camping in some place that's warm. Let's think about these outcomes. So this one is no snow, but it's only one day. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
We're assuming you're not going to go camping in some type of alpine. You're camping in some place that's warm. Let's think about these outcomes. So this one is no snow, but it's only one day. This is two days without snow. So we can circle that one. This is three days without snow. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
So this one is no snow, but it's only one day. This is two days without snow. So we can circle that one. This is three days without snow. So we can circle that one. All of these have snow. This is one day without snow, so we're not going to do this one. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
This is three days without snow. So we can circle that one. All of these have snow. This is one day without snow, so we're not going to do this one. This is two days without snow. And this is three days without snow. And so four of the equally likely outcomes satisfy this constraint. | Probability from compound sample space Statistics and probability 7th grade Khan Academy.mp3 |
Find the range and the mid-range of the following sets of numbers. So what the range tells us is essentially how spread apart these numbers are. And the way that you calculate it is you just take the difference between the largest of these numbers and the smallest of these numbers. And so if we look at the largest of these numbers, I'll circle it in magenta, it looks like it is 94. 94 is larger than every other number here. So that's the largest of the numbers. And from that we want to subtract the smallest of the numbers. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And so if we look at the largest of these numbers, I'll circle it in magenta, it looks like it is 94. 94 is larger than every other number here. So that's the largest of the numbers. And from that we want to subtract the smallest of the numbers. And the smallest of the numbers in our set right over here is 65. So you want to subtract 65 from 94. And this is equal to, let's see, if this was 95 minus 65 would be 30. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And from that we want to subtract the smallest of the numbers. And the smallest of the numbers in our set right over here is 65. So you want to subtract 65 from 94. And this is equal to, let's see, if this was 95 minus 65 would be 30. 94 is one less than that, so it is 29. So the larger this number is, that means the more spread out, the larger the difference between the largest and the smallest number. The smaller this is, that means the tighter the range, just to use the word itself, of the numbers actually are. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And this is equal to, let's see, if this was 95 minus 65 would be 30. 94 is one less than that, so it is 29. So the larger this number is, that means the more spread out, the larger the difference between the largest and the smallest number. The smaller this is, that means the tighter the range, just to use the word itself, of the numbers actually are. So that's the range. The mid-range is one way of thinking, to some degree, of kind of central tendency. So mid-range. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
The smaller this is, that means the tighter the range, just to use the word itself, of the numbers actually are. So that's the range. The mid-range is one way of thinking, to some degree, of kind of central tendency. So mid-range. And what you do with the mid-range is you take the average of the largest number and the smallest number. So here we took the difference, that's the range. The mid-range would be the average of these two numbers. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So mid-range. And what you do with the mid-range is you take the average of the largest number and the smallest number. So here we took the difference, that's the range. The mid-range would be the average of these two numbers. So it would be 94 plus 65. When we talk about average, I'm talking about the arithmetic mean, over 2. So this is going to be what? | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
The mid-range would be the average of these two numbers. So it would be 94 plus 65. When we talk about average, I'm talking about the arithmetic mean, over 2. So this is going to be what? 90 plus 60 is 150. 150 plus 4 plus 5 is 159. 159 divided by 2 is equal to, 150 divided by 2 is 75. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is going to be what? 90 plus 60 is 150. 150 plus 4 plus 5 is 159. 159 divided by 2 is equal to, 150 divided by 2 is 75. 9 divided by 2 is 4 and a half, so this would be 79.5. So it's one kind of way of thinking about the middle of these numbers. Another way is obviously the arithmetic mean, where you actually take the arithmetic mean of everything here. | Finding the range and mid-range Descriptive statistics Probability and Statistics Khan Academy.mp3 |
In the last several videos, we did some fairly hairy mathematics, and you might have even skipped them. But we got to a pretty neat result. We got to a formula for the slope and y-intercept of the best-fitting regression line when you measure the error by the squared distance to that line. And our formula is, and I'll just rewrite it here just so we have something neat to look at. So the slope of that line is going to be the mean of x's times the mean of the y's minus the mean of the xy's. And don't worry, this seems really confusing. We're going to actually do an example of this in a few seconds. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And our formula is, and I'll just rewrite it here just so we have something neat to look at. So the slope of that line is going to be the mean of x's times the mean of the y's minus the mean of the xy's. And don't worry, this seems really confusing. We're going to actually do an example of this in a few seconds. Divided by the mean of x squared minus the mean of the x squared's. And if this looks a little different than what you see in your statistics class or your textbook, you might see this swapped around. If you multiply both the numerator and the denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's, all of that over the mean of the x squared's minus the mean of the x's squared. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
We're going to actually do an example of this in a few seconds. Divided by the mean of x squared minus the mean of the x squared's. And if this looks a little different than what you see in your statistics class or your textbook, you might see this swapped around. If you multiply both the numerator and the denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's, all of that over the mean of the x squared's minus the mean of the x's squared. These are obviously the same thing. They're just multiplying the numerator and the denominator by negative 1, which is the same thing as multiplying the whole thing by 1. And of course, whatever you get for y, oh sorry, whatever you get for m, you can then just substitute back in this, back over here to get your b. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
If you multiply both the numerator and the denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's, all of that over the mean of the x squared's minus the mean of the x's squared. These are obviously the same thing. They're just multiplying the numerator and the denominator by negative 1, which is the same thing as multiplying the whole thing by 1. And of course, whatever you get for y, oh sorry, whatever you get for m, you can then just substitute back in this, back over here to get your b. Your b is going to be equal to the mean of the y's minus your m, whatever m value you got over here. Let me write that in yellow so it's very clear. You solved for the m value here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And of course, whatever you get for y, oh sorry, whatever you get for m, you can then just substitute back in this, back over here to get your b. Your b is going to be equal to the mean of the y's minus your m, whatever m value you got over here. Let me write that in yellow so it's very clear. You solved for the m value here. Minus m times the mean of the x's. And this is all you need. So let's actually put that into practice. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
You solved for the m value here. Minus m times the mean of the x's. And this is all you need. So let's actually put that into practice. So let's say I have 3 points and I'm going to make sure that these points aren't collinear. Because otherwise it wouldn't be interesting. So let me draw 3 points over here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So let's actually put that into practice. So let's say I have 3 points and I'm going to make sure that these points aren't collinear. Because otherwise it wouldn't be interesting. So let me draw 3 points over here. Let's say that one point is the point 1, 1, 2. So this is 1, 2. So we have the point right over here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So let me draw 3 points over here. Let's say that one point is the point 1, 1, 2. So this is 1, 2. So we have the point right over here. We have the point 1, 2. And then we also have the point, let's say we also have the point, oh I don't know, let's say we also have the point 2, 1. Let's say we have the point 2, 1. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So we have the point right over here. We have the point 1, 2. And then we also have the point, let's say we also have the point, oh I don't know, let's say we also have the point 2, 1. Let's say we have the point 2, 1. And then let's say we also have the point 3, I don't know, let's do something a little bit crazy. Let's do 3, 4. So 3, well let's do it over here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
Let's say we have the point 2, 1. And then let's say we also have the point 3, I don't know, let's do something a little bit crazy. Let's do 3, 4. So 3, well let's do it over here. Let's do 4, 3 just so we can actually fit it on the page. So 4, 3 is going to be something right over here. So this is 4, 3. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So 3, well let's do it over here. Let's do 4, 3 just so we can actually fit it on the page. So 4, 3 is going to be something right over here. So this is 4, 3. So those are our 3 points and what we want to do is find the best fitting regression line, which we suspect is going to look something, we'll see what it looks like, but I suspect it's going to look something like that. We'll see what it actually looks like using our formulas, which we have proven. So a good place to start is just to calculate these things ahead of time and then just substitute them back into the equation. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So this is 4, 3. So those are our 3 points and what we want to do is find the best fitting regression line, which we suspect is going to look something, we'll see what it looks like, but I suspect it's going to look something like that. We'll see what it actually looks like using our formulas, which we have proven. So a good place to start is just to calculate these things ahead of time and then just substitute them back into the equation. So what's the mean of our x's? The mean of our x's is going to be 1 plus, I'll do the same colors, 1 plus 2 plus 4 divided by, I'll do this in a neutral color, the mean of our x's divided by 3. And what's this going to be? | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So a good place to start is just to calculate these things ahead of time and then just substitute them back into the equation. So what's the mean of our x's? The mean of our x's is going to be 1 plus, I'll do the same colors, 1 plus 2 plus 4 divided by, I'll do this in a neutral color, the mean of our x's divided by 3. And what's this going to be? 1 plus 2 is 3 plus 4 is 7 divided by 3. It is equal to 7 over 3. Now, what is the mean of our y's? | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And what's this going to be? 1 plus 2 is 3 plus 4 is 7 divided by 3. It is equal to 7 over 3. Now, what is the mean of our y's? The mean of our y's, once again I'm going to do this in a neutral color, the mean of our y's is equal to 2 plus 1 plus 3, all of that over 3. So this is 2 plus 1 is 3 plus 3 is 6 divided by 3 is equal to 2. This is 6 divided by 3 is equal to 2. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
Now, what is the mean of our y's? The mean of our y's, once again I'm going to do this in a neutral color, the mean of our y's is equal to 2 plus 1 plus 3, all of that over 3. So this is 2 plus 1 is 3 plus 3 is 6 divided by 3 is equal to 2. This is 6 divided by 3 is equal to 2. Now, what is the mean of our xy's? Well, over here, it's going to be, so our first xy over here is 1 times 2 plus 2 times 1 plus 4 times 3. And we have 3 of these xy's, so divided by 3. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
This is 6 divided by 3 is equal to 2. Now, what is the mean of our xy's? Well, over here, it's going to be, so our first xy over here is 1 times 2 plus 2 times 1 plus 4 times 3. And we have 3 of these xy's, so divided by 3. So what's this going to be equal to? 2 plus 2, which is 4, 4 plus 12, which is 16. So it's going to be 16 over 3. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And we have 3 of these xy's, so divided by 3. So what's this going to be equal to? 2 plus 2, which is 4, 4 plus 12, which is 16. So it's going to be 16 over 3. Did I get that right? 4 plus 12, yep, 16 over 3. And then the last one we have to calculate is the mean of the x squareds. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So it's going to be 16 over 3. Did I get that right? 4 plus 12, yep, 16 over 3. And then the last one we have to calculate is the mean of the x squareds. So what's the mean of the x squareds? The first x squared is just going to be 1 squared, this 1 squared right over here, plus this 2 squared, plus 2 squared right over here, plus this 4 squared, plus this 4 squared. And we have 3 data points again. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And then the last one we have to calculate is the mean of the x squareds. So what's the mean of the x squareds? The first x squared is just going to be 1 squared, this 1 squared right over here, plus this 2 squared, plus 2 squared right over here, plus this 4 squared, plus this 4 squared. And we have 3 data points again. So this is 1 plus 4, which is 5, plus 16. 5 plus 16 is equal to 21 over 3, which is equal to 7. So that worked out to a pretty neat number. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And we have 3 data points again. So this is 1 plus 4, which is 5, plus 16. 5 plus 16 is equal to 21 over 3, which is equal to 7. So that worked out to a pretty neat number. So let's actually find our m's and our b's. So our slope, our optimal slope for our regression line, the mean of the x's, it's going to be 7 thirds, 7 over 3, times the mean of the y's, the mean of the y's is 2, minus the mean of the xy's, well that's 16 over 3, 16 over 3, and then all of that, all of that over the mean of the x's, the mean of the x's is 7 thirds squared. So 7 over 3 squared, minus the mean of the x squared, so it's going to be minus this 7 right over here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So that worked out to a pretty neat number. So let's actually find our m's and our b's. So our slope, our optimal slope for our regression line, the mean of the x's, it's going to be 7 thirds, 7 over 3, times the mean of the y's, the mean of the y's is 2, minus the mean of the xy's, well that's 16 over 3, 16 over 3, and then all of that, all of that over the mean of the x's, the mean of the x's is 7 thirds squared. So 7 over 3 squared, minus the mean of the x squared, so it's going to be minus this 7 right over here. And now we just have to do a little bit of mathematics here. I'm tempted to get out my calculator, but I'll resist the temptation. It's nice to keep things as fractions. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So 7 over 3 squared, minus the mean of the x squared, so it's going to be minus this 7 right over here. And now we just have to do a little bit of mathematics here. I'm tempted to get out my calculator, but I'll resist the temptation. It's nice to keep things as fractions. So let's see if I can calculate this. So this is going to be equal to, this is 14 over 3, 14 over 3 minus 16 over 3, all of that over, this is 49, 49 over 9, right? 7 thirds squared is 49 over 9. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
It's nice to keep things as fractions. So let's see if I can calculate this. So this is going to be equal to, this is 14 over 3, 14 over 3 minus 16 over 3, all of that over, this is 49, 49 over 9, right? 7 thirds squared is 49 over 9. And then minus 7, if I wanted to express that as something over 9, that's the same thing as 63, that's the same thing as 63 over 9. And so in our numerator, we get negative 2 thirds, negative 2 over 3, and then in our denominator, what's 49 minus 63? That's negative, let's see, that's negative 14, that's negative 14 over 9, and this is the same thing as negative 2 thirds times 9 over 14, negative 9 over 14, and then if we divide the numerator and denominator by 3, well the negatives are going to cancel out first of all, we divide by 3, that becomes a 1, that becomes a 3, divide by 2, becomes a 1, that becomes a 7. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
7 thirds squared is 49 over 9. And then minus 7, if I wanted to express that as something over 9, that's the same thing as 63, that's the same thing as 63 over 9. And so in our numerator, we get negative 2 thirds, negative 2 over 3, and then in our denominator, what's 49 minus 63? That's negative, let's see, that's negative 14, that's negative 14 over 9, and this is the same thing as negative 2 thirds times 9 over 14, negative 9 over 14, and then if we divide the numerator and denominator by 3, well the negatives are going to cancel out first of all, we divide by 3, that becomes a 1, that becomes a 3, divide by 2, becomes a 1, that becomes a 7. So our slope is 3 sevenths, not too bad. Now we can go back and figure out our y intercept. So let's figure out our y intercept using this right over here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
That's negative, let's see, that's negative 14, that's negative 14 over 9, and this is the same thing as negative 2 thirds times 9 over 14, negative 9 over 14, and then if we divide the numerator and denominator by 3, well the negatives are going to cancel out first of all, we divide by 3, that becomes a 1, that becomes a 3, divide by 2, becomes a 1, that becomes a 7. So our slope is 3 sevenths, not too bad. Now we can go back and figure out our y intercept. So let's figure out our y intercept using this right over here. So our y intercept, b, is going to be equal to the mean of the y's, the mean of the y's is 2, minus our slope, we just figured out our slope to be 3 sevenths, so minus 3 sevenths, times the mean of the x's, which is 7 thirds, times 7 thirds. Well these just are the reciprocal of each other, so they cancel out, that just becomes 1. So our y intercept is literally just 2 minus 1, so it equals 1. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So let's figure out our y intercept using this right over here. So our y intercept, b, is going to be equal to the mean of the y's, the mean of the y's is 2, minus our slope, we just figured out our slope to be 3 sevenths, so minus 3 sevenths, times the mean of the x's, which is 7 thirds, times 7 thirds. Well these just are the reciprocal of each other, so they cancel out, that just becomes 1. So our y intercept is literally just 2 minus 1, so it equals 1. So we have the equation for our line. Our regression line is going to be y is equal to, we figured out m, m is 3 sevenths, y is equal to 3 sevenths x, plus our y intercept is 1, plus 1. And we are done. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So our y intercept is literally just 2 minus 1, so it equals 1. So we have the equation for our line. Our regression line is going to be y is equal to, we figured out m, m is 3 sevenths, y is equal to 3 sevenths x, plus our y intercept is 1, plus 1. And we are done. We are done. So let's actually try to graph this. So our y intercept is going to be 1, it's going to be right over there, and the slope of our line is 3 sevenths. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
And we are done. We are done. So let's actually try to graph this. So our y intercept is going to be 1, it's going to be right over there, and the slope of our line is 3 sevenths. So for every 7 we run, we rise 3. Another way to think of it, for every 3 and a half we run, we rise 1 and a half. So we are going to go 1 and a half right over here. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So our y intercept is going to be 1, it's going to be right over there, and the slope of our line is 3 sevenths. So for every 7 we run, we rise 3. Another way to think of it, for every 3 and a half we run, we rise 1 and a half. So we are going to go 1 and a half right over here. So this line, if you were to graph it, and obviously I'm hand drawing it, so it's not going to be that exact, is going to look like that right over there. It actually won't go directly through that line, so I don't want to give you that impression. So it might look something like this. | Regression line example Regression Probability and Statistics Khan Academy.mp3 |
So we've defined two random variables here. The first random variable, X, is the weight of the cereal in a random box of our favorite cereal, Mattheys, a random closed box of our favorite cereal, Mattheys. And we know a few other things about it. We know what the expected value of X is. It is equal to 16 ounces. In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
We know what the expected value of X is. It is equal to 16 ounces. In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certain weight, then the company that produces it just throws out that box. And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certain weight, then the company that produces it just throws out that box. And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. But once again, there's going to be some variation. Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. But once again, there's going to be some variation. Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So given all of this information, what I wanna do is, let's just say I take a random box of mathies, and I take a random filled bowl, and I wanna think about the combined weight in the closed box and the filled bowl. So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So given all of this information, what I wanna do is, let's just say I take a random box of mathies, and I take a random filled bowl, and I wanna think about the combined weight in the closed box and the filled bowl. So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value of Y, and so it would be 16 plus four ounces. In this case, this would be equal to 20 ounces. But what about the variation? | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value of Y, and so it would be 16 plus four ounces. In this case, this would be equal to 20 ounces. But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum? | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer here and a three-ouncer here, and it is 18 ounces. And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer here and a three-ouncer here, and it is 18 ounces. And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were from their respective means. So hopefully this gives you an intuition for why this makes sense. But let me ask you another question. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were from their respective means. So hopefully this gives you an intuition for why this makes sense. But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be? Would you subtract the variances of each of the random variables here? | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be? Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here. That would be the lowest value that you could take on. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here. That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know this. Let me just write it using this sigma notation. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know this. Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64. The standard deviation of Y is 0.6. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64. The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. You add these two up, and you are going to get one. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. You add these two up, and you are going to get one. So the variance of the sum is one. And then if you take the square root of both of these, you get the standard deviation of the sum is also going to be one. And that just happened to work out because we're dealing with the scenario where the variance, where the square root of one is, well, one. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Let's say I'm trying to judge how many years of experience we have at the Khan Academy, or on average, how many years of experience we have. And in particular, the particular type of average we'll focus on is the arithmetic mean. So I go and I survey the folks there. And let's say this was when Khan Academy was a smaller organization, when there were only five people in the organization. And I find, and I'm surveying the entire population, so years of experience, the entire population of Khan Academy, because that's what I care about, years of experience at our organization, at Khan Academy. This is when we had five people. And I were to go, we're now 36 people. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And let's say this was when Khan Academy was a smaller organization, when there were only five people in the organization. And I find, and I'm surveying the entire population, so years of experience, the entire population of Khan Academy, because that's what I care about, years of experience at our organization, at Khan Academy. This is when we had five people. And I were to go, we're now 36 people. I don't want to date this video too much. But let's say I go and I say, OK, there's one person straight out of college. They have one year of experience, or recently out of college, somebody with three years of experience, someone with five years of experience, someone with seven years of experience, and someone very experienced, or reasonably experienced, with 14 years of experience. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And I were to go, we're now 36 people. I don't want to date this video too much. But let's say I go and I say, OK, there's one person straight out of college. They have one year of experience, or recently out of college, somebody with three years of experience, someone with five years of experience, someone with seven years of experience, and someone very experienced, or reasonably experienced, with 14 years of experience. So based on this data point, and this is our population for years of experience. I'm assuming that we only have five people in the organization at this point. What would be the population mean for the years of experience? | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
They have one year of experience, or recently out of college, somebody with three years of experience, someone with five years of experience, someone with seven years of experience, and someone very experienced, or reasonably experienced, with 14 years of experience. So based on this data point, and this is our population for years of experience. I'm assuming that we only have five people in the organization at this point. What would be the population mean for the years of experience? What is the mean years of experience for my population? Well, we can just calculate that. Our mean experience, and I'm going to denote it with mu, because we're talking about the population now. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
What would be the population mean for the years of experience? What is the mean years of experience for my population? Well, we can just calculate that. Our mean experience, and I'm going to denote it with mu, because we're talking about the population now. This is a parameter for the population. It's going to be equal to the sum from our first data point, so data point one, all the way to data point, in this case, data point five. We have five data points. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Our mean experience, and I'm going to denote it with mu, because we're talking about the population now. This is a parameter for the population. It's going to be equal to the sum from our first data point, so data point one, all the way to data point, in this case, data point five. We have five data points. So we're going to take all from the first data point, the second data point, the third data point, all the way to the fifth. So this is going to be equal to x1 plus x, and I'm going to divide it all by the number of data points I have, plus x2, plus x3, plus x4, plus x sub 5, subscript 5, all of that over 5. And as we said, this is a very fancy way of saying, I'm going to sum up all of these things, I'm going to sum up all of these things, and then divide by the number of things we have. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We have five data points. So we're going to take all from the first data point, the second data point, the third data point, all the way to the fifth. So this is going to be equal to x1 plus x, and I'm going to divide it all by the number of data points I have, plus x2, plus x3, plus x4, plus x sub 5, subscript 5, all of that over 5. And as we said, this is a very fancy way of saying, I'm going to sum up all of these things, I'm going to sum up all of these things, and then divide by the number of things we have. So let's do that. Get the calculator out. So I'm going to add them all up. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And as we said, this is a very fancy way of saying, I'm going to sum up all of these things, I'm going to sum up all of these things, and then divide by the number of things we have. So let's do that. Get the calculator out. So I'm going to add them all up. So I'm going to add them all up. 1 plus 3 plus 5. I really don't need a calculator for this. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So I'm going to add them all up. So I'm going to add them all up. 1 plus 3 plus 5. I really don't need a calculator for this. Plus 7 plus 14. So that's five data points, and I'm going to divide by 5. And I get 6. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
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