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Our confidence interval would be our sample mean. So it would be the mean of our difference, the mean of our difference plus or minus. Now we don't know the population standard deviation, so we're going to use our sample standard deviation. And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. Now we know most of this data here, and let me just write it down over here. We know the mean, the sample mean right over here is 6.8, so it's going to be 6.8 plus or minus. And now what will be our critical value here? | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. Now we know most of this data here, and let me just write it down over here. We know the mean, the sample mean right over here is 6.8, so it's going to be 6.8 plus or minus. And now what will be our critical value here? Well, we want to have a 95% confidence interval. And what's our degrees of freedom? Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
And now what will be our critical value here? Well, we want to have a 95% confidence interval. And what's our degrees of freedom? Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. And so we're ready to use a t-table. So this is a truncated t-table that I could fit on my screen here. And so there's a couple of ways to think about it. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. And so we're ready to use a t-table. So this is a truncated t-table that I could fit on my screen here. And so there's a couple of ways to think about it. Here they actually give us the confidence level, and the reason why that corresponds to a tail probability of.025 is if you take the middle 95% of a distribution, you're going to have 2.5% on either end. That's going to be your tail probability. So that's all that's going on over there. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
And so there's a couple of ways to think about it. Here they actually give us the confidence level, and the reason why that corresponds to a tail probability of.025 is if you take the middle 95% of a distribution, you're going to have 2.5% on either end. That's going to be your tail probability. So that's all that's going on over there. So we're going to be in this column right over here. And which degree of freedom do we use, or degrees of freedom? Well, it's gonna be four degrees of freedom. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So that's all that's going on over there. So we're going to be in this column right over here. And which degree of freedom do we use, or degrees of freedom? Well, it's gonna be four degrees of freedom. Our sample size is five, five minus one is four. So this is going to be our critical value, 2.776. So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Well, it's gonna be four degrees of freedom. Our sample size is five, five minus one is four. So this is going to be our critical value, 2.776. So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. And then we're going to divide that by the square root of our sample size. So the square root of our sample size, we already wrote a five in there. Sometimes I just write an N there. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. And then we're going to divide that by the square root of our sample size. So the square root of our sample size, we already wrote a five in there. Sometimes I just write an N there. And so what is this going to be equal to? First, let's calculate just the margin of error right over here. So this is going to be 2.776 times 1.64 divided by the square root of five. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Sometimes I just write an N there. And so what is this going to be equal to? First, let's calculate just the margin of error right over here. So this is going to be 2.776 times 1.64 divided by the square root of five. And we get a margin of error of approximately 2.036. So this is going to be 6.8 plus or minus 2.036. It's approximately equal to that, where this is our margin of error. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So this is going to be 2.776 times 1.64 divided by the square root of five. And we get a margin of error of approximately 2.036. So this is going to be 6.8 plus or minus 2.036. It's approximately equal to that, where this is our margin of error. And if we actually wanted to write out the interval, we could just take 6.8 minus this and 6.8 plus that. So let's do that again with the calculator. So 6.8 minus 2.036 is equal to 4.764. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
It's approximately equal to that, where this is our margin of error. And if we actually wanted to write out the interval, we could just take 6.8 minus this and 6.8 plus that. So let's do that again with the calculator. So 6.8 minus 2.036 is equal to 4.764. So our confidence interval starts at 4.764 approximately. And it goes to, let's see, I could actually do this one in my head. If I add 2.036 to 6.8, that is going to be 8.836. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So 6.8 minus 2.036 is equal to 4.764. So our confidence interval starts at 4.764 approximately. And it goes to, let's see, I could actually do this one in my head. If I add 2.036 to 6.8, that is going to be 8.836. Now, how would we interpret this confidence interval right over here? One way to interpret it is to say that we are 95% confident that this interval captures the true mean difference in snaps for these friends. We could also say that there appears to be a difference in the mean number of snaps, since zero is not captured in this interval. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So let's just get to the punch line. Let's solve for the optimal m and b. And just based on what we did in the last videos, there's two ways to do that. We actually now know two points that lie on that line, so we can literally find the slope of that line and then the y-intercept, the b there. Or we could just say it's a solution to this system of equations, and they're actually mathematically equivalent. So let's solve for m first. And if we want to solve for m, we want to cancel out the b. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
We actually now know two points that lie on that line, so we can literally find the slope of that line and then the y-intercept, the b there. Or we could just say it's a solution to this system of equations, and they're actually mathematically equivalent. So let's solve for m first. And if we want to solve for m, we want to cancel out the b. So let me rewrite this top equation just the way it's written over here. So we have m times the mean of the x squareds plus b times the mean of... Actually, we could even do it better than that. One step better than that is to actually, based on the work we did in the last video, we can just subtract this bottom equation from this top equation. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
And if we want to solve for m, we want to cancel out the b. So let me rewrite this top equation just the way it's written over here. So we have m times the mean of the x squareds plus b times the mean of... Actually, we could even do it better than that. One step better than that is to actually, based on the work we did in the last video, we can just subtract this bottom equation from this top equation. So let me subtract it, or let's add the negative. So if I make this negative, this is negative, this is negative, what do we get? We get m times the mean of the x's minus the mean of the x squareds over the mean of x, over the mean of the x's. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
One step better than that is to actually, based on the work we did in the last video, we can just subtract this bottom equation from this top equation. So let me subtract it, or let's add the negative. So if I make this negative, this is negative, this is negative, what do we get? We get m times the mean of the x's minus the mean of the x squareds over the mean of x, over the mean of the x's. The plus b and the negative b cancel out, is equal to the mean of the y's minus the mean of the x y's over the mean of the x's. And then we can divide both sides of the equation by this. And so we get m is equal to the mean of the y's minus the mean of the x y's over the mean of the x's over this. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
We get m times the mean of the x's minus the mean of the x squareds over the mean of x, over the mean of the x's. The plus b and the negative b cancel out, is equal to the mean of the y's minus the mean of the x y's over the mean of the x's. And then we can divide both sides of the equation by this. And so we get m is equal to the mean of the y's minus the mean of the x y's over the mean of the x's over this. The mean of the x's minus the mean of the x squareds over the mean of the x's. Now notice, this is the exact same thing that you would get if you found the slope between these two points over here. Change in y, so the difference between that y and that y is that right over there, over the change in x's. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
And so we get m is equal to the mean of the y's minus the mean of the x y's over the mean of the x's over this. The mean of the x's minus the mean of the x squareds over the mean of the x's. Now notice, this is the exact same thing that you would get if you found the slope between these two points over here. Change in y, so the difference between that y and that y is that right over there, over the change in x's. The change in that x, that x minus that x is exactly this over here. Now to simplify it, we can multiply both the numerator and the denominator by the mean in x. The mean of the x's. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
Change in y, so the difference between that y and that y is that right over there, over the change in x's. The change in that x, that x minus that x is exactly this over here. Now to simplify it, we can multiply both the numerator and the denominator by the mean in x. The mean of the x's. And I do that just so we don't have this in the denominator both places. And so if we multiply the numerator by the mean of the x's, we get the mean of the x's times the mean of the y's minus the mean of the x y's. All of that over, mean of the x's times the mean of the x's is just going to be the mean of the x's squared minus over here you have the mean of the x squared. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
The mean of the x's. And I do that just so we don't have this in the denominator both places. And so if we multiply the numerator by the mean of the x's, we get the mean of the x's times the mean of the y's minus the mean of the x y's. All of that over, mean of the x's times the mean of the x's is just going to be the mean of the x's squared minus over here you have the mean of the x squared. And that's what we get for m. And then if we want to solve for b, we literally can just substitute back into either equation, but this equation right here is simpler. And so if we wanted to solve for b there, we can solve for b in terms of m. We just subtract m times the mean of x's from both sides. We get b is equal to the mean of the y's minus m times the mean of the x's. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
All of that over, mean of the x's times the mean of the x's is just going to be the mean of the x's squared minus over here you have the mean of the x squared. And that's what we get for m. And then if we want to solve for b, we literally can just substitute back into either equation, but this equation right here is simpler. And so if we wanted to solve for b there, we can solve for b in terms of m. We just subtract m times the mean of x's from both sides. We get b is equal to the mean of the y's minus m times the mean of the x's. So what you do is you take your data points, you find the mean of the x's, the mean of the y's, the mean of the x y's, the mean of the x's squared. You find your m. Once you find your m, then you can substitute back in here and you find your b. And then you have your actual optimal line. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
We get b is equal to the mean of the y's minus m times the mean of the x's. So what you do is you take your data points, you find the mean of the x's, the mean of the y's, the mean of the x y's, the mean of the x's squared. You find your m. Once you find your m, then you can substitute back in here and you find your b. And then you have your actual optimal line. And we're done. So these are the two big formula takeaways for our optimal line. What I'm going to do in the next video, and this is where you can kind of, if anyone was skipping up to this point, the next video is where they should reengage. | Proof (part 4) minimizing squared error to regression line Khan Academy.mp3 |
Let's do another problem from the normal distribution section of ck12.org's AP Statistics book. And I'm using theirs because it's open source. It's actually quite a good book. The problems are, I think, good practice for us. So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
The problems are, I think, good practice for us. So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. 9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Not 9.5. 9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
And the standard deviation is equal to 1.1 grams. Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
That's my normal distribution. I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
This is our mean right there. That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So this is our mean plus one standard deviation. This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
That's what the empirical rule tells us. Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Let me draw that out. Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. I think you get the idea. I'm not a computer. 9.5 is the mean. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
This height should be the same as that height there. I think you get the idea. I'm not a computer. 9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean. So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
One standard deviation above the mean. So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
They told us the standard deviation is 1.1. That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
One standard deviation below the mean. One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Doing it on the other side. One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
I said mass because kilograms is actually a unit of mass. But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Let's do part B. Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
This is two standard deviations below. This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So that is this area way out there. I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. This little small area. So what is that probability? So let's turn back to our empirical rule. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So it's this long tail out here. This little small area. So what is that probability? So let's turn back to our empirical rule. Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So let's turn back to our empirical rule. Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
We know the area between minus three standard deviations and plus three standard deviations. We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
We know this area right here. Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. So what do we have left over for the two tails? Remember, there are two tails. This is one of them. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
I mean, almost all of them. So what do we have left over for the two tails? Remember, there are two tails. This is one of them. And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
This is one of them. And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%. And this right here is going to be 0.15%. So the probability of having a one-year-old baby girl in the US that is more than 12.8 kilograms, if you assume a perfect normal distribution, is the area under this curve, the area that is more than three standard deviations above the mean. | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
So we have a type of statistical study described here. I encourage you to pause this video, read it, and see if you can figure out, is this a sample study? Is it an observational study? Is it an experiment? And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
Is it an experiment? And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study? Observational, or is this an experiment? | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study? Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here? | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference. If the farmer is petting the cows and treating them humanely and doing other things, it might not matter whether the farmer names them or not. Likewise, if you take some less nice farmers who hit their cows and they have really inhumane conditions, it might not make a difference whether they name the cows or not. And so it's very important that you, from the observational studies, you might, if they're well-constructed, you might be able to make a, you might be able to say there's a correlation. | Worked example identifying observational study Study design AP Statistics Khan Academy.mp3 |
If you're not, I encourage you to review the videos on that. And we've already done some hypothesis testing with the chi-squared statistic. And we've even done some hypothesis testing based on two-way tables. And now we're going to extend that by thinking about a chi-squared test for association between two variables. So let's say that we suspect that someone's foot length is related to their hand length, that these things are not independent. Well, what we can do is set up a hypothesis test. And remember, the null hypothesis in a hypothesis test is to always assume no news. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And now we're going to extend that by thinking about a chi-squared test for association between two variables. So let's say that we suspect that someone's foot length is related to their hand length, that these things are not independent. Well, what we can do is set up a hypothesis test. And remember, the null hypothesis in a hypothesis test is to always assume no news. So what we could say is here is that there's no association, no association between foot and hand length. Another way to think about it is that they are independent. And oftentimes what we're doing is called a chi-squared test for independence. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And remember, the null hypothesis in a hypothesis test is to always assume no news. So what we could say is here is that there's no association, no association between foot and hand length. Another way to think about it is that they are independent. And oftentimes what we're doing is called a chi-squared test for independence. And then our alternative hypothesis would be our suspicion there is an association. There is an association. So foot and hand length are not independent. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And oftentimes what we're doing is called a chi-squared test for independence. And then our alternative hypothesis would be our suspicion there is an association. There is an association. So foot and hand length are not independent. So what we can then do is go to a population and we can randomly sample it. And so let's say we randomly sample 100 folks. And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
So foot and hand length are not independent. So what we can then do is go to a population and we can randomly sample it. And so let's say we randomly sample 100 folks. And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. And we also do that for the feet. And we tabulate all of the data. And this is the data that we actually get. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. And we also do that for the feet. And we tabulate all of the data. And this is the data that we actually get. Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity. In a chi-squared test for homogeneity, we sample from two different populations, or we look at two different groups, and we see whether the distribution of a certain variable amongst those two different groups is the same. Here we are just sampling from one group, but we're thinking about two different variables for that one group. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And this is the data that we actually get. Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity. In a chi-squared test for homogeneity, we sample from two different populations, or we look at two different groups, and we see whether the distribution of a certain variable amongst those two different groups is the same. Here we are just sampling from one group, but we're thinking about two different variables for that one group. We're thinking about feet length and we're thinking about hand length. And so you can see here that 11 folks had both their right hand longer and their right foot longer. Three folks had their right hand longer but their left foot was longer. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Here we are just sampling from one group, but we're thinking about two different variables for that one group. We're thinking about feet length and we're thinking about hand length. And so you can see here that 11 folks had both their right hand longer and their right foot longer. Three folks had their right hand longer but their left foot was longer. And then eight folks had their right hand longer but both feet were the same. Likewise, we had nine people where their left foot and hand was longer, but you had two people where the left hand was longer but the right foot was longer. And we could go through all of these. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Three folks had their right hand longer but their left foot was longer. And then eight folks had their right hand longer but both feet were the same. Likewise, we had nine people where their left foot and hand was longer, but you had two people where the left hand was longer but the right foot was longer. And we could go through all of these. But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length. So to help us do that, I'm gonna make a total of our columns here and also a total of our rows here. Let me draw a line here so we know what is going on. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
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