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You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not j... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that inte... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would mak... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval ... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the cur... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like i... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famo... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be te... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no wid... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than th... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe tha... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be ro... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's t... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So for example, one driver drives one hour a day. Two drivers drive two hours a day. One driver drives three hours a day. It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a da... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount i... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count t... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is a... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 4... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you want... | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
So now I'm going to think about, I'm going to take a fair coin, and I'm going to flip it three times. And I want to find the probability of at least one head out of the three flips. So the easiest way to think about this is how many equally likely possibilities there are. In the last video we saw, if we flip a coin thr... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
In the last video we saw, if we flip a coin three times, there's eight possibilities. For the first flip, there's two possibilities. Second flip, there's two possibilities. And in the third flip, there are two possibilities. So 2 times 2 times 2, there are eight equally likely possibilities if I'm flipping a coin three... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
And in the third flip, there are two possibilities. So 2 times 2 times 2, there are eight equally likely possibilities if I'm flipping a coin three times. Now how many of those possibilities have at least one head? Well, we drew all of the possibilities over here, so we just have to count how many of these have at leas... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
Well, we drew all of the possibilities over here, so we just have to count how many of these have at least one head. So that's 1, 2, 3, 4, 5, 6, 7. So 7 of these have at least one head in them, and this last one does not. So 7 of the 8 have at least one head. Now you're probably thinking, OK, Sal, you were able to do i... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
So 7 of the 8 have at least one head. Now you're probably thinking, OK, Sal, you were able to do it by writing out all of the possibilities, but that would be really hard if I said at least one head out of 20 flips. This would work well because I only had three flips. So let me make it clear. This is in three flips. Th... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
So let me make it clear. This is in three flips. This would have been a lot harder to do or more time-consuming to do if I had 20 flips. Is there some shortcut here? Is there some other way to think about it? And you couldn't just do it in some simple way. You can't just say, oh, probability of heads times probability ... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
Is there some shortcut here? Is there some other way to think about it? And you couldn't just do it in some simple way. You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore. Or you could get heads again, but you do... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore. Or you could get heads again, but you don't have to. So it becomes a little bit more complicated. But there is an easy way to think about it where you could use... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
But there is an easy way to think about it where you could use this methodology right over here. You'll actually see this on a lot of exams where they make it seem like a harder problem, but if you just think about it the right way, all of a sudden it becomes simpler. One way to think about it is the probability of at ... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
If we got all tails, then we don't have at least one head. So these two things are equivalent. The probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips. Let me write in three flips. So what's the probability of not getting all tails? Well, ... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
Let me write in three flips. So what's the probability of not getting all tails? Well, that's going to be 1 minus the probability of getting all tails. And since it's three flips, it's the probability of tails, tails, and tails. Because any of the other situations are going to have at least one head in them. And that's... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
And since it's three flips, it's the probability of tails, tails, and tails. Because any of the other situations are going to have at least one head in them. And that's all of the other possibilities. And this is the only other leftover possibility. If you add them together, you're going to get one. Let me write it thi... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
And this is the only other leftover possibility. If you add them together, you're going to get one. Let me write it this way. The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive. This is all of the possible circumstances. So your chances of getting either not all tai... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive. This is all of the possible circumstances. So your chances of getting either not all tails or all tails, and these are mutual exclusives, so we can add them, so the probability of not all tails or the probability o... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
These are mutual exclusives. You're either going to have not all tails, which means a head shows up, or you're going to have all tails, but you can't have both of these things happening. And since they're mutual exclusives, and you're saying the probability of this or this happening, you can add their probabilities. An... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
And this is essentially all of the possible events. So this is essentially, if you combine these, this is the probability of any of the events happening, and that's going to be a 1 or 100% chance. So another way to think about it is the probability of not all tails is going to be 1 minus the probability of all tails. S... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
So that's what we did right over here. And the probability of all tails is pretty straightforward. That's the probability of it's going to be 1 half, because you have a 1 half chance of getting a tails on the first flip, times, let me write it here so it becomes a little clearer, so this is going to be 1 minus the prob... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
And then 1 half times 1 half times 1 half, this is going to be 1 eighth. And then 1 minus 1 eighth, or 8 eighths minus 1 eighth, is going to be equal to 7 eighths. So we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem, we can say, let's say we have 10 ... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
Well, we use the same idea, this is the same thing, this is going to be equal to the probability of not all tails in 10 flips. So we're just saying the probability of not getting all of the flips going to be tails, all of the flips is tails, not all tails in 10 flips. And this is going to be 1 minus the probability of ... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
So it's 1 minus 10 tails in a row. And so this is going to be equal to, this part right over here, let me write this. So this is going to be this 1, let me just rewrite it, this is equal to 1 minus, and this part is going to be, well, 1 tail, another tail, so it's 1 half times 1 half, and I'm going to do this 10 times.... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
Let me write this a little neater, because I need a 1 half, so that's 5, 6, 7, 8, 9, and 10. And so we really just have to, the numerator is going to be 1. So this is going to be 1, this is going to be equal to 1, let me do it in that same color of green. This is going to be equal to 1 minus, our numerator, you just ha... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
This is going to be equal to 1 minus, our numerator, you just have 1 times itself 10 times, so that's 1. And then on the denominator, you have 2 times 2 is 4, 4 times 2 is 8, 16, 32, 64, 128, 256, 512, 1,024. This is the same exact same thing as 1 is 1024 over 1024 minus 1 over 1024, which is equal to 1023, or 1,023, o... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
So if you flip a coin 10 times in a row, a fair coin, you're probability of getting at least one heads in that 10 flips is pretty high. It's 1,023 over 1,024, and you can get a calculator out to figure that out in terms of a percentage. Actually, let me just do that just for fun. So if we have 1,023 divided by 1,024, t... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
So if we have 1,023 divided by 1,024, that gives us, you have a 99.9% chance that you're going to have at least one heads. So this is, if we round, this is equal to 99.9% chance. And I rounded a little bit. It's actually even slightly higher than that. And this is a pretty powerful tool, or a pretty powerful way to thi... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
It's actually even slightly higher than that. And this is a pretty powerful tool, or a pretty powerful way to think about it, because it would have taken you forever to write all of the scenarios down. In fact, there would have been 1,024 scenarios to write down. So doing this exercise for 10 flips would have taken up ... | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
What proportion of exam scores are higher than Ludwig's score? Give your answer correct to four decimal places. So let's just visualize what's going on here. So the scores are normally distributed. So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looki... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So the scores are normally distributed. So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough. And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
And once again, this is just very rough. And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludw... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that? T... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, ... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of a... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of all, let's figure out the z-score for 47.5. So let's see. We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So let's see. We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5. So that's how much more above the mean. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5. So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three? | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 st... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So her... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So here we go. And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we w... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hund... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scor... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. S... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than ... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062. So that's the proportion. If you thought of it in percent, it would be.62% scores higher than Ludwig. And that makes sense, because Ludwig scored over two standard devi... | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
And we talked about different scenarios. We could use a z-table plus the true population standard deviation, and that actually would construct pretty valid confidence intervals. But the problem is you don't know the population standard deviation. And so you might try to use a z-table to find your critical values, plus ... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
And so you might try to use a z-table to find your critical values, plus the sample standard deviation. But what we talked about is that this doesn't actually do a good job of calculating our confidence intervals. And we're going to see that experimentally in a few seconds. And so instead, we have something called a t-... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
And so instead, we have something called a t-statistic, where if we want our critical value, we use a t-table instead of a z-table. And then we use that in conjunction with our sample standard deviation. And then all of a sudden, we are actually going to have pretty good confidence intervals. To make this a little bit ... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
To make this a little bit more real, let's look at a simulation. So this is a scratch pad on Khan Academy made by Khan Academy user Charlotte Allen. And the whole point there is to see what our confidence intervals look like with these different scenarios. So let's say we have a true population mean of 2.0 for some, le... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
So let's say we have a true population mean of 2.0 for some, let's say it's the average number, the mean number of apples people eat a day. The true population mean is two. That seems high, but maybe it's in a certain country that has a lot of apples. And let's say we know that the population standard deviation is 0.5.... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
And let's say we know that the population standard deviation is 0.5. And we're gonna create confidence intervals with the goal of having a 95% confidence level. And we're gonna take sample sizes of 12. So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it. And so let'... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it. And so let's just draw a bunch of samples here. And so we do see that it looks like it is roughly 95% when we keep making these samples and constructing these confidence intervals, that 95% of the time, these conf... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
So these look like good confidence intervals. But what we've talked about is, normally when you're doing this type of thing, this type of inferential statistics, you don't know the population standard deviation. You don't know sigma. So instead, you might be tempted to use z with our sample standard deviations. But if ... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
So instead, you might be tempted to use z with our sample standard deviations. But if you look at that for these exact same samples we just calculated, notice now when we did it over and over again, we've done this 625 times, in this scenario where we keep calculating the confidence intervals with z and s, the true pop... | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
So let's say that we have two random variables, X and Y, and they are completely independent. They are independent random variables. And I'm just going to go over a little bit of notation here. If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing ... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is th... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the s... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y. There's a variance of Y. Now, you may or may... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third ... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y. What is the expected value of Z going... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is th... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm run... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus th... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is ... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expect... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the differe... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements. So you could view this as being equal to, just using this over here, the s... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same th... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is. Now, what is the expected value of negative Y righ... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared. Ever... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is ju... | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
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