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So for example, in this, let me draw a grid here, just to make it a little bit neater. So let me draw a line there, and then a line right over there. Let me draw, actually, several of these, just so that we can really do this a little bit clearer. So let me draw a full grid. All right. And then let me draw the vertical lines. Only a few more left. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So let me draw a full grid. All right. And then let me draw the vertical lines. Only a few more left. There we go. Now, all of this top row, these are the outcomes where I roll a 1 on the first die. So I roll a 1 on the first die. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Only a few more left. There we go. Now, all of this top row, these are the outcomes where I roll a 1 on the first die. So I roll a 1 on the first die. These are all of those outcomes. And this would be I run a 1 on the second die, but I'll fill that in later. These are all the outcomes where I roll a 2 on the first die. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So I roll a 1 on the first die. These are all of those outcomes. And this would be I run a 1 on the second die, but I'll fill that in later. These are all the outcomes where I roll a 2 on the first die. This is where I roll a 3 on the first die. 4, I think you get the idea, on the first die. And then a 5 on the first die. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
These are all the outcomes where I roll a 2 on the first die. This is where I roll a 3 on the first die. 4, I think you get the idea, on the first die. And then a 5 on the first die. 5. And then finally, this last row is all the outcomes where I roll a 6 on the first die. Now, we can go through the columns. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then a 5 on the first die. 5. And then finally, this last row is all the outcomes where I roll a 6 on the first die. Now, we can go through the columns. And this first column is where we roll a 1 on the second die. This is where we roll a 2 on the second die. Let's draw that out, write it out, fill in the chart. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Now, we can go through the columns. And this first column is where we roll a 1 on the second die. This is where we roll a 2 on the second die. Let's draw that out, write it out, fill in the chart. Here's where we roll a 3 on the second die. This is a comma that I'm doing between the two numbers. Here's where we have a 4. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's draw that out, write it out, fill in the chart. Here's where we roll a 3 on the second die. This is a comma that I'm doing between the two numbers. Here's where we have a 4. And then here's where we roll a 5 on the second die. Just filling this in. And then filling this in. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Here's where we have a 4. And then here's where we roll a 5 on the second die. Just filling this in. And then filling this in. This last column is where we roll a 6 on the second die. Now, every one of these represents a possible outcome. This outcome is where we roll a 1 on the first die and a 1 on the second die. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then filling this in. This last column is where we roll a 6 on the second die. Now, every one of these represents a possible outcome. This outcome is where we roll a 1 on the first die and a 1 on the second die. This outcome is where we roll a 3 on the first die, a 2 on the second die. This outcome is where we roll a 4 on the first die and a 5 on the second die. And you can see here, there are 36 possible outcomes. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
This outcome is where we roll a 1 on the first die and a 1 on the second die. This outcome is where we roll a 3 on the first die, a 2 on the second die. This outcome is where we roll a 4 on the first die and a 5 on the second die. And you can see here, there are 36 possible outcomes. 6 times 6 possible outcomes. Now, with this out of the way, how many of these outcomes satisfy our criteria, satisfy the criteria of rolling doubles on two six-sided dice? How many of these outcomes are essentially described by our event? | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And you can see here, there are 36 possible outcomes. 6 times 6 possible outcomes. Now, with this out of the way, how many of these outcomes satisfy our criteria, satisfy the criteria of rolling doubles on two six-sided dice? How many of these outcomes are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling a 1 and a 1. It's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
How many of these outcomes are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling a 1 and a 1. It's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6 events satisfy this event or are the outcomes that are consistent with this event. Now, given that, let's answer our question. What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
It's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6 events satisfy this event or are the outcomes that are consistent with this event. Now, given that, let's answer our question. What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? Well, the probability is going to be equal to the number of outcomes that satisfy our criteria, or the number of outcomes for this event, which are 6. We just figured that out. Over the total number of outcomes, over the size of our sample space. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? Well, the probability is going to be equal to the number of outcomes that satisfy our criteria, or the number of outcomes for this event, which are 6. We just figured that out. Over the total number of outcomes, over the size of our sample space. So this right over here, we have 36 total outcomes. So we have 36 outcomes. And if you simplify this, 6 over 36 is the same thing as 1, 6. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Ludwig got a score of 47.5 points on the exam. 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 looking. The mean is 40 points. So that would be 40 points right over there. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. And so this would be 43. This would be 37 right over here. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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? 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, the entire area under the curve is one. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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, 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 all, let's figure out the z-score for 47.5. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. We know what that's gonna be. That's 7.5. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. But how many standard deviations is that? Well, each standard deviation is three. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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 here we go. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. So what I have here, so I have 2.5. So I am going to be in this row. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 scores higher than Ludwig, but you'd be wrong. 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. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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. 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 gonna be the proportion that scores more than him. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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 a certain z-score. So this gives us right over here, 0.0062. So that's the proportion. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
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 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 deviations, two and a half standard deviations above the mean. So our answer here is 0.0062. | Standard normal table for proportion above AP Statistics Khan Academy.mp3 |
In a previous video, we talked about trying to estimate a population mean with a sample mean, and then constructing a confidence interval about that sample mean. 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 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. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
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 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-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. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
And we're going to see that experimentally in a few seconds. 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 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. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
And then all of a sudden, we are actually going to have pretty good confidence intervals. 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, 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. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
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, 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. 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. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
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. 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'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 confidence intervals contain our true population mean. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
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'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 confidence intervals contain our true population mean. 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. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
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 confidence intervals contain our true population mean. 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 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 population mean is contained in the intervals only 92.2% of the time. And we could keep going. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
You don't know sigma. 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 population mean is contained in the intervals only 92.2% of the time. And we could keep going. So we have a much lower hit rate than we would hope to have if we were actually using z and sigma. Now what's neat is if we use t, use a t-table, notice this is getting much closer. And this is neat because with a t-table and something that we can actually get from the sample, the sample standard deviation, we're actually able to have a pretty close hit rate to what we would have if we actually knew the population standard deviation. | Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3 |
I want to build on what we did on the last video a little bit. So let's say we have two random variables. So I have random variable X, and let me draw its probability distribution. And actually, it doesn't have to be normal, but I'll just draw it as a normal distribution. So this is the distribution of random variable X. This is the population mean of random variable X. And then it has some type of standard deviation. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And actually, it doesn't have to be normal, but I'll just draw it as a normal distribution. So this is the distribution of random variable X. This is the population mean of random variable X. And then it has some type of standard deviation. Or actually, let me just focus on the variance. So it has some variance right here for random variable X. Now let's say, let me just write this is X, the distribution for X. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And then it has some type of standard deviation. Or actually, let me just focus on the variance. So it has some variance right here for random variable X. Now let's say, let me just write this is X, the distribution for X. And let's say we have another random variable, random variable Y. Let's do the same thing for it. Let's draw its distribution. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Now let's say, let me just write this is X, the distribution for X. And let's say we have another random variable, random variable Y. Let's do the same thing for it. Let's draw its distribution. And let me draw the parameters for that distribution. So it has some true mean, some population mean for the random variable Y. And it has some variance right over here. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Let's draw its distribution. And let me draw the parameters for that distribution. So it has some true mean, some population mean for the random variable Y. And it has some variance right over here. So it has some variance to this distribution. And I've drawn it roughly normal. Once again, we don't have to assume that it's normal, because we're going to assume when we go to the next level that when we take the samples, we're taking enough samples that the central limit theorem will actually apply. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And it has some variance right over here. So it has some variance to this distribution. And I've drawn it roughly normal. Once again, we don't have to assume that it's normal, because we're going to assume when we go to the next level that when we take the samples, we're taking enough samples that the central limit theorem will actually apply. But with that said, let's think about the sampling distributions of each of these random variables. So let's think about the sampling distribution of the sample mean of X. When the sample size, and let's say the sample size over here is going to be equal to n. And actually over here, I'm going, well I'll stay in green right now. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Once again, we don't have to assume that it's normal, because we're going to assume when we go to the next level that when we take the samples, we're taking enough samples that the central limit theorem will actually apply. But with that said, let's think about the sampling distributions of each of these random variables. So let's think about the sampling distribution of the sample mean of X. When the sample size, and let's say the sample size over here is going to be equal to n. And actually over here, I'm going, well I'll stay in green right now. So what is that going to look like? Well it's going to be some distribution. And now we're assuming that n is a fairly large number. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
When the sample size, and let's say the sample size over here is going to be equal to n. And actually over here, I'm going, well I'll stay in green right now. So what is that going to look like? Well it's going to be some distribution. And now we're assuming that n is a fairly large number. So this is going to be a normal distribution, or can be approximated with a normal distribution. Notice I drew it having a, let me shift it over a little bit. I'm going to draw it a little bit narrow, because we learned from the central limit theorem that the standard deviation of this thing, that the standard deviation, so let me draw the mean. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And now we're assuming that n is a fairly large number. So this is going to be a normal distribution, or can be approximated with a normal distribution. Notice I drew it having a, let me shift it over a little bit. I'm going to draw it a little bit narrow, because we learned from the central limit theorem that the standard deviation of this thing, that the standard deviation, so let me draw the mean. So the population mean of the sampling distribution is going to be, we're going to denote it with this X bar. That tells us the distribution of the means when the sample size is n. And we know that this is going to be the same thing as the population mean for that random variable. And we know from the central limit theorem that the variance of the sampling distribution, or often called the standard error of the mean, is going to be equal to the population variance divided by this n right over here. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
I'm going to draw it a little bit narrow, because we learned from the central limit theorem that the standard deviation of this thing, that the standard deviation, so let me draw the mean. So the population mean of the sampling distribution is going to be, we're going to denote it with this X bar. That tells us the distribution of the means when the sample size is n. And we know that this is going to be the same thing as the population mean for that random variable. And we know from the central limit theorem that the variance of the sampling distribution, or often called the standard error of the mean, is going to be equal to the population variance divided by this n right over here. And if you wanted the standard deviation of this, you just take the square root of both sides. Now let's do the same thing for random variable y. So let's take the sampling distribution of the sample mean, but here we're talking about y, random variable y. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And we know from the central limit theorem that the variance of the sampling distribution, or often called the standard error of the mean, is going to be equal to the population variance divided by this n right over here. And if you wanted the standard deviation of this, you just take the square root of both sides. Now let's do the same thing for random variable y. So let's take the sampling distribution of the sample mean, but here we're talking about y, random variable y. And let's just say it has a different sample size. It doesn't have to be a different one, but it just shows you that it doesn't have to be the same. So it has a sample size. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So let's take the sampling distribution of the sample mean, but here we're talking about y, random variable y. And let's just say it has a different sample size. It doesn't have to be a different one, but it just shows you that it doesn't have to be the same. So it has a sample size. Let's say it has a sample size of m. So let me draw its distribution right over here. Once again, it'll be a narrower distribution than the population distribution. And it will be approximately normal, assuming that we have a large enough sample size. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So it has a sample size. Let's say it has a sample size of m. So let me draw its distribution right over here. Once again, it'll be a narrower distribution than the population distribution. And it will be approximately normal, assuming that we have a large enough sample size. And its mean, the sampling distribution of the sample mean, is going to be the same thing as the population mean. We've seen that multiple times. Same thing as the population mean. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And it will be approximately normal, assuming that we have a large enough sample size. And its mean, the sampling distribution of the sample mean, is going to be the same thing as the population mean. We've seen that multiple times. Same thing as the population mean. And its variance, so the variance over here, so the variance for the sample means, or the standard error of the mean, actually this isn't the standard error. This is the, I guess you could, well, standard error would be the square root of this. So if I call this the standard error mean, that's wrong. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Same thing as the population mean. And its variance, so the variance over here, so the variance for the sample means, or the standard error of the mean, actually this isn't the standard error. This is the, I guess you could, well, standard error would be the square root of this. So if I call this the standard error mean, that's wrong. The standard error of the mean is the square root of this. It's the standard deviation. This is the variance of the mean. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So if I call this the standard error mean, that's wrong. The standard error of the mean is the square root of this. It's the standard deviation. This is the variance of the mean. The variance of the mean, don't want to confuse you. So the variance of the mean here is going to be the exact same thing. It's going to be the variance of the population divided by our sample size. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
This is the variance of the mean. The variance of the mean, don't want to confuse you. So the variance of the mean here is going to be the exact same thing. It's going to be the variance of the population divided by our sample size. And everything we've done so far is complete review. It's a little different, because I'm actually doing it with two different random variables. And I'm doing it with two different random variables for a reason. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
It's going to be the variance of the population divided by our sample size. And everything we've done so far is complete review. It's a little different, because I'm actually doing it with two different random variables. And I'm doing it with two different random variables for a reason. Because now I'm going to define a new random variable. I'm now going to define a new random variable. That is, well, we could just call it z. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And I'm doing it with two different random variables for a reason. Because now I'm going to define a new random variable. I'm now going to define a new random variable. That is, well, we could just call it z. We'll just call it z. But z is equal to the difference of our sample means. And let me stay with the colors. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
That is, well, we could just call it z. We'll just call it z. But z is equal to the difference of our sample means. And let me stay with the colors. It's equal to the x sample mean minus the y sample mean. So what does that really mean? Well, to get a sample mean, or at least for this distribution, you're taking n samples from this population over here. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And let me stay with the colors. It's equal to the x sample mean minus the y sample mean. So what does that really mean? Well, to get a sample mean, or at least for this distribution, you're taking n samples from this population over here. Maybe n is 10. You're taking 10 samples and finding its mean. That sample mean is a random variable. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Well, to get a sample mean, or at least for this distribution, you're taking n samples from this population over here. Maybe n is 10. You're taking 10 samples and finding its mean. That sample mean is a random variable. You could view that sample mean. Let's say you take 10 samples from here and you get 9.2 when you find their mean. That 9.2 can be viewed as a sample from this distribution right over here. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
That sample mean is a random variable. You could view that sample mean. Let's say you take 10 samples from here and you get 9.2 when you find their mean. That 9.2 can be viewed as a sample from this distribution right over here. Same thing if this right here is m. Or if m right here is 12. You're taking 12 samples, taking its mean. And that sample mean, maybe it's 15.2, could be viewed as a sample from this distribution, as a sample from the sampling distribution. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
That 9.2 can be viewed as a sample from this distribution right over here. Same thing if this right here is m. Or if m right here is 12. You're taking 12 samples, taking its mean. And that sample mean, maybe it's 15.2, could be viewed as a sample from this distribution, as a sample from the sampling distribution. So what z is, z is a random variable where you're taking n samples from this distribution up here, this population distribution, taking its mean. Then you're taking m samples from this population distribution up here, taking its mean, and then finding the difference between that mean and that mean. So it's another random variable. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And that sample mean, maybe it's 15.2, could be viewed as a sample from this distribution, as a sample from the sampling distribution. So what z is, z is a random variable where you're taking n samples from this distribution up here, this population distribution, taking its mean. Then you're taking m samples from this population distribution up here, taking its mean, and then finding the difference between that mean and that mean. So it's another random variable. But what is the distribution of z? What is going to be the distribution of z? So let's draw it. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So it's another random variable. But what is the distribution of z? What is going to be the distribution of z? So let's draw it. Let's draw it like this. Well, there's a couple of things we immediately know about z. And we kind of came up with this in the last video. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So let's draw it. Let's draw it like this. Well, there's a couple of things we immediately know about z. And we kind of came up with this in the last video. So the mean of z, instead of writing z, I'm just going to write the mean of x. Let me do that same shade of green. The mean of x bar, which is the mean of x minus, or a sample from the sampling distribution of x, or the sample mean of x, minus the sample mean of y. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And we kind of came up with this in the last video. So the mean of z, instead of writing z, I'm just going to write the mean of x. Let me do that same shade of green. The mean of x bar, which is the mean of x minus, or a sample from the sampling distribution of x, or the sample mean of x, minus the sample mean of y. So the mean of this is going to be equal to, and we saw this in the last video. In fact, I think I still have the work up here. Yeah, I still have the work right up here. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
The mean of x bar, which is the mean of x minus, or a sample from the sampling distribution of x, or the sample mean of x, minus the sample mean of y. So the mean of this is going to be equal to, and we saw this in the last video. In fact, I think I still have the work up here. Yeah, I still have the work right up here. The mean of the difference is going to be the difference of the means. The mean of the difference is the same thing as the difference of the means. So the mean of this new distribution right over here is going to be the same thing as it's going to be the mean of our sample mean minus the mean of our sample mean of y. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Yeah, I still have the work right up here. The mean of the difference is going to be the difference of the means. The mean of the difference is the same thing as the difference of the means. So the mean of this new distribution right over here is going to be the same thing as it's going to be the mean of our sample mean minus the mean of our sample mean of y. And this might seem a little abstract in this video. In the next video, we're actually going to do this with concrete numbers. And hopefully, it'll make a little bit more sense. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So the mean of this new distribution right over here is going to be the same thing as it's going to be the mean of our sample mean minus the mean of our sample mean of y. And this might seem a little abstract in this video. In the next video, we're actually going to do this with concrete numbers. And hopefully, it'll make a little bit more sense. Just so you know where we're going with this, the point of this is so that we can eventually do some inferential statistics about differences of means. How likely is a difference of means of two samples, random chance or not random chance? Or what is a confidence interval of the difference of means? | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And hopefully, it'll make a little bit more sense. Just so you know where we're going with this, the point of this is so that we can eventually do some inferential statistics about differences of means. How likely is a difference of means of two samples, random chance or not random chance? Or what is a confidence interval of the difference of means? That's what this is all building up to. So anyway, we know the mean of this distribution right over here. And what's the variance of this distribution? | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Or what is a confidence interval of the difference of means? That's what this is all building up to. So anyway, we know the mean of this distribution right over here. And what's the variance of this distribution? And we came up with that result in the last video. If we're taking essentially the difference of two random variables, the variance is going to be the sum of those two random variables. And the whole point of that video is to show you that, hey, it's not the difference of the variances. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And what's the variance of this distribution? And we came up with that result in the last video. If we're taking essentially the difference of two random variables, the variance is going to be the sum of those two random variables. And the whole point of that video is to show you that, hey, it's not the difference of the variances. It's the sum of the variances. So the variance of this new distribution, and I haven't drawn the distribution yet, the variance of this new distribution, I'll just write x bar minus y bar, is going to be equal to the sum of the variances of each of these distributions. The variance of x bar plus the variance of y bar. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And the whole point of that video is to show you that, hey, it's not the difference of the variances. It's the sum of the variances. So the variance of this new distribution, and I haven't drawn the distribution yet, the variance of this new distribution, I'll just write x bar minus y bar, is going to be equal to the sum of the variances of each of these distributions. The variance of x bar plus the variance of y bar. Now, actually, let me just draw this here, just so we can visualize another distribution. Although, all I'm going to draw is another normal distribution. So this is its mean. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
The variance of x bar plus the variance of y bar. Now, actually, let me just draw this here, just so we can visualize another distribution. Although, all I'm going to draw is another normal distribution. So this is its mean. So the mean over here, let me scroll down a little bit. So the mean over here, mean of x bar minus y bar, is going to be equal to the difference of these means over here. So I have to rewrite it. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So this is its mean. So the mean over here, let me scroll down a little bit. So the mean over here, mean of x bar minus y bar, is going to be equal to the difference of these means over here. So I have to rewrite it. And then let me draw the curve. And notice, I'm drawing a fatter curve. I'm drawing a fatter curve than either one. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So I have to rewrite it. And then let me draw the curve. And notice, I'm drawing a fatter curve. I'm drawing a fatter curve than either one. And why am I doing that? Because the variance here is the sum of the variances here. So we're going to have a fatter curve. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
I'm drawing a fatter curve than either one. And why am I doing that? Because the variance here is the sum of the variances here. So we're going to have a fatter curve. It's going to have a bigger variance or a bigger standard deviation than either of these. So then we have some variance here. Variance of x bar minus y bar. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
So we're going to have a fatter curve. It's going to have a bigger variance or a bigger standard deviation than either of these. So then we have some variance here. Variance of x bar minus y bar. Now, what are these in terms of the original population distribution? Well, we came up with those results right over here. We know what the standard deviation. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
Variance of x bar minus y bar. Now, what are these in terms of the original population distribution? Well, we came up with those results right over here. We know what the standard deviation. We know that this thing is the same thing as the variance of the population distribution divided by n. We've done this multiple, multiple times. So this is going to be equal to, what's this going to be equal to? This part right here is the same thing as the variance of our population distribution. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
We know what the standard deviation. We know that this thing is the same thing as the variance of the population distribution divided by n. We've done this multiple, multiple times. So this is going to be equal to, what's this going to be equal to? This part right here is the same thing as the variance of our population distribution. And the x just means this is for random variable x. But there's no bar on top. This is the actual population distribution, not the sampling distribution of the sample mean. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
This part right here is the same thing as the variance of our population distribution. And the x just means this is for random variable x. But there's no bar on top. This is the actual population distribution, not the sampling distribution of the sample mean. So that divided by n. And then if we want the variance of the sampling distribution for y, let me do that in a different color. I'll use blue because that was what we were using for the y random variable. That's going to be equal to this thing over here. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
This is the actual population distribution, not the sampling distribution of the sample mean. So that divided by n. And then if we want the variance of the sampling distribution for y, let me do that in a different color. I'll use blue because that was what we were using for the y random variable. That's going to be equal to this thing over here. And we've done this multiple times. Same exact logic as this. The population distribution for y divided by m. And so once again, I'll just write this out front. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
That's going to be equal to this thing over here. And we've done this multiple times. Same exact logic as this. The population distribution for y divided by m. And so once again, I'll just write this out front. This is the variance of the differences of the sample means. And if you wanted the standard deviation of the differences of the sample means, you just have to take the square root of both sides of this. If you take the square root of this, you get the standard deviation of the difference of the sample means is equal to the square root of the population distribution of x, or the variance of the population distribution of x divided by n plus the variance of the population distribution of y divided by m. And then the whole reason why I've even done this, and this is just neat because it kind of looks a little bit like a distance formula. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
The population distribution for y divided by m. And so once again, I'll just write this out front. This is the variance of the differences of the sample means. And if you wanted the standard deviation of the differences of the sample means, you just have to take the square root of both sides of this. If you take the square root of this, you get the standard deviation of the difference of the sample means is equal to the square root of the population distribution of x, or the variance of the population distribution of x divided by n plus the variance of the population distribution of y divided by m. And then the whole reason why I've even done this, and this is just neat because it kind of looks a little bit like a distance formula. And I'll kind of throw that out there as we get more sophisticated with our statistics and try to visualize what all of this kind of stuff means in more advanced topics. But the whole point of this is now we can make inferences about a difference of means. If we have two samples and we want to say, and we take the means of both of those samples and we find some difference, we can make some conclusions about how likely that difference was just by chance. | Difference of sample means distribution Probability and Statistics Khan Academy.mp3 |
And one of these things that you'll find in probability is that you can always do a more interesting problem. 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 three times, there's eight possibilities. For the first flip, there's two possibilities. Second flip, there's two possibilities. | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
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 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 times. Now how many of those possibilities have at least one head? | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
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 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 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. | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
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 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 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. | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
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 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. This would have been a lot harder to do or more time-consuming to do if I had 20 flips. | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
This would work well because I only had three flips. 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. | Coin flipping probability Probability and Statistics Khan Academy.mp3 |
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