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This is our t distribution. So if you randomly pick a t value from this t distribution, it has a 95% chance of being within this far from the mean. Or maybe we should write it this way. If I pick a random t value, if I take a random t statistic, there's a 95% chance that a random t statistic is going to be less than 2.262 and greater than negative 2.262. 95% chance. Now, when we took this sample, we can also derive a random t statistic from this. We have our sample mean and our sample standard deviation. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
If I pick a random t value, if I take a random t statistic, there's a 95% chance that a random t statistic is going to be less than 2.262 and greater than negative 2.262. 95% chance. Now, when we took this sample, we can also derive a random t statistic from this. We have our sample mean and our sample standard deviation. Our sample mean here is 17.17. Figured that out in the last video. Just add these up, divide by 10. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
We have our sample mean and our sample standard deviation. Our sample mean here is 17.17. Figured that out in the last video. Just add these up, divide by 10. And our sample standard deviation here is 2.98. So the t statistic that we can derive from this information right over here, so let me write it over here. The t statistic that we can derive from this. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Just add these up, divide by 10. And our sample standard deviation here is 2.98. So the t statistic that we can derive from this information right over here, so let me write it over here. The t statistic that we can derive from this. And you can view this t statistic as being a random sample from a t distribution. A t distribution with 9 degrees of freedom. So the t statistic that we can derive from that is going to be our mean, 17.17, minus the true mean of our population. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
The t statistic that we can derive from this. And you can view this t statistic as being a random sample from a t distribution. A t distribution with 9 degrees of freedom. So the t statistic that we can derive from that is going to be our mean, 17.17, minus the true mean of our population. Or actually, we say the true mean of our sampling distribution, which is also going to be the same as the true mean of our population. That's our population mean over there. Divided by s, which is 2.98, over the square root of our number of samples. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So the t statistic that we can derive from that is going to be our mean, 17.17, minus the true mean of our population. Or actually, we say the true mean of our sampling distribution, which is also going to be the same as the true mean of our population. That's our population mean over there. Divided by s, which is 2.98, over the square root of our number of samples. We've seen this multiple times. This right here is the t statistic. So by taking this sample, you can say that we've randomly sampled a t statistic from this 9 degree of freedom t distribution. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Divided by s, which is 2.98, over the square root of our number of samples. We've seen this multiple times. This right here is the t statistic. So by taking this sample, you can say that we've randomly sampled a t statistic from this 9 degree of freedom t distribution. So there's a 95% chance that this thing right over here is going to be less than 2.262 and greater than negative 2.262. So the 95% probability still applies to this right here. Now, we just have to do some math, calculate these things. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So by taking this sample, you can say that we've randomly sampled a t statistic from this 9 degree of freedom t distribution. So there's a 95% chance that this thing right over here is going to be less than 2.262 and greater than negative 2.262. So the 95% probability still applies to this right here. Now, we just have to do some math, calculate these things. Let me get my calculator out. Let me just calculate this denominator right over here. So we have 2.98 divided by the square root of 10. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now, we just have to do some math, calculate these things. Let me get my calculator out. Let me just calculate this denominator right over here. So we have 2.98 divided by the square root of 10. So that's 0.9423. So what I'm going to do is I'm going to multiply both sides of this equation by this expression right over here. So if I do that, let me just do that right over. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So we have 2.98 divided by the square root of 10. So that's 0.9423. So what I'm going to do is I'm going to multiply both sides of this equation by this expression right over here. So if I do that, let me just do that right over. So if I multiply this entire, this is really two equations, or two inequalities, I should say. That this quantity is greater than this quantity and that this quantity is greater than that quantity. But we can operate on all of them at the same time, this entire inequality. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if I do that, let me just do that right over. So if I multiply this entire, this is really two equations, or two inequalities, I should say. That this quantity is greater than this quantity and that this quantity is greater than that quantity. But we can operate on all of them at the same time, this entire inequality. So what we want to do is multiply this entire inequality by this value right over here. And we just calculated that that value, let me write it over here, that 2.98, I'll write it over here, 2.98 over the square root of 10 is equal to 0.942. So if I multiply this entire inequality by 0.942, I get, on this left-hand side over here, I have negative 2.262 times 0.942. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
But we can operate on all of them at the same time, this entire inequality. So what we want to do is multiply this entire inequality by this value right over here. And we just calculated that that value, let me write it over here, that 2.98, I'll write it over here, 2.98 over the square root of 10 is equal to 0.942. So if I multiply this entire inequality by 0.942, I get, on this left-hand side over here, I have negative 2.262 times 0.942. And it's a positive number that we're multiplying the whole inequality by, so the inequality signs are still going to be in the same direction, is less than, well, we're multiplying this whole expression by the same expression in the denominator, so it'll cancel out. So we're just going to be less than 17.17 minus our population mean, which is going to be less than 2.262 times, once again, 0.942. Let me scroll over to the right a little bit. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if I multiply this entire inequality by 0.942, I get, on this left-hand side over here, I have negative 2.262 times 0.942. And it's a positive number that we're multiplying the whole inequality by, so the inequality signs are still going to be in the same direction, is less than, well, we're multiplying this whole expression by the same expression in the denominator, so it'll cancel out. So we're just going to be less than 17.17 minus our population mean, which is going to be less than 2.262 times, once again, 0.942. Let me scroll over to the right a little bit. 0.942. Just to be clear, I'm just multiplying both or all three sides of this inequality by this number right over here. In the middle, this cancels out. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let me scroll over to the right a little bit. 0.942. Just to be clear, I'm just multiplying both or all three sides of this inequality by this number right over here. In the middle, this cancels out. So if I multiply, I'll just write it right here, 0.942, 0.942, 0.942, this and this is the same number, so that's why those cancel out. And now let's hit the calculator to figure out what these numbers are. So if we have the 0.942 times 2.262, so we're going to say times 2.262 is 2.13. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
In the middle, this cancels out. So if I multiply, I'll just write it right here, 0.942, 0.942, 0.942, this and this is the same number, so that's why those cancel out. And now let's hit the calculator to figure out what these numbers are. So if we have the 0.942 times 2.262, so we're going to say times 2.262 is 2.13. So this is going to be, so this number right over here on the right-hand side, this number on the right-hand side is 2.13. This number on the left is just the negative of that, so it's negative 2.13. And then we still have our inequalities is going to be less than 17.17 minus the mean, which is less than 2.13. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if we have the 0.942 times 2.262, so we're going to say times 2.262 is 2.13. So this is going to be, so this number right over here on the right-hand side, this number on the right-hand side is 2.13. This number on the left is just the negative of that, so it's negative 2.13. And then we still have our inequalities is going to be less than 17.17 minus the mean, which is less than 2.13. Now what I want to do is I actually want to solve for this mean. And I don't like that negative sign in the mean. I'd rather have this swapped around. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
And then we still have our inequalities is going to be less than 17.17 minus the mean, which is less than 2.13. Now what I want to do is I actually want to solve for this mean. And I don't like that negative sign in the mean. I'd rather have this swapped around. I'd rather have the mean minus 17.17. So what I'm going to do is multiply this entire inequality by negative 1. If you do that, if you multiply the entire thing times negative 1, this quantity right here, this negative 2.13 will become a positive 2.13. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
I'd rather have this swapped around. I'd rather have the mean minus 17.17. So what I'm going to do is multiply this entire inequality by negative 1. If you do that, if you multiply the entire thing times negative 1, this quantity right here, this negative 2.13 will become a positive 2.13. But since we are multiplying an inequality by a negative number, you have to swap the inequality sign. So this less than will become a greater than. This negative mu will become a positive mu. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
If you do that, if you multiply the entire thing times negative 1, this quantity right here, this negative 2.13 will become a positive 2.13. But since we are multiplying an inequality by a negative number, you have to swap the inequality sign. So this less than will become a greater than. This negative mu will become a positive mu. This positive 17.17 will become a negative 17.17. We have to swap this inequality sign as well. And this positive 2.13 will become a negative 2.13. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
This negative mu will become a positive mu. This positive 17.17 will become a negative 17.17. We have to swap this inequality sign as well. And this positive 2.13 will become a negative 2.13. And we're almost there. We just want to solve for mu, have this inequality expressed in terms of mu. So what we can do is now just add 17.17 to all three sides of this inequality. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
And this positive 2.13 will become a negative 2.13. And we're almost there. We just want to solve for mu, have this inequality expressed in terms of mu. So what we can do is now just add 17.17 to all three sides of this inequality. And we are left with 2.13 plus 17.17 is greater than mu minus 17.17 plus 17.17 is just going to be mu, which is greater than. So this is greater than mu, which is greater than negative 2.13 plus 17.17. Or a more natural way to write it, since we actually have a bunch of greater than signs, that this is actually the largest number. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So what we can do is now just add 17.17 to all three sides of this inequality. And we are left with 2.13 plus 17.17 is greater than mu minus 17.17 plus 17.17 is just going to be mu, which is greater than. So this is greater than mu, which is greater than negative 2.13 plus 17.17. Or a more natural way to write it, since we actually have a bunch of greater than signs, that this is actually the largest number. And this is actually the smallest number. And this over here is actually the largest number. It's actually flip. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Or a more natural way to write it, since we actually have a bunch of greater than signs, that this is actually the largest number. And this is actually the smallest number. And this over here is actually the largest number. It's actually flip. You can just rewrite this inequality the other way. So now we can write. Well, actually, let's just figure out what these values are. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
It's actually flip. You can just rewrite this inequality the other way. So now we can write. Well, actually, let's just figure out what these values are. So we have 2.13 plus 17.17. So that is the high end of our range. So that is 19.3. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Well, actually, let's just figure out what these values are. So we have 2.13 plus 17.17. So that is the high end of our range. So that is 19.3. So this value right over here. So this is 19. Let me do it in that same color. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
So that is 19.3. So this value right over here. So this is 19. Let me do it in that same color. This value right here is 19.3 is going to be greater than mu, which is going to be greater than negative 2.13 plus 17.17. Or we could have 17.17 minus 2.13, which gives us 15.04. And remember, the whole thing, all of this, we started with, there was a 95% chance that a random t statistic will fall in this interval. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let me do it in that same color. This value right here is 19.3 is going to be greater than mu, which is going to be greater than negative 2.13 plus 17.17. Or we could have 17.17 minus 2.13, which gives us 15.04. And remember, the whole thing, all of this, we started with, there was a 95% chance that a random t statistic will fall in this interval. We had a random t statistic. And all we did is a bunch of math. So there's a 95% chance that any of these steps are true. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
And remember, the whole thing, all of this, we started with, there was a 95% chance that a random t statistic will fall in this interval. We had a random t statistic. And all we did is a bunch of math. So there's a 95% chance that any of these steps are true. And so there's a 95% chance that this is true. There's a 95% chance that the true population mean, which is the same thing as the mean of the sampling distribution of the sample mean, there's a 95% chance, or we're confident that there's a 95% chance, that it will fall in this interval. And we're done. | T-statistic confidence interval Inferential statistics Probability and Statistics Khan Academy.mp3 |
And as you can see, X can take on only a finite number of values, zero, one, two, three, or four, and so because there's a finite number of values here, we would call this a discrete random variable. And you can see that this is a valid probability distribution because the combined probability is one. .1 plus 0.15 plus 0.4 plus 0.25 plus 0.1 is one, and none of these are negative probabilities, which wouldn't have made sense. But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. | Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3 |
But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. | Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3 |
This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. | Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3 |
So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1. | Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3 |
Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. | Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3 |
So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. How can you have 2.1 workouts in a week? What is 0.1 of a workout? Well, this isn't saying that in a given week, you would expect me to work out exactly 2.1 times, but this is valuable because you could say, well, in 10 weeks, you would expect me to do roughly 21 workouts. | Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3 |
At Hogwarts, there are four houses, Gryffindor, Hufflepuff, Ravenclaw, and Slytherin. The bar chart below shows the number of house points that each house received today. How many house points did Hufflepuff receive? So if we look at this bar chart, we can assume that this column right over here is Hufflepuff, because it's the only one that starts with an H. This will be Gryffindor. This is Ravenclaw. And this must be Slytherin. And so if we look at Huff and Puff's bar chart, it looks like they have three points scored today. | Reading bar charts basic example Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So if we look at this bar chart, we can assume that this column right over here is Hufflepuff, because it's the only one that starts with an H. This will be Gryffindor. This is Ravenclaw. And this must be Slytherin. And so if we look at Huff and Puff's bar chart, it looks like they have three points scored today. So let's put three points. Let's try one more. So here, they're saying how many house points did Slytherin receive? | Reading bar charts basic example Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
A club of nine people wants to choose a board of three officers. A president, a vice president, and a secretary. How many ways are there to choose the board from the nine people? Now, we're going to assume that one person can't hold more than one office. That if I'm picked for president, that I'm no longer a valid person for vice president or secretary. So let's just think about the three different positions. So you have the president, you have the vice president, VP, and then you have the secretary. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
Now, we're going to assume that one person can't hold more than one office. That if I'm picked for president, that I'm no longer a valid person for vice president or secretary. So let's just think about the three different positions. So you have the president, you have the vice president, VP, and then you have the secretary. Now, let's say that we go for the president first. It actually doesn't matter. Let's say we're picking the president slot first and we haven't appointed any other slots yet. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
So you have the president, you have the vice president, VP, and then you have the secretary. Now, let's say that we go for the president first. It actually doesn't matter. Let's say we're picking the president slot first and we haven't appointed any other slots yet. How many possibilities are there for president? Well, the club has nine people, so there's nine possibilities for president. Now, we're going to pick one of those nine. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's say we're picking the president slot first and we haven't appointed any other slots yet. How many possibilities are there for president? Well, the club has nine people, so there's nine possibilities for president. Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices, right? Because someone's going to be president. So one of the nine is going to be president. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices, right? Because someone's going to be president. So one of the nine is going to be president. There's nine possibilities, but one of the nine is going to be president. So you take that person aside. He or she is now the president. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
So one of the nine is going to be president. There's nine possibilities, but one of the nine is going to be president. So you take that person aside. He or she is now the president. How many people are left to be vice president? Well, now there's only eight possible candidates for vice president. Eight possibilities. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
He or she is now the president. How many people are left to be vice president? Well, now there's only eight possible candidates for vice president. Eight possibilities. Now, he or she also goes aside. Now, how many people are left for secretary? Well, now there's only seven possibilities for secretary. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
Eight possibilities. Now, he or she also goes aside. Now, how many people are left for secretary? Well, now there's only seven possibilities for secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the nine for president times the eight for vice president times the seven for secretary. You didn't have to do it this way. You could have picked secretary first and there would have been nine choices. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, now there's only seven possibilities for secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the nine for president times the eight for vice president times the seven for secretary. You didn't have to do it this way. You could have picked secretary first and there would have been nine choices. And then you could have picked vice president. There would have still been eight choices. Then you could have picked president last and there would have only been seven choices. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
You could have picked secretary first and there would have been nine choices. And then you could have picked vice president. There would have still been eight choices. Then you could have picked president last and there would have only been seven choices. But either way, you would have gotten nine times eight times seven. And that is, let's see, nine times eight is 72. 72 times seven is 14. | Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3 |
A nutritionist wants to estimate the average caloric content of the burritos at a popular restaurant. They obtain a random sample of 14 burritos and measure their caloric content. Their sample data are roughly symmetric with a mean of 700 calories and a standard deviation of 50 calories. Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos? So pause this video and see if you can figure it out. All right, what's going on here? So there's a population of burritos here. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos? So pause this video and see if you can figure it out. All right, what's going on here? So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but doesn't know the true population parameter here, the population mean. And so they take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but doesn't know the true population parameter here, the population mean. And so they take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. And they wanna use this data to construct a 95% confidence interval. And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. And they wanna use this data to construct a 95% confidence interval. And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. And if we use that, if we used sigma, which is a population parameter, then we could use a z-statistic right over here. We would use a z-distribution. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. And if we use that, if we used sigma, which is a population parameter, then we could use a z-statistic right over here. We would use a z-distribution. But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be? | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
We would use a z-distribution. But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus, plus or minus. So what would be our critical value for a 95% confidence interval? | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus, plus or minus. So what would be our critical value for a 95% confidence interval? Well, we will just get out our t-table. And with a t-table, remember, you have to care about degrees of freedom. And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
So what would be our critical value for a 95% confidence interval? Well, we will just get out our t-table. And with a t-table, remember, you have to care about degrees of freedom. And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. So that's gonna be 14 minus one is equal to 13. So we have 13 degrees of freedom that we have to keep in mind when we look at our t-table. So let's look at our t-table. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. So that's gonna be 14 minus one is equal to 13. So we have 13 degrees of freedom that we have to keep in mind when we look at our t-table. So let's look at our t-table. So 95% confidence interval and 13 degrees of freedom. So degrees of freedom right over here. So we have 13 degrees of freedom. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
So let's look at our t-table. So 95% confidence interval and 13 degrees of freedom. So degrees of freedom right over here. So we have 13 degrees of freedom. So that is this row right over here. And if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see if I'll draw it really small, little small distribution right over here. So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1⁄2% at the tails, 2 1⁄2%. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
So we have 13 degrees of freedom. So that is this row right over here. And if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see if I'll draw it really small, little small distribution right over here. So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1⁄2% at the tails, 2 1⁄2%. So what you wanna look for is a tail probability of 2 1⁄2%. So that is this right over here,.025, that's 2 1⁄2%. And so there you go, that is our critical value, 2.160. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1⁄2% at the tails, 2 1⁄2%. So what you wanna look for is a tail probability of 2 1⁄2%. So that is this right over here,.025, that's 2 1⁄2%. And so there you go, that is our critical value, 2.160. So this, so this part right over here, so this is going to be two, let me do that in a darker color. This is going to be 2.160 times, what's our sample standard deviation? It's 50 over the square root of n, square root of 14. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
And so there you go, that is our critical value, 2.160. So this, so this part right over here, so this is going to be two, let me do that in a darker color. This is going to be 2.160 times, what's our sample standard deviation? It's 50 over the square root of n, square root of 14. So all of our choices have the 700 there. So we just need to figure out what our margin of error, this part of it, and we could use a calculator for that. Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
It's 50 over the square root of n, square root of 14. So all of our choices have the 700 there. So we just need to figure out what our margin of error, this part of it, and we could use a calculator for that. Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here, I think. 28.86, so this part right over here is approximately 28.86. That's our margin of error. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here, I think. 28.86, so this part right over here is approximately 28.86. That's our margin of error. And we see out of all of these choices here, if we round to the nearest tenth, that'd be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
That's our margin of error. And we see out of all of these choices here, if we round to the nearest tenth, that'd be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. So there we have it. We have established our 95% confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval? | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. So there we have it. We have established our 95% confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval? And here we have to think, well, did we take a random sample and they tell us that they obtained a random sample of 14 burritos, so we check that one. Is the sampling distribution roughly normal? | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval? And here we have to think, well, did we take a random sample and they tell us that they obtained a random sample of 14 burritos, so we check that one. Is the sampling distribution roughly normal? Well, either you take, if you take over 30 samples, then it would be, but here we only took 14. But they do tell us that the sample data is roughly symmetric. And so if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
Is the sampling distribution roughly normal? Well, either you take, if you take over 30 samples, then it would be, but here we only took 14. But they do tell us that the sample data is roughly symmetric. And so if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal. And then the last condition is the independence condition. And here, if we aren't sampling with replacement, and it doesn't look like we are, if we're not sampling with replacement, this has to be less than 10% of, this has to be less than 10% of the population of burritos. And we're assuming that there's going to be more than 140 burritos that the universe, that the population, that this popular restaurant makes. | Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3 |
In previous videos, we took this bivariate data and we calculated the correlation coefficient. And just as a bit of a review, we have the formula here. And it looks a bit intimidating, but in that video we saw all it is is an average of the product of the z-scores for each of those pairs. And as we said, if r is equal to one, you have a perfect positive correlation. If r is equal to negative one, you have a perfect negative correlation. And if r is equal to zero, you don't have a correlation. But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
And as we said, if r is equal to one, you have a perfect positive correlation. If r is equal to negative one, you have a perfect negative correlation. And if r is equal to zero, you don't have a correlation. But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points. So before I do that, let's just visualize some of the statistics that we have here for these data points. We clearly have the four data points plotted, but let's plot the statistics for x. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points. So before I do that, let's just visualize some of the statistics that we have here for these data points. We clearly have the four data points plotted, but let's plot the statistics for x. So the sample mean and the sample standard deviation for x are here in red. And actually, let me box these off in red so that you know that's what is going on here. So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
We clearly have the four data points plotted, but let's plot the statistics for x. So the sample mean and the sample standard deviation for x are here in red. And actually, let me box these off in red so that you know that's what is going on here. So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. So we have x equals two right over here. And then this is one sample standard deviation above the mean. This is one sample standard deviation below the mean. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. So we have x equals two right over here. And then this is one sample standard deviation above the mean. This is one sample standard deviation below the mean. And then we could do the same thing for the y variables. So the mean is three. And this is one sample standard deviation for y above the mean. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
This is one sample standard deviation below the mean. And then we could do the same thing for the y variables. So the mean is three. And this is one sample standard deviation for y above the mean. And this is one sample standard deviation for y below the mean. And visualizing these means, especially their intersection, and also their standard deviations, will help us build an intuition for the equation of the least squares line. So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
And this is one sample standard deviation for y above the mean. And this is one sample standard deviation for y below the mean. And visualizing these means, especially their intersection, and also their standard deviations, will help us build an intuition for the equation of the least squares line. So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. For the regression line, we'll put a little hat over it. So this, you would literally say y hat. This tells you that this is a regression line that we're trying to fit to these points. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. For the regression line, we'll put a little hat over it. So this, you would literally say y hat. This tells you that this is a regression line that we're trying to fit to these points. First, what is going to be the slope? Well, the slope is going to be r times the ratio between the sample standard deviation in the y direction over the sample standard deviation in the x direction. This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
This tells you that this is a regression line that we're trying to fit to these points. First, what is going to be the slope? Well, the slope is going to be r times the ratio between the sample standard deviation in the y direction over the sample standard deviation in the x direction. This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept? Well, like you first learned in Algebra I, you can calculate the y-intercept if you already know the slope by saying, well, what point is definitely going to be on my line? And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept? Well, like you first learned in Algebra I, you can calculate the y-intercept if you already know the slope by saying, well, what point is definitely going to be on my line? And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. So you're definitely going to go through that point. So before I even calculate for this particular example where in previous videos we calculated the r to be 0.946, or roughly equal to that, let's just think about what's going on. So our least squares line is definitely going to go through that point. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. So you're definitely going to go through that point. So before I even calculate for this particular example where in previous videos we calculated the r to be 0.946, or roughly equal to that, let's just think about what's going on. So our least squares line is definitely going to go through that point. Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x. So if you were to start at this point, and if you were to run your standard deviation of x and rise your standard deviation of y, well, with a perfect positive correlation, your line would look like this. And that makes a lot of sense because you're looking at your spread of y over your spread of x. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So our least squares line is definitely going to go through that point. Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x. So if you were to start at this point, and if you were to run your standard deviation of x and rise your standard deviation of y, well, with a perfect positive correlation, your line would look like this. And that makes a lot of sense because you're looking at your spread of y over your spread of x. If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x. That has parallels to when you first learn about slope. Change in y over change in x. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
And that makes a lot of sense because you're looking at your spread of y over your spread of x. If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x. That has parallels to when you first learn about slope. Change in y over change in x. Here you're seeing the, you could say the average spread in y over the average spread in x. And this would be the case when r is one. So let me write that down. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
Change in y over change in x. Here you're seeing the, you could say the average spread in y over the average spread in x. And this would be the case when r is one. So let me write that down. This would be the case if r is equal to one. What if r were equal to negative one? It would look like this. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So let me write that down. This would be the case if r is equal to one. What if r were equal to negative one? It would look like this. That would be our line if we had a perfect negative correlation. Now what if r were zero? Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
It would look like this. That would be our line if we had a perfect negative correlation. Now what if r were zero? Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. So you would just go through that right over there. But now let's think about this scenario. In this scenario, our r is 0.946. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. So you would just go through that right over there. But now let's think about this scenario. In this scenario, our r is 0.946. So we have a fairly strong correlation. This is pretty close to one. And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
In this scenario, our r is 0.946. So we have a fairly strong correlation. This is pretty close to one. And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? Well, you would move up r times the standard deviation of y. And as we said, if r was one, you would get all the way up to this perfect correlation line but here it's 0.946. So you would get up about 95% of the way to that. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? Well, you would move up r times the standard deviation of y. And as we said, if r was one, you would get all the way up to this perfect correlation line but here it's 0.946. So you would get up about 95% of the way to that. And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points. I'm not proving it here in this video. But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So you would get up about 95% of the way to that. And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points. I'm not proving it here in this video. But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. It actually makes intuitive sense. Let's calculate it for this particular set of data. M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. It actually makes intuitive sense. Let's calculate it for this particular set of data. M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. We can get our calculator out to calculate that. So we have 0.946 times 2.160 divided by 0.816. It gets us to 2.50. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. We can get our calculator out to calculate that. So we have 0.946 times 2.160 divided by 0.816. It gets us to 2.50. Let's just round to the nearest hundredth for simplicity here. So this is approximately equal to 2.50. And so how do we figure out the y-intercept? | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
It gets us to 2.50. Let's just round to the nearest hundredth for simplicity here. So this is approximately equal to 2.50. And so how do we figure out the y-intercept? Well, remember, we go through this point. So we're going to have 2.50 times our x-mean. So our x-mean is two times two. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
And so how do we figure out the y-intercept? Well, remember, we go through this point. So we're going to have 2.50 times our x-mean. So our x-mean is two times two. Remember, this right over here is our x-mean. Plus b is going to be equal to our y-mean. Our y-mean, we see right over here, is three. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So our x-mean is two times two. Remember, this right over here is our x-mean. Plus b is going to be equal to our y-mean. Our y-mean, we see right over here, is three. And so what do we get? We get three is equal to five plus b. Five plus b. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
Our y-mean, we see right over here, is three. And so what do we get? We get three is equal to five plus b. Five plus b. And so what is b? Well, if you subtract five from both sides, you get b is equal to negative two. And so there you have it. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
Five plus b. And so what is b? Well, if you subtract five from both sides, you get b is equal to negative two. And so there you have it. The equation for our regression line. We deserve a little bit of a drum roll here. We would say y-hat, the hat tells us that this is the equation for a regression line, is equal to 2.50 times x minus two. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
So let's ask ourselves some interesting questions about alphabets in the English language. And in case you don't remember, or are in the mood to count, there are 26 alphabets. So if you go A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, and Z, you get 26. 26 alphabets. Now let's ask some interesting questions. So given that there are 26 alphabets in the English language, how many possible three-letter words are there? And we're not gonna be thinking about phonetics or how hard it is to pronounce it. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
26 alphabets. Now let's ask some interesting questions. So given that there are 26 alphabets in the English language, how many possible three-letter words are there? And we're not gonna be thinking about phonetics or how hard it is to pronounce it. So for example, the word, the word zigget would be a legitimate word in this example. Or the word, the word, the word skudge would be a legitimate word in this example. So how many possible three-letter words are there in the English language? | Possible three letter words Probability and Statistics Khan Academy.mp3 |
And we're not gonna be thinking about phonetics or how hard it is to pronounce it. So for example, the word, the word zigget would be a legitimate word in this example. Or the word, the word, the word skudge would be a legitimate word in this example. So how many possible three-letter words are there in the English language? I encourage you to pause the video and try to think about it. All right, I assume you've had a go at it. So let's just think about it. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
So how many possible three-letter words are there in the English language? I encourage you to pause the video and try to think about it. All right, I assume you've had a go at it. So let's just think about it. For three-letter words, there's three spaces. So how many possibilities are there for the first one? Well, there's 26 possible letters for the first one. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
So let's just think about it. For three-letter words, there's three spaces. So how many possibilities are there for the first one? Well, there's 26 possible letters for the first one. Anything from A to Z would be completely fine. Now how many possibilities for the second one? And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
Well, there's 26 possible letters for the first one. Anything from A to Z would be completely fine. Now how many possibilities for the second one? And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. But that's not the case right over here, because we can repeat letters. I didn't say that all the letters had to be different. So for example, the word, the word, ha, would also be a legitimate word in our example right over here. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. But that's not the case right over here, because we can repeat letters. I didn't say that all the letters had to be different. So for example, the word, the word, ha, would also be a legitimate word in our example right over here. So we have 26 possibilities for the second letter, and we have 26 possibilities for the third letter. So we're going to have, and I don't know what this is, 26 to the third power possibilities, or 26 times 26 times 26, and you can figure out what that is. That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
So for example, the word, the word, ha, would also be a legitimate word in our example right over here. So we have 26 possibilities for the second letter, and we have 26 possibilities for the third letter. So we're going to have, and I don't know what this is, 26 to the third power possibilities, or 26 times 26 times 26, and you can figure out what that is. That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. Now let's ask a different question. What if we said, how many possible three-letter words are there if we want all different letters? So we want all different letters. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
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