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So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width? | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width? The width here, if we're going from 2.9 to 3.1, the width is going to be 0.2 wide. And so we could approximate this area by approximating this rectangle, the area of the rectangle. 0.2 times 0.2, that would give us an area of 0.04. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And this density curve doesn't look like the ones we typically see that are a little bit curvier, but this is a little easier for us to work with and figure out areas. And so they ask us to find the percent of the area under the density curve where x is more than two. So what area represents when x is more than two? So this is when x is equal to two. So they're talking about this area right over here. And so we need to figure out the percent of the total area under the curve that this blue area actually represents. So first let's find the total area under the density curve. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
So this is when x is equal to two. So they're talking about this area right over here. And so we need to figure out the percent of the total area under the curve that this blue area actually represents. So first let's find the total area under the density curve. And the density only has area, the density curve only has area from x equals one to x equals three. So it does amount to finding the area of this larger trapezoid. So let me highlight this trapezoid in red. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
So first let's find the total area under the density curve. And the density only has area, the density curve only has area from x equals one to x equals three. So it does amount to finding the area of this larger trapezoid. So let me highlight this trapezoid in red. So we wanna find the area of this trapezoid right over here. And then that should be equal to one because all density curves have an area of one under the total curve. So let's first verify that. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
So let me highlight this trapezoid in red. So we wanna find the area of this trapezoid right over here. And then that should be equal to one because all density curves have an area of one under the total curve. So let's first verify that. So there's a couple of ways to think about it. We could split it up into two shapes or you could just use the formula for an area of a trapezoid. In fact, let's use the formula for an area of a trapezoid. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
So let's first verify that. So there's a couple of ways to think about it. We could split it up into two shapes or you could just use the formula for an area of a trapezoid. In fact, let's use the formula for an area of a trapezoid. The formula for an area of a trapezoid is you would take the average of, you would take the average of this length, let me do that in another color, this length and this length, and then multiply that times the base. So the average of this length and this length, let's see, this is 0.25, 0.25 plus this height, 0.75 divided by two, so that's the average of those two sides, times the base, times this right over here, which is two. And so this is going to give us, as it should have, 0.25 plus 0.75, which is equal to one. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
In fact, let's use the formula for an area of a trapezoid. The formula for an area of a trapezoid is you would take the average of, you would take the average of this length, let me do that in another color, this length and this length, and then multiply that times the base. So the average of this length and this length, let's see, this is 0.25, 0.25 plus this height, 0.75 divided by two, so that's the average of those two sides, times the base, times this right over here, which is two. And so this is going to give us, as it should have, 0.25 plus 0.75, which is equal to one. So the area under the entire density curve is one, which we need to be true for this to be a density curve. Now let's think about what percentage of that area is represented in blue right over here. Well, we could do the same thing. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
And so this is going to give us, as it should have, 0.25 plus 0.75, which is equal to one. So the area under the entire density curve is one, which we need to be true for this to be a density curve. Now let's think about what percentage of that area is represented in blue right over here. Well, we could do the same thing. We could say, all right, this is a trapezoid. We wanna take the average of this side and this side and multiply it times the base. So this side is 0.5 high, 0.5, plus 0.75, 0.75 high, and we're gonna take the average of that, divided by two, times the base. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
Well, we could do the same thing. We could say, all right, this is a trapezoid. We wanna take the average of this side and this side and multiply it times the base. So this side is 0.5 high, 0.5, plus 0.75, 0.75 high, and we're gonna take the average of that, divided by two, times the base. Well, the base going from two to three is only equal to, is equal to one, so times one. And so this is going to give us 1.25, 1.25 over two. And what is that going to be equal to? | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
So this side is 0.5 high, 0.5, plus 0.75, 0.75 high, and we're gonna take the average of that, divided by two, times the base. Well, the base going from two to three is only equal to, is equal to one, so times one. And so this is going to give us 1.25, 1.25 over two. And what is that going to be equal to? Well, that would be the same thing as 0. what? Let's see, 0.625. Did I do that right? | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
And what is that going to be equal to? Well, that would be the same thing as 0. what? Let's see, 0.625. Did I do that right? Yep, if I multiply two times this, I would get 1.25. So the percent of the area under the density curve where x is more than two, this is the decimal expression of it, but if we wanted to write it as a percent, it would be 62.5%. Let's do another example. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
Did I do that right? Yep, if I multiply two times this, I would get 1.25. So the percent of the area under the density curve where x is more than two, this is the decimal expression of it, but if we wanted to write it as a percent, it would be 62.5%. Let's do another example. Consider the density curve below, all right, we have another one of these somewhat angular density curves. Find the percent of the area under the density curve where x is more than three. So we're talking about, see, this is where x is equal to three, x is more than three, we're talking about this entire area right over here. | Worked example finding area under density curves AP Statistics Khan Academy.mp3 |
So we're told that Amanda Young wants to win some prizes. A cereal company is giving away a prize in each box of cereal, and they advertise, collect all six prizes. Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get? No, we don't know that. | Random number list to run experiment Probability AP Statistics Khan Academy (2).mp3 |
He may choose the same number both times. If his ticket matches the two numbers in one letter drawn in order, he wins the grand prize and receives $10,405. If just his letter matches, but one or both of his numbers do not match, he wins the small prize of $100. Under any other outcome, he loses and receives nothing. The game costs him $5 to play. So under any other outcome, he loses and receives nothing. He has chosen the ticket 0 for R. So we're assuming he's paying the $5 to play and he picks the ticket 0 for R. So let's say we define a random variable X. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Under any other outcome, he loses and receives nothing. The game costs him $5 to play. So under any other outcome, he loses and receives nothing. He has chosen the ticket 0 for R. So we're assuming he's paying the $5 to play and he picks the ticket 0 for R. So let's say we define a random variable X. Let's say that this random variable is the net profit from playing this lottery game. What is the expected from playing 0 for R? So M is particular ticket right over here. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
He has chosen the ticket 0 for R. So we're assuming he's paying the $5 to play and he picks the ticket 0 for R. So let's say we define a random variable X. Let's say that this random variable is the net profit from playing this lottery game. What is the expected from playing 0 for R? So M is particular ticket right over here. So let's just say X is the random variable, it's the net profit from playing this ticket. What I want to think about in this video is what is the expected value of that? What is the expected net profit from playing 0 for R? | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So M is particular ticket right over here. So let's just say X is the random variable, it's the net profit from playing this ticket. What I want to think about in this video is what is the expected value of that? What is the expected net profit from playing 0 for R? I encourage you to pause the video and think through it on your own. So let's think about what expected value is. It's the probability of each of those outcomes times the net profit from those outcomes. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
What is the expected net profit from playing 0 for R? I encourage you to pause the video and think through it on your own. So let's think about what expected value is. It's the probability of each of those outcomes times the net profit from those outcomes. So there's the probability of the grand prize. Let me do that in that red color. So there is the probability of getting the grand prize. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
It's the probability of each of those outcomes times the net profit from those outcomes. So there's the probability of the grand prize. Let me do that in that red color. So there is the probability of getting the grand prize. And now what would times his net payoff from the grand prize? What would that be? Well, he gets $10,405, but that's not his net payoff or his net profit, I should say. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So there is the probability of getting the grand prize. And now what would times his net payoff from the grand prize? What would that be? Well, he gets $10,405, but that's not his net payoff or his net profit, I should say. His net profit is what he gets minus what he paid to play. So he paid $5 to play. So that's that. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Well, he gets $10,405, but that's not his net payoff or his net profit, I should say. His net profit is what he gets minus what he paid to play. So he paid $5 to play. So that's that. So plus the probability of getting the small prize times the payoff of the small prize, which is going to be $100 or times the net profit, I guess, if you get the small prize. So you get a payoff of $100 minus you have to pay $5 to play. And then finally you have the probability of neither. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So that's that. So plus the probability of getting the small prize times the payoff of the small prize, which is going to be $100 or times the net profit, I guess, if you get the small prize. So you get a payoff of $100 minus you have to pay $5 to play. And then finally you have the probability of neither. So you're essentially not winning. And there in that situation, what is the net profit? Well, in that situation, your net profit is negative 5. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
And then finally you have the probability of neither. So you're essentially not winning. And there in that situation, what is the net profit? Well, in that situation, your net profit is negative 5. You paid $5 and you got nothing in return. So to figure out the expected value, you just have to figure out these probabilities. So what's the probability of the grand prize? | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Well, in that situation, your net profit is negative 5. You paid $5 and you got nothing in return. So to figure out the expected value, you just have to figure out these probabilities. So what's the probability of the grand prize? Do that over here. Probability of grand prize. Well, the probability that he gets the first letter right is 1 in 10. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So what's the probability of the grand prize? Do that over here. Probability of grand prize. Well, the probability that he gets the first letter right is 1 in 10. There's 10 digits there. Probability that he gets the second letter right is 1 in 10. These are all independent. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Well, the probability that he gets the first letter right is 1 in 10. There's 10 digits there. Probability that he gets the second letter right is 1 in 10. These are all independent. And probability he gets the letter right, there's 26 equally likely letters that might be in the actual one. So he has a 1 in 26 chance of that one as well. So the probability of the grand prize is 1 in 2,600. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
These are all independent. And probability he gets the letter right, there's 26 equally likely letters that might be in the actual one. So he has a 1 in 26 chance of that one as well. So the probability of the grand prize is 1 in 2,600. So this is 1 in 2,600. Now what's the probability of getting the small prize? Well, let's see. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So the probability of the grand prize is 1 in 2,600. So this is 1 in 2,600. Now what's the probability of getting the small prize? Well, let's see. He has a 1 in 26 chance. The small prize is getting the letter right, but not getting both of the numbers right. So he has a 1 in 26 chance of getting the letter right. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Well, let's see. He has a 1 in 26 chance. The small prize is getting the letter right, but not getting both of the numbers right. So he has a 1 in 26 chance of getting the letter right. But we're not done here just with the 1 in 26. Because this 1 in 26, this includes all the scenarios where he gets the letter right, including the scenarios where he wins the grand prize, where he gets the letter and he gets the two numbers right. So what we need to do is we need to subtract out the situation, the probability of getting the letter and the two numbers right. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So he has a 1 in 26 chance of getting the letter right. But we're not done here just with the 1 in 26. Because this 1 in 26, this includes all the scenarios where he gets the letter right, including the scenarios where he wins the grand prize, where he gets the letter and he gets the two numbers right. So what we need to do is we need to subtract out the situation, the probability of getting the letter and the two numbers right. And we already know what that is. It's 1 in 2,600. So it's 1 in 26 minus 1 in 2,600. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So what we need to do is we need to subtract out the situation, the probability of getting the letter and the two numbers right. And we already know what that is. It's 1 in 2,600. So it's 1 in 26 minus 1 in 2,600. The reason why I have to subtract out this 2,600 is he has a 1 in 26 chance of getting this letter right. So that includes the scenario where he gets everything right. But the small prize is only where you get the letter and one or none of these. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So it's 1 in 26 minus 1 in 2,600. The reason why I have to subtract out this 2,600 is he has a 1 in 26 chance of getting this letter right. So that includes the scenario where he gets everything right. But the small prize is only where you get the letter and one or none of these. If you get both of these, then you're in the grand prize case. So you essentially have to subtract out the probability that you won the grand prize, that you got all three of them, to figure out the probability of the small prize. Now, what's the probability of essentially losing? | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
But the small prize is only where you get the letter and one or none of these. If you get both of these, then you're in the grand prize case. So you essentially have to subtract out the probability that you won the grand prize, that you got all three of them, to figure out the probability of the small prize. Now, what's the probability of essentially losing? The probability of neither. Well, it's just kind of, you know, that's everything else. So it would be 1 minus these probabilities right over here. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Now, what's the probability of essentially losing? The probability of neither. Well, it's just kind of, you know, that's everything else. So it would be 1 minus these probabilities right over here. So it would be 1 minus the probability of the small prize minus the probability of the grand. These are the possible outcomes, so they have to add up to 1 or 100%. So this is 1 minus probability of small minus probability of large. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So it would be 1 minus these probabilities right over here. So it would be 1 minus the probability of the small prize minus the probability of the grand. These are the possible outcomes, so they have to add up to 1 or 100%. So this is 1 minus probability of small minus probability of large. Or I should say, not grand prize. Grand prize. So let's fill this in. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So this is 1 minus probability of small minus probability of large. Or I should say, not grand prize. Grand prize. So let's fill this in. So the probability of the small one, this right over here, I'm using that red too much, this right over here is 1 in 26 minus 1 in 2,600. And then this right over here is 1 minus the small, which is 1 in 26 minus 1 in 2,600. Minus 1 in 2,600. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So let's fill this in. So the probability of the small one, this right over here, I'm using that red too much, this right over here is 1 in 26 minus 1 in 2,600. And then this right over here is 1 minus the small, which is 1 in 26 minus 1 in 2,600. Minus 1 in 2,600. And this simplifies to, let's see, this is 1 minus 1 over 26 plus 1 in 2,600, plus or minus 1 in 2,600. These cancel and you're left with 1 in 1 in 2,600. Now, why does this make sense? | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Minus 1 in 2,600. And this simplifies to, let's see, this is 1 minus 1 over 26 plus 1 in 2,600, plus or minus 1 in 2,600. These cancel and you're left with 1 in 1 in 2,600. Now, why does this make sense? Well, the way you lose, the way you get nothing, is if you get the letter wrong. So you have a 1 in 26 chance of getting the letter right, and then you're going to be in one of these two categories. Or you have a 1 minus 1 in 26, which is equal to 25 out of 26. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Now, why does this make sense? Well, the way you lose, the way you get nothing, is if you get the letter wrong. So you have a 1 in 26 chance of getting the letter right, and then you're going to be in one of these two categories. Or you have a 1 minus 1 in 26, which is equal to 25 out of 26. You have a 25 in 26 chance of getting the letter wrong, in which case you get nothing, in which case you completely lose. So let's just take our calculator out and calculate this. And we'll round to the nearest penny here. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Or you have a 1 minus 1 in 26, which is equal to 25 out of 26. You have a 25 in 26 chance of getting the letter wrong, in which case you get nothing, in which case you completely lose. So let's just take our calculator out and calculate this. And we'll round to the nearest penny here. So let's see, it is going to be 1,2,600. So 1 divided by 2,600 times, let's see, 10,004 minus 5 is going to be 10,400. Times 10,400, that's your net profit when you win the grand prize. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
And we'll round to the nearest penny here. So let's see, it is going to be 1,2,600. So 1 divided by 2,600 times, let's see, 10,004 minus 5 is going to be 10,400. Times 10,400, that's your net profit when you win the grand prize. And then you're going to have plus 1 divided by 26 minus 1 divided by 2,600 times your net profit for the small prize, 100 minus 5, which is 95. And then finally, plus 25,26. So 25 divided by 26, actually I'll put parentheses around here just to make it consistent. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
Times 10,400, that's your net profit when you win the grand prize. And then you're going to have plus 1 divided by 26 minus 1 divided by 2,600 times your net profit for the small prize, 100 minus 5, which is 95. And then finally, plus 25,26. So 25 divided by 26, actually I'll put parentheses around here just to make it consistent. So 25 divided by 26 times that net payoff, when you get nothing, well you have to pay out $5 and you've got nothing in return, times negative 5. Actually I don't know if it's going to recognize that as times, so I'll just write times negative 5, and then we delete that. And we deserve a drum roll now. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
So 25 divided by 26, actually I'll put parentheses around here just to make it consistent. So 25 divided by 26 times that net payoff, when you get nothing, well you have to pay out $5 and you've got nothing in return, times negative 5. Actually I don't know if it's going to recognize that as times, so I'll just write times negative 5, and then we delete that. And we deserve a drum roll now. We get a expected net profit of playing as $2.81, if we round up to the nearest penny. So this is all going to be equal to $2.81. And so this is actually a very unusual lottery game where you have a positive expected net profit as a player. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
And we deserve a drum roll now. We get a expected net profit of playing as $2.81, if we round up to the nearest penny. So this is all going to be equal to $2.81. And so this is actually a very unusual lottery game where you have a positive expected net profit as a player. Usually the person operating the lottery, the state, who are the casino, whoever it is, they're the ones who have the expected net profit and then the player has the expected net loss. But this actually would make rational sense to play, which is not the case with most lottery games. That if by playing you actually expect a $2.81 net profit. | Expected profit from lottery ticket Probability and Statistics Khan Academy.mp3 |
And these are going to be in some ways similar to the conditions for inference that we thought about when we were doing hypothesis testing and confidence intervals for means and for proportions, but there's also going to be a few new conditions. So to help us remember these conditions, you might want to think about the LINER acronym, L-I-N-E-R, and if it isn't obvious to you, this almost is linear. Liner, with an A, it would be linear, and this is valuable because remember, we're thinking about linear regression. So the L right over here actually does stand for linear, and here, the condition is is that the actual relationship in the population between your X and Y variables actually is a linear relationship. So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
So the L right over here actually does stand for linear, and here, the condition is is that the actual relationship in the population between your X and Y variables actually is a linear relationship. So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you could be thinking about your 10% rule that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you could be thinking about your 10% rule that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks like this. And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks like this. And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. So I'll leave you there. It's good to know it will show up on some exams, but many times when it comes to problem solving, in an introductory statistics class, they will tell you, hey, just assume the conditions for inference have been met, or what are the conditions for inference? But they're not going to actually make you prove, for example, the normal or the equal variance condition. | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Each employee receives an annual rating, the best of which is exceeds expectations. Management claimed that 10% of employees earned this rating but Jules suspected it was actually less common. She obtained an anonymous random sample of 10 ratings for employees on her team. She wants to use the sample data to test her null hypothesis that the true proportion is 10% versus her alternative hypothesis that the true proportion is less than 10% where P is the proportion of all employees on her team who earned exceeds expectations. Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
She wants to use the sample data to test her null hypothesis that the true proportion is 10% versus her alternative hypothesis that the true proportion is less than 10% where P is the proportion of all employees on her team who earned exceeds expectations. Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis. | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then we want to calculate a p value. | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then we want to calculate a p value. And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less than or equal to the one that you just calculated for a sample size of n equals 10 given that your null hypothesis is true. And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
And then we want to calculate a p value. And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less than or equal to the one that you just calculated for a sample size of n equals 10 given that your null hypothesis is true. And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition? | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition? | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition? The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10. So they need to be greater than or equal to 10. Now, what are they for this particular scenario? | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
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