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Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes. Or, since these are independent events, you can multiply. You see that there are 12 outcomes because there's 12 possible outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So right over here, we have a fairly simple least squares regression. We're trying to fit four points. And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate. Maybe some type of nonlinear model, some type of nonlinear curve might better fit the data, or the relationship between the y and the x is nonlinear. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
We're told that a board game has players roll two three-sided dice. These exist, and actually I looked it up, they do exist and they're actually fascinating. And subtract the numbers showing on the faces. The game only looks at non-negative differences. For example, if a player rolls a one and a three, the difference is two. Let D represent the difference in a given roll. Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
The game only looks at non-negative differences. For example, if a player rolls a one and a three, the difference is two. Let D represent the difference in a given roll. Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. All right, now let's work through it together. So let's just think about all of the scenarios for the two die. So let me draw a little table here. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. All right, now let's work through it together. So let's just think about all of the scenarios for the two die. So let me draw a little table here. So let me do it like that. And let me do it like this. And then let me put a little divider right over here. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
So let me draw a little table here. So let me do it like that. And let me do it like this. And then let me put a little divider right over here. And for this top, this is going to be die one, and then this is going to be die two. Die one can take on one, two, or three. And die two could be one, two, or three. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
And then let me put a little divider right over here. And for this top, this is going to be die one, and then this is going to be die two. Die one can take on one, two, or three. And die two could be one, two, or three. And so let me finish making this a bit of a table here. And what we wanna do is look at the difference, but the non-negative difference. So we'll always subtract the lower die from the higher die. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
And die two could be one, two, or three. And so let me finish making this a bit of a table here. And what we wanna do is look at the difference, but the non-negative difference. So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a one and a one. Now, what if I roll a two and a one? | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a one and a one. Now, what if I roll a two and a one? Well, here the difference is going to be two minus one, which is one. Here the difference is three minus one, which is two. Now, what about right over here? | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
Now, what if I roll a two and a one? Well, here the difference is going to be two minus one, which is one. Here the difference is three minus one, which is two. Now, what about right over here? Well, here the higher die is two, the lower one is one right over here. So two minus one is one, two minus two is zero. And now this is going to be the higher roll. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
Now, what about right over here? Well, here the higher die is two, the lower one is one right over here. So two minus one is one, two minus two is zero. And now this is going to be the higher roll. Die one is gonna have the higher roll in this scenario. Three minus two is one. And then right over here, three minus one is two. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
And now this is going to be the higher roll. Die one is gonna have the higher roll in this scenario. Three minus two is one. And then right over here, three minus one is two. Now, if die one rolls a two, die two rolls a three. Die three is higher, three minus two is one. And then three minus three is zero. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
And then right over here, three minus one is two. Now, if die one rolls a two, die two rolls a three. Die three is higher, three minus two is one. And then three minus three is zero. So we've come up with all of the scenarios, and we can see that we're either gonna end up with a zero or one or a two when we look at the positive difference. So there's a scenario of getting a zero, a one, or a two. Those are the different differences that we could actually get. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
And then three minus three is zero. So we've come up with all of the scenarios, and we can see that we're either gonna end up with a zero or one or a two when we look at the positive difference. So there's a scenario of getting a zero, a one, or a two. Those are the different differences that we could actually get. And so let's think about the probability of each of them. What's the probability that the difference is zero? Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
Those are the different differences that we could actually get. And so let's think about the probability of each of them. What's the probability that the difference is zero? Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. So it's gonna be three out of nine or 1 3rd. What about a difference of, let me use blue, one? Well, we could see there are one, two, three, four of the nine scenarios have that. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. So it's gonna be three out of nine or 1 3rd. What about a difference of, let me use blue, one? Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a 4 9th probability. And then last but not least, a difference of two. Well, there's two out of the nine scenarios that have that. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a 4 9th probability. And then last but not least, a difference of two. Well, there's two out of the nine scenarios that have that. So there is a 2 9th probability right over there. And we're done. We've constructed the theoretical probability distribution of D. | Theoretical probability distribution example tables Probability & combinatorics.mp3 |
And we want to get a sense of how these students feel about the quality of math instruction at this school. So we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. So anyway, this is a very high-level overview of how you could think about sampling. You wanna go random because it lowers the probability of their introducing some bias into it. And then these are some techniques. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And what I'm hoping to do in this video is get a little bit of practice interpreting this. And what I have here are five different statements. And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
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