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Sponge, huh, these are all, I don't see a pattern just yet. Let's look at the choices. The subset consists of all outcomes where your friend does not win. All outcomes where your friend does not win. Well, that's not true, because look, outcome one, my friend wins. Water puts out fire. So we're not going to select this first choice.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
All outcomes where your friend does not win. Well, that's not true, because look, outcome one, my friend wins. Water puts out fire. So we're not going to select this first choice. So let's see, the subset consists of all the outcomes where your friend wins or there's a tie. So let's see, where the friend wins or there's a tie. Well, outcome three, this is an outcome where I would win, or you or whoever your is, whoever they're talking about.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
So we're not going to select this first choice. So let's see, the subset consists of all the outcomes where your friend wins or there's a tie. So let's see, where the friend wins or there's a tie. Well, outcome three, this is an outcome where I would win, or you or whoever your is, whoever they're talking about. This is one where the friend doesn't win, because fire burns sponge. So I'm not going to select that one either. Choice three, the subset consists of all of the outcome where you win or there is a tie.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
Well, outcome three, this is an outcome where I would win, or you or whoever your is, whoever they're talking about. This is one where the friend doesn't win, because fire burns sponge. So I'm not going to select that one either. Choice three, the subset consists of all of the outcome where you win or there is a tie. Well, we just said outcome one, I don't win that. My friend wins that. Water puts out the fire.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
Choice three, the subset consists of all of the outcome where you win or there is a tie. Well, we just said outcome one, I don't win that. My friend wins that. Water puts out the fire. Now let's look at the last choice. The subset consists of all of the outcomes where there is not a tie. All right, so this is interesting.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
Water puts out the fire. Now let's look at the last choice. The subset consists of all of the outcomes where there is not a tie. All right, so this is interesting. Because look, outcome two, there is a tie. Outcome six, there is a tie. Outcome nine, there is a tie.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
All right, so this is interesting. Because look, outcome two, there is a tie. Outcome six, there is a tie. Outcome nine, there is a tie. And there's actually only three scenarios where there's a tie. Either it's fire, fire, water, water, or sponge, sponge. And those are the ones that are not selected.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
Outcome nine, there is a tie. And there's actually only three scenarios where there's a tie. Either it's fire, fire, water, water, or sponge, sponge. And those are the ones that are not selected. So all of these, someone is going to win. Outcome one, three, four, five, seven, or eight. So definitely, definitely go with that one.
Describing subsets of sample spaces exercise Probability and Statistics Khan Academy.mp3
Let's say that the set B, let me do this in a different color. Let's say that the set B is composed of 1, 7, and 18. And let's say that the set C is composed of 18, 7, 1, and 19. Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset. And this is a notation right over here. This is a subset.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset. And this is a notation right over here. This is a subset. B is a subset of A. So let me write that down. B is subset of A.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
This is a subset. B is a subset of A. So let me write that down. B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further. And we could say that B is a strict, or sometimes said, a proper subset of A. And the way you do that is, essentially, you can almost imagine that this is kind of a less than or equal sign.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further. And we could say that B is a strict, or sometimes said, a proper subset of A. And the way you do that is, essentially, you can almost imagine that this is kind of a less than or equal sign. And then you kind of cross out the equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member of A, but everything that's in A is not a member of B. So let me write this.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
And the way you do that is, essentially, you can almost imagine that this is kind of a less than or equal sign. And then you kind of cross out the equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member of A, but everything that's in A is not a member of B. So let me write this. This is B. B is a strict or proper subset. So for example, we can write that A is a subset of A.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
So let me write this. This is B. B is a strict or proper subset. So for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
So for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see. C contains a 1. It contains a 7. It contains an 18.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
Well, let's see. C contains a 1. It contains a 7. It contains an 18. So every member of B is indeed a member of C. So this right over here is true. Now, can we write that C is a subset of A? Let's see.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
It contains an 18. So every member of B is indeed a member of C. So this right over here is true. Now, can we write that C is a subset of A? Let's see. Every element of C needs to be in A. So A has an 18. It has a 7.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
Let's see. Every element of C needs to be in A. So A has an 18. It has a 7. It has a 1. But it does not have a 19. So once again, this right over here is false.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
It has a 7. It has a 1. But it does not have a 19. So once again, this right over here is false. Now, we could write B is a subset of C, or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this, and then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B. And this is just another way of saying that B is a subset of A.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
So once again, this right over here is false. Now, we could write B is a subset of C, or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this, and then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B. And this is just another way of saying that B is a subset of A. But the way you could think about this is A contains every element that is in B. And it might contain more. It might contain exactly every element.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
And this is just another way of saying that B is a subset of A. But the way you could think about this is A contains every element that is in B. And it might contain more. It might contain exactly every element. Because you can kind of view this as you kind of have the equal symbol there if you were to view this as greater than or equal. They're not quite exactly the same thing. But we know already that we could also write A is a strict superset of B, which means that A contains everything B has and then some.
Subset, strict subset, and superset Probability and Statistics Khan Academy.mp3
In the last video, we came up with a 95% confidence interval for the mean weight loss between the low-fat group and the control group. In this video, I actually want to do a hypothesis test to see really the test if this data makes us believe that the low-fat diet actually does anything at all. And to do that, let's set up our null and alternative hypotheses. So our null hypothesis should be that, hey, this low-fat diet does nothing. And if the low-fat diet does nothing, that means that the mean, the population mean on our low-fat diet minus the population mean on our control should be equal to 0. And this is a completely equivalent statement to saying that the mean of the sampling distribution of our low-fat diet minus the mean of the sampling distribution of our control should be equal to 0. And that's because we've seen this multiple times.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So our null hypothesis should be that, hey, this low-fat diet does nothing. And if the low-fat diet does nothing, that means that the mean, the population mean on our low-fat diet minus the population mean on our control should be equal to 0. And this is a completely equivalent statement to saying that the mean of the sampling distribution of our low-fat diet minus the mean of the sampling distribution of our control should be equal to 0. And that's because we've seen this multiple times. The mean of your sampling distribution is going to be the same thing as your population mean. So this is the same thing as that. That is the same thing as that.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
And that's because we've seen this multiple times. The mean of your sampling distribution is going to be the same thing as your population mean. So this is the same thing as that. That is the same thing as that. Or another way of saying it is if we think about the distribution, if we think about the mean of the distribution of the difference of the sample means, and we focused on this in the last video, that that should be equal to 0. Because this thing right over here is the same thing as that right over there. So that is our null hypothesis.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
That is the same thing as that. Or another way of saying it is if we think about the distribution, if we think about the mean of the distribution of the difference of the sample means, and we focused on this in the last video, that that should be equal to 0. Because this thing right over here is the same thing as that right over there. So that is our null hypothesis. And our alternative hypothesis is just going to be, our alternative hypothesis, I'll write it over here, our alternative hypothesis is just that it actually does do something, that our mean, and actually let's say that it actually has an improvement. So that would mean that we have more weight loss. So if we have the mean of group 1, the population mean of group 1 minus the population mean of group 2 should be greater than 0.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So that is our null hypothesis. And our alternative hypothesis is just going to be, our alternative hypothesis, I'll write it over here, our alternative hypothesis is just that it actually does do something, that our mean, and actually let's say that it actually has an improvement. So that would mean that we have more weight loss. So if we have the mean of group 1, the population mean of group 1 minus the population mean of group 2 should be greater than 0. So this is going to be a one-tailed distribution. Or another way we could view it is that the mean of the difference of the distributions, x1 minus x2, is going to be greater than 0. These are equivalent statements.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So if we have the mean of group 1, the population mean of group 1 minus the population mean of group 2 should be greater than 0. So this is going to be a one-tailed distribution. Or another way we could view it is that the mean of the difference of the distributions, x1 minus x2, is going to be greater than 0. These are equivalent statements. Because we know that this is the same thing as this, which is the same thing as this, which is what I wrote right over here. Now, to do any type of hypothesis test, we have to decide on a level of significance. We have to say, what we're going to do is, we're going to assume that our null hypothesis is correct.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
These are equivalent statements. Because we know that this is the same thing as this, which is the same thing as this, which is what I wrote right over here. Now, to do any type of hypothesis test, we have to decide on a level of significance. We have to say, what we're going to do is, we're going to assume that our null hypothesis is correct. And then given, with that assumption that the null hypothesis is correct, we're going to see what is the probability of getting this sample data right over here. And if that probability is below some threshold, we will reject the null hypothesis in favor of the alternative hypothesis. Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
We have to say, what we're going to do is, we're going to assume that our null hypothesis is correct. And then given, with that assumption that the null hypothesis is correct, we're going to see what is the probability of getting this sample data right over here. And if that probability is below some threshold, we will reject the null hypothesis in favor of the alternative hypothesis. Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha. And here, we're going to decide for a significance level of 95%. Or another way to think about it, we want there to be, assuming that the null hypothesis is correct, we want there to be no more than a 5% chance of getting this result here, or no more than a 5% chance of incorrectly rejecting the null hypothesis when it is actually true, or that would be a type 1 error. So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha. And here, we're going to decide for a significance level of 95%. Or another way to think about it, we want there to be, assuming that the null hypothesis is correct, we want there to be no more than a 5% chance of getting this result here, or no more than a 5% chance of incorrectly rejecting the null hypothesis when it is actually true, or that would be a type 1 error. So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis. And go, less than a 5% probability, given the null hypothesis is true, then we're going to reject the null hypothesis in favor of the alternative. So let's think about this. So we have the null hypothesis.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis. And go, less than a 5% probability, given the null hypothesis is true, then we're going to reject the null hypothesis in favor of the alternative. So let's think about this. So we have the null hypothesis. Let me draw a distribution over here. The null hypothesis says that the mean of our differences, so the mean of the differences of the sampling distributions should be equal to 0. Now, in that situation, what is going to be our critical region here?
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So we have the null hypothesis. Let me draw a distribution over here. The null hypothesis says that the mean of our differences, so the mean of the differences of the sampling distributions should be equal to 0. Now, in that situation, what is going to be our critical region here? Well, we need a result, so we need some critical z-value here, some critical z-score, or some critical, I should actually say some critical value here, because this isn't a normalized standard. This isn't a normalized normal distribution. But so there's some critical value here.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
Now, in that situation, what is going to be our critical region here? Well, we need a result, so we need some critical z-value here, some critical z-score, or some critical, I should actually say some critical value here, because this isn't a normalized standard. This isn't a normalized normal distribution. But so there's some critical value here. The hardest thing in statistics is getting the wording right. There's some critical value here that the probability of getting a sample from this distribution above that value is only 5%. So we just need to figure out what this critical value is.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
But so there's some critical value here. The hardest thing in statistics is getting the wording right. There's some critical value here that the probability of getting a sample from this distribution above that value is only 5%. So we just need to figure out what this critical value is. And if our value is larger than that critical value, then we can reject the null hypothesis, because that means the probability of getting this is less than 5%. We could reject the null hypothesis and go with the alternative hypothesis. And to figure out that critical value, and remember, once again, we can use z-scores, and we can assume this is a normal distribution, because our sample size is large for either of those samples.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So we just need to figure out what this critical value is. And if our value is larger than that critical value, then we can reject the null hypothesis, because that means the probability of getting this is less than 5%. We could reject the null hypothesis and go with the alternative hypothesis. And to figure out that critical value, and remember, once again, we can use z-scores, and we can assume this is a normal distribution, because our sample size is large for either of those samples. We have a sample size of 100. And to figure that out, we just have to figure out, the first step is to say, well, if we just look at a normalized normal distribution like this, what is your critical z-value We're getting a result above that z-value only has a 5% chance, and to do that, so this is actually cumulative, so this whole area right over here is going to be 95% chance. We can just look at the z-table.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
And to figure out that critical value, and remember, once again, we can use z-scores, and we can assume this is a normal distribution, because our sample size is large for either of those samples. We have a sample size of 100. And to figure that out, we just have to figure out, the first step is to say, well, if we just look at a normalized normal distribution like this, what is your critical z-value We're getting a result above that z-value only has a 5% chance, and to do that, so this is actually cumulative, so this whole area right over here is going to be 95% chance. We can just look at the z-table. We're going to look for 95%, because this is a one-tailed case, so let's look for 95%. This is the closest thing. We want to err on the side of being a little bit maybe to the right of this.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
We can just look at the z-table. We're going to look for 95%, because this is a one-tailed case, so let's look for 95%. This is the closest thing. We want to err on the side of being a little bit maybe to the right of this. So let's say 95.05 is pretty good. So that's 1.65. So this critical z-value is equal to 1.65.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
We want to err on the side of being a little bit maybe to the right of this. So let's say 95.05 is pretty good. So that's 1.65. So this critical z-value is equal to 1.65. Or another way to view it is, this distance right here is going to be 1.65 standard deviations. I know my writing is really small. I'm just saying the standard deviation of that distribution.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So this critical z-value is equal to 1.65. Or another way to view it is, this distance right here is going to be 1.65 standard deviations. I know my writing is really small. I'm just saying the standard deviation of that distribution. So what is the standard deviation of that distribution? We actually calculated it in the last video, and I'll recalculate it here. The standard deviation of our distribution, of the difference of the mean, the distribution of the difference of the sample means, is going to be equal to the square root of the variance of our first population.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
I'm just saying the standard deviation of that distribution. So what is the standard deviation of that distribution? We actually calculated it in the last video, and I'll recalculate it here. The standard deviation of our distribution, of the difference of the mean, the distribution of the difference of the sample means, is going to be equal to the square root of the variance of our first population. Now the variance of our first population, we don't know it, but we can estimate it with our sample standard deviation. If you take your sample standard deviation, 4.67, and you square it, you get your sample variance. So this is the variance, this is our best estimate, of the variance of the population.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
The standard deviation of our distribution, of the difference of the mean, the distribution of the difference of the sample means, is going to be equal to the square root of the variance of our first population. Now the variance of our first population, we don't know it, but we can estimate it with our sample standard deviation. If you take your sample standard deviation, 4.67, and you square it, you get your sample variance. So this is the variance, this is our best estimate, of the variance of the population. And we want to divide that by the sample size. And then plus our best estimate of the variance of the population of group 2, which is 4.04 squared, the sample standard deviation of group 2 squared, that gives us the variance, divided by 100. And this gives us, I did it before in the last, maybe it's still sitting on my calculator.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So this is the variance, this is our best estimate, of the variance of the population. And we want to divide that by the sample size. And then plus our best estimate of the variance of the population of group 2, which is 4.04 squared, the sample standard deviation of group 2 squared, that gives us the variance, divided by 100. And this gives us, I did it before in the last, maybe it's still sitting on my calculator. So, yep, it's still sitting on the calculator. It's this quantity right up here, 4.67 squared divided by 100, plus 4.04 squared divided by 100, so it's 0.617. So this right here is going to be, this right here is 0.617.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
And this gives us, I did it before in the last, maybe it's still sitting on my calculator. So, yep, it's still sitting on the calculator. It's this quantity right up here, 4.67 squared divided by 100, plus 4.04 squared divided by 100, so it's 0.617. So this right here is going to be, this right here is 0.617. So this distance right here, this distance right here is going to be 1.65 times 0.617. So let's figure out what that is. So let's take 0.617, and we can even add, well, I'll just leave it there, times 1.65.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So this right here is going to be, this right here is 0.617. So this distance right here, this distance right here is going to be 1.65 times 0.617. So let's figure out what that is. So let's take 0.617, and we can even add, well, I'll just leave it there, times 1.65. So it's 1.02. I'll go with 1.02. This distance right here is 1.02.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So let's take 0.617, and we can even add, well, I'll just leave it there, times 1.65. So it's 1.02. I'll go with 1.02. This distance right here is 1.02. So what this tells us is that there is only a 5% chance that the difference, if we assume that the diet actually does nothing, there's only a 5% chance of having a difference between these two means, the means of these two samples, to have a difference of more than 1.02. There's only a 5% chance of that. Well, the mean that we actually got is 1.91.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
This distance right here is 1.02. So what this tells us is that there is only a 5% chance that the difference, if we assume that the diet actually does nothing, there's only a 5% chance of having a difference between these two means, the means of these two samples, to have a difference of more than 1.02. There's only a 5% chance of that. Well, the mean that we actually got is 1.91. The mean that we actually got is 1.91. So that's sitting out here someplace. So it definitely falls in this critical region.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
Well, the mean that we actually got is 1.91. The mean that we actually got is 1.91. So that's sitting out here someplace. So it definitely falls in this critical region. The probability of getting this, assuming that the null hypothesis is correct, is less than 5%. So that is, it's smaller probability than our significance level. Actually, let me be very clear.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
So it definitely falls in this critical region. The probability of getting this, assuming that the null hypothesis is correct, is less than 5%. So that is, it's smaller probability than our significance level. Actually, let me be very clear. The significance level, this alpha right here, the significance level, I don't want to give you the wrong, the significance level needs to be 5%, not the 95%. I think I might have said it here, but I wrote down the wrong number there. I subtracted it from 1 by accident, probably in my head.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
Actually, let me be very clear. The significance level, this alpha right here, the significance level, I don't want to give you the wrong, the significance level needs to be 5%, not the 95%. I think I might have said it here, but I wrote down the wrong number there. I subtracted it from 1 by accident, probably in my head. But anyway, the significance level is 5%. The probability, given that the null hypothesis is true, the probability of getting the result that we got, the probability of getting that difference, is less than our significance level. It is less than 5%.
Hypothesis test for difference of means Probability and Statistics Khan Academy.mp3
Mom is always grouchy when it rains, Adam's brother said to him. So Adam decided to figure out if the statement was actually true. For the next year, he charted every time it rained and every time his mom was grouchy. What he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. Some of his data are shown in the table below. Fill in the missing values from the frequency table.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
What he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. Some of his data are shown in the table below. Fill in the missing values from the frequency table. And let's see, we have this raining days, not raining days, and the total days that he kept data for. And then he tabulated on, say, the raining day, whether his mom was grouchy or not grouchy, and on a not raining day, whether his mom was grouchy or not grouchy. And there was a total of 35 days it rained, 330 days that it didn't rain.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
Fill in the missing values from the frequency table. And let's see, we have this raining days, not raining days, and the total days that he kept data for. And then he tabulated on, say, the raining day, whether his mom was grouchy or not grouchy, and on a not raining day, whether his mom was grouchy or not grouchy. And there was a total of 35 days it rained, 330 days that it didn't rain. And then 73 times his mom was grouchy and 292 times his mom was not grouchy. So the first thing that we said, well, how do we figure this out? We have these four boxes here.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
And there was a total of 35 days it rained, 330 days that it didn't rain. And then 73 times his mom was grouchy and 292 times his mom was not grouchy. So the first thing that we said, well, how do we figure this out? We have these four boxes here. It's not clear that we have enough information to fill it out just with this table. But we have to remember what they told us. They told us that his mom being grouchy and it raining were entirely independent events.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
We have these four boxes here. It's not clear that we have enough information to fill it out just with this table. But we have to remember what they told us. They told us that his mom being grouchy and it raining were entirely independent events. Another way of saying that is the probability of his, let me do this in a color that you're more likely to see. Another way of saying that, so independent events, that means that the probability, my pen is acting up a little bit, probability that mom is grouchy, so let me write that. Mom, my pen is really, mom is grouchy given it is raining, it shouldn't really matter whether it's raining.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
They told us that his mom being grouchy and it raining were entirely independent events. Another way of saying that is the probability of his, let me do this in a color that you're more likely to see. Another way of saying that, so independent events, that means that the probability, my pen is acting up a little bit, probability that mom is grouchy, so let me write that. Mom, my pen is really, mom is grouchy given it is raining, it shouldn't really matter whether it's raining. It should just be the same thing as the probability of mom being grouchy in general. So what does that tell us? Well, we can figure out the probability that mom is grouchy in general.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
Mom, my pen is really, mom is grouchy given it is raining, it shouldn't really matter whether it's raining. It should just be the same thing as the probability of mom being grouchy in general. So what does that tell us? Well, we can figure out the probability that mom is grouchy in general. She's grouchy 73 out of 365 days. So the probability that mom is grouchy in general is going to be 73 divided by 365. Or at least just based on the data we have, that's the best estimate that mom is grouchy, of the probability that mom is grouchy.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
Well, we can figure out the probability that mom is grouchy in general. She's grouchy 73 out of 365 days. So the probability that mom is grouchy in general is going to be 73 divided by 365. Or at least just based on the data we have, that's the best estimate that mom is grouchy, of the probability that mom is grouchy. It's the percentage of days that she's been grouchy. So that is.2. So based on the data, the best estimate of the probability of mom being grouchy is.2 or 20%.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
Or at least just based on the data we have, that's the best estimate that mom is grouchy, of the probability that mom is grouchy. It's the percentage of days that she's been grouchy. So that is.2. So based on the data, the best estimate of the probability of mom being grouchy is.2 or 20%. And so we should have the probability of mom being grouchy given that it's raining, should be 20% as well. So this number, so given that it's raining, we should also have 20% of the time mom is grouchy because these are independent events. It shouldn't matter whether it's raining or not.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
So based on the data, the best estimate of the probability of mom being grouchy is.2 or 20%. And so we should have the probability of mom being grouchy given that it's raining, should be 20% as well. So this number, so given that it's raining, we should also have 20% of the time mom is grouchy because these are independent events. It shouldn't matter whether it's raining or not. This should be 20%, this should be, she should be grouchy 20% of the time that it's raining and she should be grouchy 20% of the time that it's not raining. That would be consistent with the data saying that these were entirely independent events. So what is 20% of 35?
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
It shouldn't matter whether it's raining or not. This should be 20%, this should be, she should be grouchy 20% of the time that it's raining and she should be grouchy 20% of the time that it's not raining. That would be consistent with the data saying that these were entirely independent events. So what is 20% of 35? Well, 20% is 1 5th. 1 5th of 35 is 7. And once again, all I did is I said 20% of 35 is 7.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
So what is 20% of 35? Well, 20% is 1 5th. 1 5th of 35 is 7. And once again, all I did is I said 20% of 35 is 7. And if that's 7, then 35 minus 7, that's gonna be 28 right over there. And then if this is 7, then 73 minus 7 is going to be 66. And 333, I guess there's a couple of ways we could do it.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
And once again, all I did is I said 20% of 35 is 7. And if that's 7, then 35 minus 7, that's gonna be 28 right over there. And then if this is 7, then 73 minus 7 is going to be 66. And 333, I guess there's a couple of ways we could do it. We could take, actually we could just take 292 minus 28 is going to be, let's see, 292 minus 8 would be 284 minus another 264, 264. Do the numbers all add up? Yes, 66 plus 264 is 330.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
And 333, I guess there's a couple of ways we could do it. We could take, actually we could just take 292 minus 28 is going to be, let's see, 292 minus 8 would be 284 minus another 264, 264. Do the numbers all add up? Yes, 66 plus 264 is 330. So the key realization here is, what he's saying he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. That means that the probability of his mom being grouchy, it shouldn't matter whether it's raining or not.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
Yes, 66 plus 264 is 330. So the key realization here is, what he's saying he found was very interesting. Rainy days and his mom being grouchy were entirely independent events. That means that the probability of his mom being grouchy, it shouldn't matter whether it's raining or not. It should just be, it should be the same probability of whether it's raining or not. And our best estimate of the probability of his mom being grouchy is on the total days is 20%. And so if the data is backing up that it's independent events, then the best way to fill this out would be the probability of his mom being grouchy on a rainy day or a not rainy day should be the same.
Filling out frequency table for independent events Probability and Statistics Khan Academy.mp3
The monthly data on ticket sales is shown below. What are the best and worst months for cruise sales? So what they've given us this diagram, this is usually called a pie chart or pie graph because it looks like a pie that's sliced up into a bunch of pieces. Sometimes this is called a circle graph, but pie graph is much more common. And then they say it's monthly ticket sales. So each of these slices represent the sales in a given month. So for example, this blue slice over here represents the sales in January.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
Sometimes this is called a circle graph, but pie graph is much more common. And then they say it's monthly ticket sales. So each of these slices represent the sales in a given month. So for example, this blue slice over here represents the sales in January. And the way that a pie chart is set up, each slice is bigger or smaller depending on what fraction of the whole it represents. So for example, they're telling us in January, they sold 18% of the total year's ticket sales in January. So if you add up all of these percentages, it should add up to 100%.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
So for example, this blue slice over here represents the sales in January. And the way that a pie chart is set up, each slice is bigger or smaller depending on what fraction of the whole it represents. So for example, they're telling us in January, they sold 18% of the total year's ticket sales in January. So if you add up all of these percentages, it should add up to 100%. And not only do they tell us that they sold 18%, but the slice of this pie should be 18% of the area of the entire pie. It is literally 18% of the pie. If you were to eat this slice, you would have eaten 18% of the pie.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
So if you add up all of these percentages, it should add up to 100%. And not only do they tell us that they sold 18%, but the slice of this pie should be 18% of the area of the entire pie. It is literally 18% of the pie. If you were to eat this slice, you would have eaten 18% of the pie. Now with that out of the way, let's think about their questions. What are the best and the worst months for cruise sales? So the best month is obviously the month where they sell, where they have the largest or the largest percentage of their tickets were sold.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
If you were to eat this slice, you would have eaten 18% of the pie. Now with that out of the way, let's think about their questions. What are the best and the worst months for cruise sales? So the best month is obviously the month where they sell, where they have the largest or the largest percentage of their tickets were sold. And actually I started with January, and this is what's neat about pie graphs. You wouldn't even have to look at the numbers. January just jumps out as the biggest slice of pie.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
So the best month is obviously the month where they sell, where they have the largest or the largest percentage of their tickets were sold. And actually I started with January, and this is what's neat about pie graphs. You wouldn't even have to look at the numbers. January just jumps out as the biggest slice of pie. If you didn't even see the numbers, if you couldn't even read, and you just looked at this and someone said, what is the largest slice of pie? You would immediately say, hey, this is clearly the largest slice of pie right over there. And so that is actually the best month for cruise sales because they sold 18%.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
January just jumps out as the biggest slice of pie. If you didn't even see the numbers, if you couldn't even read, and you just looked at this and someone said, what is the largest slice of pie? You would immediately say, hey, this is clearly the largest slice of pie right over there. And so that is actually the best month for cruise sales because they sold 18%. You see this 18% is larger than all of the other percentages over here. But it's clear just by looking at the graph. This is the largest slice.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
And so that is actually the best month for cruise sales because they sold 18%. You see this 18% is larger than all of the other percentages over here. But it's clear just by looking at the graph. This is the largest slice. Now what's the worst month for cruise ticket sales? The worst month? Well then we just have to find the thinnest slice of pie.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
This is the largest slice. Now what's the worst month for cruise ticket sales? The worst month? Well then we just have to find the thinnest slice of pie. And if we look over here, the slices of pie get pretty thin out down here. This is in the summer in June and July and in May. But the smallest are actually July and June.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
Well then we just have to find the thinnest slice of pie. And if we look over here, the slices of pie get pretty thin out down here. This is in the summer in June and July and in May. But the smallest are actually July and June. And this is where the numbers become useful because when you just look at it by, you know, when you just eyeball it, you're not sure, hey, are these exactly the same or do they just look exactly the same? And that's where these numbers are valuable. And based on the data they've given us, it looks like these are tied for the worst in terms of ticket sales.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
But the smallest are actually July and June. And this is where the numbers become useful because when you just look at it by, you know, when you just eyeball it, you're not sure, hey, are these exactly the same or do they just look exactly the same? And that's where these numbers are valuable. And based on the data they've given us, it looks like these are tied for the worst in terms of ticket sales. In both of these months, they sell only 3% in each month. So the worst months for cruise sales are July and June. July and June are tied for the worst.
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
And based on the data they've given us, it looks like these are tied for the worst in terms of ticket sales. In both of these months, they sell only 3% in each month. So the worst months for cruise sales are July and June. July and June are tied for the worst. The best is clearly January. And then after January, the next best, they're not really asking us that, but since we have the pie chart in front of us, might as well ask ourselves that. What's the next best?
Reading pie graphs (circle graphs) Applying mathematical reasoning Pre-Algebra Khan Academy.mp3
We've already been introduced to the chi-squared statistic in other videos. Now we're going to use it for a test for homogeneity. And homogeneity or homogeneity, in everyday language, this means how similar things are. And that's what we're essentially going to test here. We're gonna look at two different groups and see whether the distributions of those groups for a certain variable are similar or not. And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains?
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And that's what we're essentially going to test here. We're gonna look at two different groups and see whether the distributions of those groups for a certain variable are similar or not. And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, humanities, or neither? And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, humanities, or neither? And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed folks. And then the alternative hypothesis, well, no, there is a difference. So there is a difference.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed folks. And then the alternative hypothesis, well, no, there is a difference. So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent. They liked them equally.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent. They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the expected data would be if we are assuming that the null hypothesis is true, that there's no difference in preference between right and left-handed folks. This is the right-handed column. This is the left-handed column.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the expected data would be if we are assuming that the null hypothesis is true, that there's no difference in preference between right and left-handed folks. This is the right-handed column. This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from this total column. Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from this total column. Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60?
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. And these, once again, they add up to 60.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM. 40% of 40, that is 16.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM. 40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. And then all of these add up to 40.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition. And so these need to be truly random samples.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition. And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well. And now we're ready to calculate our chi-squared statistic. We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
So let's assume that that is the case as well. And now we're ready to calculate our chi-squared statistic. We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3
And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24. I'm running out of colors. And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected. And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12.
Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3