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And instead of just redoing the whole example again, the way that would get maximum points on the AP test, I thought it would be even more instructive to show where I went wrong. So the math in this problem was not incorrect, but if I were actually taking the AP Statistics exam, I was told that, say, for the conditions... | Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3 |
I am told that the AP graders do not like this. They do not want you to just point to a part of the passage that says that it's a random sample. Instead, what you could say, and they are functionally equivalent, but you wanna make sure that you're doing what people are looking for. Instead, what we could do is, it is s... | Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3 |
Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken. And that would be sufficient, instead of drawing the arrow to that part over there. Then I could check it off. Now, the other thing that I didn't do, I ... | Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3 |
Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down. And I really did wanna model what you would need to do on the actual AP exam is, I came, I said that we failed to reject the null hypothesis, and therefore, there's not enough evidence to suggest the alternative ... | Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3 |
The Olympic running team of Freedonia has always used Zeppo's running shoes, but their manager suspects Harpo's shoes can produce better results, which would be lower times. The manager has six runners each run two laps, one lap wearing Zeppo's and another lap wearing Harpo's. Each runner flips a coin to determine whic... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
The manager wants to test if their times when wearing Harpo's are significantly lower than their times when they wear Zeppo's. Assume that all conditions for inference were met. Which of these is the most appropriate test, an alternative hypothesis? So they're just asking us about the alternative, not even the null, bu... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
So they're just asking us about the alternative, not even the null, but we can talk about that. So pause this video and see if you can figure it out. So before I even go into this particular example, let's just make sure we understand the difference between a two-sample t-test and a paired t-test. So when we're talking... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
So when we're talking about either a two-sample t-test or a two-sample t-interval for the difference between the means, what we're doing is we're considering two populations, and you take two independent samples from those populations. And what you're trying to do is you get statistics off of these samples, and you're ... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
That's what you're trying to figure out, mu one minus mu two. A paired situation is quite different, even though they might sound the same at first. Here we're looking at just one population. And that's exactly what's happening in this situation right over here. We're trying to figure out what is the mean difference be... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
And that's exactly what's happening in this situation right over here. We're trying to figure out what is the mean difference between using Zeppo's and Harpo's shoes. So this is what we're trying to get at. The mean difference, so we could call that mu sub Zeppo's minus Harpo's. And the way that we go about doing that ... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
The mean difference, so we could call that mu sub Zeppo's minus Harpo's. And the way that we go about doing that is we take a sample, and then for each subject in the sample we perform two measurements, one where they run with the Zeppo's and one when they run with the Harpo's. And then for that sample, you can calcula... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
Zeppo's and the Harpo's. You're gonna calculate this difference for each member of your sample, and then you're gonna take the mean of all of those. So hopefully you notice that this is quite different. And so as you can imagine, here in this example we are dealing with a paired t-test. We aren't looking at two indepen... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
And so as you can imagine, here in this example we are dealing with a paired t-test. We aren't looking at two independent groups or two independent samples like you would with a two-sample t-test. And so we are in a paired t-test, and the manager wants to test if their times when wearing Harpo's are significantly lower... | Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3 |
They obtain a random sample of records from 500 cats. They find 24 of the 259 male cats have the disease, while 14 of 241 female cats have the disease. The veterinarian uses these results to test their null hypothesis that the true proportion is the same amongst the male and female cats versus the alternative hypothesi... | Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3 |
The test statistic was Z is equal to 1.46 and the P value was approximately 0.07. That's useful, they've done a lot of work for us. Assume that all conditions for inference were met. At the alpha equals 0.10 level of significance, is there sufficient evidence to conclude that a larger proportion of male cats have the d... | Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3 |
At the alpha equals 0.10 level of significance, is there sufficient evidence to conclude that a larger proportion of male cats have the disease? Pause this video and see if you can answer that. All right, remember what we do with the significance test. We assume, we assume our null hypothesis and then with that assumpt... | Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3 |
We assume, we assume our null hypothesis and then with that assumption, we look at our data and we calculate a P value. We say, hey, what is the probability of getting the data that we did assuming our null hypothesis is true? And if that is lower than our significance level, then we reject the null hypothesis because ... | Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3 |
The graphs below show the test grades of the students in Dexter's class. The first graph shows the relationship between test grades and the amount of time the student spent studying. So this is study time on this axis, and this is the test grade on this axis. And the second graph shows the relationship between test gra... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
And the second graph shows the relationship between test grades and shoe size. So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actua... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that. And it makes sens... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
I could almost fit a line that would go just like that. And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well. For some people... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard t... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply. There's a negative linear... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
So let's see which of these choices apply. There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship. The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlin... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
It looks like there's a positive linear relationship. The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't see... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable. And there isn't any type of, ... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score. That's right. And no relationship between sho... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
There is a positive linear relationship between study time and score. That's right. And no relationship between shoe size and score. Well, I'm going to go with that one. Both graphs show positive linear trends of approximately equal strength. No, not at all. This one doesn't show a linear relationship of really any str... | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give o... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is ... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data p... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 ye... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. And so half of the ages are... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then thes... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the r... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actu... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in t... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the ... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. ... | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Regulations require that product labels on containers of food that are available for sale to the public accurately state the amount of food in those containers. Specifically, if milk containers are labeled to have 128 fluid ounces, and the mean number of fluid ounces of milk in the containers is at least 128, the milk ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Variability in the filling process causes the actual contents of milk containers to be normally distributed. A random sample of 12 containers of milk was drawn from the milk processing line in a plant, and the amount of milk in each container was recorded. The sample mean and standard deviation of this sample of 12 con... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? Provide statistical justification for your answer. So pause this video and see if you can have a go at it. All right, now let's do this together. So first, let's say what we're talking about. So let me define mu... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
All right, now let's do this together. So first, let's say what we're talking about. So let me define mu, and this is going to be the mean amount amount of milk in population, population of containers, containers at the plant that we care about. And so then we can set up our hypotheses. Our null hypothesis over here is... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so then we can set up our hypotheses. Our null hypothesis over here is that we are in compliance. We could say that the mean for our population of containers is actually 128. That's our minimum we need to be in compliance. And that our alternative hypothesis is that we are not in compliance. So that's that our mean... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
That's our minimum we need to be in compliance. And that our alternative hypothesis is that we are not in compliance. So that's that our mean, the true population mean, is less than 128 fluid ounces. And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis. ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis. Now, if you're going to do a significance test, you need to set a significance level. So let's do that over here. Significance level. And if you haven't noticed, I'm doing, I'm trying to do in this vi... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Significance level. And if you haven't noticed, I'm doing, I'm trying to do in this video what would be expected of you on a test. And this is an actual question from an AP exam. So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one. And since they didn't give it one to... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one. And since they didn't give it one to us, it's important to set one ahead of time. And now we wanna check our conditions for inference. So let me do that over here. Conditions, conditions for inference. And this is t... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So let me do that over here. Conditions, conditions for inference. And this is to feel good that the sample that we're using to make our inference, to do our significance test, that it's a reasonable one to make inferences from. And so the first one is the random condition. And do we meet that? Well, they tell us here ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so the first one is the random condition. And do we meet that? Well, they tell us here it's a random sample of 12 containers of milk. If I was doing this on the AP exam, I would write it out here. So I would say in the passage or in question, in the question, they say, they say a random, a random sample of 12. And ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
If I was doing this on the AP exam, I would write it out here. So I would say in the passage or in question, in the question, they say, they say a random, a random sample of 12. And then they go on to say more things. And so I would say that meets condition, meets condition. Now the next one we wanna care about is our ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so I would say that meets condition, meets condition. Now the next one we wanna care about is our normal condition and this is to feel good that our sampling distribution is roughly normal. Now there's a couple of ways that we could do that. One is if our sample size is greater than 30 or greater than or equal to 3... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
One is if our sample size is greater than 30 or greater than or equal to 30, then we say, okay, our sampling distribution is going to be roughly normal. But in this situation, our sample size n, so sample size, sample size is less than 30, but, but there's another way to meet the normal condition and that's if the unde... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Variability in the filling process causes the actual contents of milk to be normally distributed. So we could say in passage, in passage says, and let's see, I could quote part of this. So actual contents, actual contents, and then dot, dot, dot, normally distributed, normally distributed. So that meets condition, meet... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So that meets condition, meets condition. And then the last condition we wanna think about is the independence condition, independence. And this is to feel good that the observations, the individual observations in our sample can be considered to be roughly independent. Now one way is if they were sampling with replace... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Now one way is if they were sampling with replacement, which they're not doing here. It looks like they took all 12 containers at once. But another way is if this is less than 10% of the overall population, then you could say, okay, they're gonna, you can view them as roughly independent. And so you can say didn't, did... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so you can say didn't, didn't sample with replacement, with replacement, but, but assume, assume that 12 is less than 10% of the population. And in that case, you would meet condition, meet this condition as well. So it looks like we are, we've met these three conditions that we need to make for inference, or we ca... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
They haven't given us any information to the contrary. And so now what we can do is calculate a t-statistic, and then from that, calculate our p-value, compare our p-value to our significance level, and see what kind of conclusions we can make. And so our t-statistic, right over here, and once again, if at any point yo... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Our t-statistic is going to be our sample mean minus the assumed mean from the null hypothesis. And let me, since I'm introducing this notation, this little sub zero, I'll say that's the assumed mean from my null hypothesis. So I'll do that, and then I'll divide. Ideally, if I was doing a z-statistic, I would divide by... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Ideally, if I was doing a z-statistic, I would divide by the standard deviation of the sampling distribution of the sample mean, which is often known as the standard error of the mean. But the whole reason why I'm doing a t-statistic is, well, I don't know exactly what that is, but I could estimate the standard deviati... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
But in this case, this is going to be 127.2, that is our sample mean, minus our assumed mean from our null hypothesis, minus 128, all of that over, our sample standard deviation is 2.1, divided by the square root of 12. And so this is going to be approximately equal to, got a calculator out here, and so we have, see in... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
Yeah, that looks right. Click Enter. And so this is negative, I'll say it's approximately negative 1.32. So negative 1.32. And now we can figure out our p-value, our p-value, which is the same thing as the probability of getting a t-statistic this low or lower. So we could say t is less than or equal to negative 1.32 i... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So negative 1.32. And now we can figure out our p-value, our p-value, which is the same thing as the probability of getting a t-statistic this low or lower. So we could say t is less than or equal to negative 1.32 is equal to, so I'll get my calculator back out. And so here, what I would use is I would use the cumulati... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so here, what I would use is I would use the cumulative distribution function for t-statistic, so that's that right over there. And so I do care about the left tail, so I care about the area under the curve from negative infinity up to and including negative 1.32. So let's do negative 1.32. And then my degrees of f... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And then my degrees of freedom, well, it's gonna be my sample size minus one. My sample size was 12, so that minus one is 11. And then I do paste. And so I have this tcdf from negative E99 to negative 1.32 comma 11. And actually, you'd wanna write this down on your exam if you were doing it, just so they know where you... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so I have this tcdf from negative E99 to negative 1.32 comma 11. And actually, you'd wanna write this down on your exam if you were doing it, just so they know where you got that from. And so this is, this is equal to 0.107. So let me write it. This is approximately 0.107. And it's important to say how you calculat... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So let me write it. This is approximately 0.107. And it's important to say how you calculated this. So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here. And it also might be good practice ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here. And it also might be good practice to draw your t-distribution right over here. So that's our t-distribution. That's the mean of our t-... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
That's the mean of our t-distribution. So we say that this is the area that we care about. So that is that right over there, just to make sure people know what we're talking about. And so here, now we're ready to make a conclusion. We can compare this to our significance level. And so we can say since, since 0.107 is g... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
And so here, now we're ready to make a conclusion. We can compare this to our significance level. And so we can say since, since 0.107 is greater than, our significance level is greater than 0.05, which is alpha, we fail, we fail to reject, reject the null hypothesis. And so let's just make sure we read their question ... | Free response example Significance test for a mean AP Statistics Khan Academy.mp3 |
So I've got a binomial variable x, and I'm gonna describe it in very general terms. It is the number of successes after n trials, after n trials, where the probability of success, the probability of success, success for each trial is p. And this is a safe, this is a reasonable way to describe really any random, any bin... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
The probability stays constant. We have a finite number of trials right over here. Each trial results in either a very clear success or failure. So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable? What would the expected value, expected value of x be equal to... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable? What would the expected value, expected value of x be equal to? And I will just cut to the chase and tell you the answer, and then later in this video, we'll prove it to ourselves a little bit more mathemati... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
The expected value of x, it turns out, is just going to be equal to the number of trials times the probability of success for each of those trials. And so if you wanted to make that a little bit more concrete, imagine if a trial is a free throw, taking a shot from the free throw line. Success, success is made shot, so ... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Your probability is, let me do this yellow color, your probability, this would be your free throw percentage, so let's say it's 30% or 0.3. And let's say, for the sake of argument, that we're taking 10 free throws, so n is equal to 10. So this is making it all a lot more concrete. So in this particular scenario, your e... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So in this particular scenario, your expected value, your expected value, if x is the number of made free throws after taking 10 free throws with a free throw percentage of 30%, well, based on what I just told you, it'd be n times b. It would be the number of trials times the probability of success in any one of those ... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Well, if you're taking 10 shots with a 30% free throw percentage, it actually does feel natural that I would expect to make three shots. Now, with that out of the way, let's make ourselves feel good about this mathematically, and we're gonna leverage some of our expected value properties. In particular, we're gonna lev... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
And so, assuming this right over here, let's construct a new random variable. Let's call our random variable y. And we know the following things about y. The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for thi... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for this random variable. And so, you might be seeing where this is going. You could view this random variable, it's really representing one trial. It becomes on... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
It becomes one in a success, zero when you don't have a success. And so, you could view our original random variable x right over here as being equal to y plus y, and we're gonna have 10 of these. So, we're gonna have 10 y's. In the concrete sense, you could view the random variable y as equaling one if you make a free... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
In the concrete sense, you could view the random variable y as equaling one if you make a free throw, and equaling zero if you don't make a free throw. It's really just representing one of those trials, and you can view x as the sum of n of those trials. Well, actually, let me be very clear here. I immediately went to ... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
I immediately went to the concrete, but I really should be saying n y's, because I wanna stay general right over here. So, there are n n y's right over here. This was just a particular example, but I am going to try to stay general for the rest of the video, because now we are really trying to prove this result right o... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So, let's just take the expected value of both sides. So, what is it going to be? So, we get the expected value of x is equal to, well, it's the expected value of all of this thing, but by that property right over here, this is going to be the expected value of y plus the expected value of y, plus, and we're gonna do t... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
So, we have n. And so, you could rewrite this as being equal to, so this is our n right over here. This is n times the expected value of y. Now, what is the expected value of y? Well, this is pretty straightforward. We can actually just do it directly. The expected value of y, let me just write it over here. The expect... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
Well, this is pretty straightforward. We can actually just do it directly. The expected value of y, let me just write it over here. The expected value of y is just the probability weighted outcomes, and since there's only two discrete outcomes here, it's pretty easy to calculate. We have a probability of p of getting a... | Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3 |
we've seen in the last several videos. You start off with any crazy distribution. It doesn't have to be crazy. It could be a nice normal distribution, but to really make the point that you don't have to have a normal distribution, I like to use crazy ones. So let's say you have some kind of crazy distribution that look... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
It could be a nice normal distribution, but to really make the point that you don't have to have a normal distribution, I like to use crazy ones. So let's say you have some kind of crazy distribution that looks something like that. It could look like anything. So we've seen multiple times you take samples from this cra... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So we've seen multiple times you take samples from this crazy distribution. So let's say you were to take samples of, let's say n is equal to 10. So we take 10 instances of this random variable, average them out, and then plot our average. We plot our average. We get one instance there. We keep doing that. We do that a... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
We plot our average. We get one instance there. We keep doing that. We do that again. We take 10 samples from this random variable, average them, plot them again. You plot it again, and eventually you do this a gazillion times, in theory infinite number of times, and you're going to approach the sampling distribution o... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
We do that again. We take 10 samples from this random variable, average them, plot them again. You plot it again, and eventually you do this a gazillion times, in theory infinite number of times, and you're going to approach the sampling distribution of the sample mean. And n equal 10, it's not gonna be a perfect norma... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And n equal 10, it's not gonna be a perfect normal distribution, but it's gonna be close. It'd be perfect only if n was infinity. But let's say we eventually, all of our samples, you know, we get a lot of averages that are there, that stacks up there, that stacks up there, and eventually we'll approach something that l... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And we've seen from the last video that, one, if, let's say we were to do it again, and this time let's say that n is equal to 20. One, the distribution that we get is going to be more normal, and maybe in future videos we'll delve even deeper into things like kurtosis and skew. But it's gonna be more normal, but even ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So they're all gonna have the same mean. Let's say the mean here is, you know, I don't know, let's say the mean here is five. Then the mean here is also gonna be five. The mean of our sampling distribution of the sample mean is gonna be five. And it doesn't matter what our n is. If our n is 20, it's still gonna be five... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
The mean of our sampling distribution of the sample mean is gonna be five. And it doesn't matter what our n is. If our n is 20, it's still gonna be five, but our standard deviation is gonna be less in either of these scenarios. And we saw that just by experimenting. It might look like this. It's gonna be more normal, b... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And we saw that just by experimenting. It might look like this. It's gonna be more normal, but it's gonna have a tighter standard deviation. So maybe it'll look like that. And if we did it with a even larger sample size, let me do that in a different color. If we did that with an even larger sample size, n is equal to ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So maybe it'll look like that. And if we did it with a even larger sample size, let me do that in a different color. If we did that with an even larger sample size, n is equal to 100, what we're gonna get is something that fits the normal distribution even better. We take 100 instances of this random variable, average ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
We take 100 instances of this random variable, average them, plot it. 100 instances of this random variable, average them, plot it. We just keep doing that. If we keep doing that, what we're gonna have is something that's even more normal than either of these. So it's gonna be a much closer fit to a true normal distrib... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
If we keep doing that, what we're gonna have is something that's even more normal than either of these. So it's gonna be a much closer fit to a true normal distribution. But even more obvious to the human eye, it's gonna be even tighter. So it's going to be a very low standard deviation. It's gonna look something like ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
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